# Modern PDE Techniques for Image Inpainting Carola-Bibiane Sch¨onlieb

Modern PDE Techniques for Image Inpainting Carola-Bibiane Schönlieb Girton College DAMTP, Centre for Mathematical Sciences University of Cambridge A thesis submitted for the degree of Doctor of Philosophy 15th of June 2009 I would like to dedicate this thesis to my mother, who gave me the will to be independent and always believed in me. Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where speciﬁcally indicated in the text. Carola-Bibiane Schönlieb Acknowledgements First I would like to thank my supervisor Peter A. Markowich for his support, his mentorship, his belief in me, and his friendship. Moreover, I owe thanks to my two co-supervisors, Martin Burger and Massimo Fornasier. They were always there for me, introduced me to interesting and challenging problems, and motivated me in my own creative thinking. Further, I would like to thank all my collaborators, for numerous discussions, for our fruitful work together, and for all they have taught me (alphabetically ordered): Andrea Bertozzi, Julian Fernandez Bonder, Martin Burger, Shun-Yin Chu, Massimo Fornasier, Marzena Franek, Lin He, Andreas Langer, Peter A. Markowich and Julio D. Rossi. Also, I would like to emphasize that I really enjoyed working alongside everyone in the Applied PDE (APDE) group of my supervisor. In particular, I would like to point out Marcus Wunsch, who welcomed me at the institute in Vienna and always believed in my academical excellence, Rada-Maria Weishäupl for preparing the way for other female PhD students in Peter’s group to follow her, Alexander Lorz for his humanity and common love of dogs, Marie-Therese Wolfram for her friendship and her sympathy during the last six months of my PhD studies, and Klemens Fellner and Christof Sparber for playing the ”older brothers“ role. The other members of the APDE group (alphabetically ordered) - Gonca Aki, Paolo Antonelli, Marco DiFrancesco, Guillaume Dujardin, Norayr Matevosyan and Jan-Frederik Pietschmann - have also been very helpful, supportive, and reliable mates. Leaving this group now, I know that I will miss you all very much! Moreover, I would like to thank the assistant of my supervisor, and very good friend of mine, Renate Feikes, for her honest and critical personality, her caring and friendship (no matter on which part of the globe I was), and for her very eﬃcient and reliable management of projects and paperwork. There are also a couple of people outside of the APDE group, I owe thanks to. Practically, all the academic people in Vienna, Linz and Cambridge with whom I shared some time together. Among them, I especially would like to thank Carlota Cuesta and Vera Miljanovic, who showed me how women are able to make an excellent career, while still preserving their female qualities and their happiness, Helen Lei, who taught me that life barely stays between the lines, and Massimo Fornasier, for always believing in me and my qualities, for his coaching, his friendship, and his wonderful pasta al ragù. Many thanks as well to the whole Numerical Analysis group in Cambridge, in particular Tanya Shingel, with whom I shared the last steps of the PhD course, and Arieh Iserles for the warm and caring welcome he showed to our whole group when arriving at DAMTP. During my research stays in Buenos Aires and Los Angeles, I received great hospitality. For someone who is used to a protected nest at home, it is not always easy to overcome one’s inhibitions and leave for the unknown. I am glad I did it though, and I want to thank the following people for making these stays one of the most enjoyful times in my present life (in alphabetical order): Andrea Bertozzi, Julián Fernández Bonder, Lincoln Chayes, Irene Drelichman, Ricardo Durán, Dimitris Giannakis, Fredrik Johansson, Helen Lei, Sandra Martinez, Pierre Nolin, Jan-Frederik Pietschmann, Mariana Prieto, Jesus Rosado, Julio D. Rossi, Noemi Wolanski, Marie-Therese Wolfram and Marcus Wunsch. For carefully proofreading parts of this thesis, I would like to thank (alphabetically ordered): Bertram Düring, Renate Feikes, Helen Lei and MarieTherese Wolfram. My whole family has been very supportive, especially my mother Karin Schönlieb. In my lows she encouraged me to look on the bright side of life and helped me to get up again. In my highs she was so proud of me and motivated me to keep it up. In addition, she made all my research stays abroad possible by taking care of my beloved dog Eureka-Carissima von Salmannsdorf, short Reka. Mama, I love you very much! I also would like to thank a couple of important other people in my life in particular. My sister Charlotte Schönlieb for always keeping a place for me in her inﬁnitely big heart. My nephew Manuel Michl for always viewing me as a mathematical genius. My best friend Isabella Wirth, who accompanied me from the beginnig of my mathematical studies, and always reminded me that there is something beyond work. My boyfriend and ”father“ of Reka, Herwig Rittsteiger, who always tried to give Reka additional love and care during my absence. My dear friend and breeder of Reka, Elfriede Wilﬁnger, who gave me a warm home in Vienna, and the best dog in the world. My current landlady Dorothea Heller, and her son Nicholas Heller, for making it possible for Reka and myself to be together in Cambridge, and for accepting me as a member of their family. And last but by no means least within this list, I thank Hannelore and Hans Liszt, Elisabeth Beranek, and Ingeborg Schön, for their company and support during the whole time since my birth. In the end, I would like to address Reka separately: Without her, my PhD time would have been half as much fun as it has been. I also acknowledge the ﬁnancial support provided by the following institutions and funds: the Wissenschaftskolleg (Graduiertenkolleg, Ph.D. program) of the Faculty for Mathematics at the University of Vienna (funded by the Austrian Science Fund FWF), the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts, the FFG project no. 813610 Erarbeitung neuer Algorithmen zum Image Inpainting. Further, this publication is based on work supported by Award No. KUK-I1-007-43 , made by King Abdullah University of Science and Technology (KAUST). For the hospitality and the ﬁnancial support during parts of the preparation of this work, I thank the Institute for Pure and Applied Mathematics (IPAM), UCLA. I also acknowledge the ﬁnancial support of the US Oﬃce of Naval Research grant N000140810363, and the Department of Defense, National Science Foundation Grant ACI-0321917, during my visits to UCLA. Further the author would like to thank the UCLA Mathematics Department, and Alan Van Nevel and Gary Hewer from the Naval Air Weapons Station in China Lake, CA for providing the road data in Section 5.2. Abstract Partial diﬀerential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order (which is then called the order of the PDE). Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat ﬂow, optics, electrostatics and -dynamics, elasticity, ﬂuid ﬂow and much more. In this globalized and technologically advanced age, PDEs are also extensively used for modeling social situations (e.g., models for opinion formation) and tasks in engineering (like models for semiconductors, networks, and signal and image processing tasks). In my Ph.D. thesis I study nonlinear PDEs of higher-order appearing in image processing with specialization to inpainting (i.e., image interpolation). Digital image interpolation is an important challenge in our modern computerized society: From the reconstruction of crucial information in satellite images of our earth, restoration of CT- or PET images in molecular imaging to the renovation of digital photographs and ancient artwork, digital image interpolation is ubiquitous. Motivated by these applications, I investigate certain PDEs used for these tasks. I am concerned with the mathematical analysis and the eﬃcient numerical solution of these equations as well as the concrete real world applications (like the restoration of ancient Viennese frescoes). Keywords: Partial Diﬀerential Equations, Variational Calculus, Image Processing, Numerical Analysis. Contents Notation and Symbols xi 1 Introduction 1 1.1 The Role of Image Processing in our Modern Society . . . . . . . . . . . 3 1.2 What is a Digital Image? . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Image Inpainting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Energy-Based and PDE Methods for Image Inpainting . . . . . . 7 1.3.2 Second- Versus Higher-Order Approaches . . . . . . . . . . . . . 16 1.3.3 Numerical Solution of Higher-Order Inpainting Approaches . . . 19 2 Image Inpainting With Higher-Order Equations 2.1 Cahn-Hilliard Inpainting . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.1 Existence of a Stationary Solution . . . . . . . . . . . . . . . . . 26 2.1.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.3 2.2 Neumann Boundary Conditions and the Space 38 TV-H−1 Inpainting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.1 Γ-Convergence of the Cahn-Hilliard Energy . . . . . . . . . . . . 46 2.2.2 Existence of a Stationary Solution . . . . . . . . . . . . . . . . . 48 2.2.3 Characterization of Solutions . . . . . . . . . . . . . . . . . . . . 51 2.2.4 Error Estimation and Stability Analysis With the Bregman Dis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Inpainting with LCIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.3.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 61 The Inpainting Mechanisms of Transport and Diﬀusion - A Comparison 62 2.2.5 2.4 H∂−1 (Ω) . . . . . tance 2.3 24 viii CONTENTS 3 Analysis of Higher-Order Equations 3.1 3.2 Instabilities in the Cahn-Hilliard Equation . . . . . . . . . . . . . . . . . 70 3.1.1 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.1.2 Linear Stability / Instability . . . . . . . . . . . . . . . . . . . . 81 3.1.3 Nonlinear Stability / Instability . . . . . . . . . . . . . . . . . . . 85 3.1.4 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Nonlocal Higher-Order Evolution Equations . . . . . . . . . . . . . . . . 93 3.2.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . 97 3.2.2 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2.3 Scaling the Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4 Numerical Solution of Higher-Order Inpainting Approaches 4.1 4.2 4.3 70 104 Unconditionally Stable Solvers . . . . . . . . . . . . . . . . . . . . . . . 105 4.1.1 The Convexity Splitting Idea . . . . . . . . . . . . . . . . . . . . 106 4.1.2 Cahn-Hilliard Inpainting . . . . . . . . . . . . . . . . . . . . . . . 109 4.1.3 TV-H−1 Inpainting . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.1.4 LCIS Inpainting . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.1.5 Numerical Discussion . . . . . . . . . . . . . . . . . . . . . . . . 133 A Dual Solver for TV-H−1 Minimization . . . . . . . . . . . . . . . . . . 135 4.2.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . 135 4.2.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Domain Decomposition for TV Minimization . . . . . . . . . . . . . . . 153 4.3.1 Preliminary Assumptions . . . . . . . . . . . . . . . . . . . . . . 159 4.3.2 A Convex Variational Problem and Subspace Splitting . . . . . . 162 4.3.3 Local Minimization by Lagrange Multipliers . . . . . . . . . . . . 165 4.3.4 Convergence of the Sequential Alternating Subspace Minimization 174 4.3.5 A Parallel Alternating Subspace Minimization and its Convergence180 4.3.6 Domain Decomposition for TV-L2 Minimization . . . . . . . . . . 182 4.3.7 Domain Decomposition for TV-H−1 Minimization . . . . . . . . . 197 ix CONTENTS 5 Applications 5.1 5.2 202 Restoration of Medieval Frescoes . . . . . . . . . . . . . . . . . . . . . . 202 5.1.1 Neidhart Frescoes . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.1.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Road Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.2.1 Bitwise Cahn-Hilliard Inpainting . . . . . . . . . . . . . . . . . . 212 6 Conclusion 214 A Mathematical Preliminaries 217 A.1 Distributional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 217 A.2 Subgradients and Subdiﬀerentials . . . . . . . . . . . . . . . . . . . . . . 217 A.3 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 A.4 The Space H −1 and the Inverse Laplacian ∆−1 . . . . . . . . . . . . . . 218 A.5 Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . 219 References 241 x Notation and Symbols Function Spaces and Norms For Ω an open and bounded subset of Rd we deﬁne the following real-valued function spaces. Rd R+ BV (Ω) BV − w∗ C m (Ω) Lp (Ω) h·, ·i2 k·k2 L∞ (Ω) Lploc (Ω) W p,q (Ω) W0p,q (Ω) The Euclidean space of dimension d with the Euclidean norm | · |. the non-negative real numbers. Space of functions of bounded variation with seminorm |Df | (Ω), the total variation of f in Ω. The weak∗ topology of BV (Ω). The space of functions on Ω, which are m−times continuously diﬀerentiable. With 1 ≤ p < functions R ∞: p Space of Lebesque measurable p f such that Ω |f | dx < ∞. The space L (Ω) is a Banach 1/p R space with corresponding norm kf kLp (Ω) = Ω |f |p dx . In the case p = 2 it is a Hilbert space with corresponding R inner product hf, giL2 (Ω) = Ω f · g dx. := h·, ·iL2 (Ω) := k·kL2 (Ω) for the norm in L2 (Ω) Space of Lebesque measurable functions f such that there exists a constant C with |f (x)| ≤ C, a.e. x ∈ Ω. The space L∞ (Ω) is a Banach space with corresponding norm kf kL∞ (Ω) = supx∈Ω {|f (x)|}. Lploc (Ω) = {f : Ω → R : f ∈ Lp (D) for each D ⋐ Ω}. With 1 ≤ p, q ≤ ∞: Sobolev space of functions f ∈ Lq (Ω) such that all derivatives up to order p belong to Lq (Ω). The space W p,q (Ω) is a Banach space with norm kf kW p,q (Ω) = 1/q Pp R k q dx , where Dk f denotes the k−th disk=1 Ω D f tributional derivative of f , cf. Appendix A.1 {f ∈ W p,q (Ω) : f |∂Ω = 0}. xi H p (Ω) k·k1 H0p (Ω) H −1 (Ω) k·k−1 h·, ·i−1 H∂−1 (Ω) W p,2 (Ω). This is a Hilbert with corresponding inner R Pp space k product hf, giH p (Ω) = k=1 Ω D f · Dk g dx. For this special Sobolev space we write k·kH p (Ω) := k·kW p,2 (Ω) for its corresponding norm. := k·kH 1 (Ω) W0p,2 (Ω). ∗ 1 H01 (Ω) , i.e., the dual space H 0 (Ω) with correspond of ing norm k·kH − 1(Ω) = ∇∆−1 ·2 and inner product h·, ·iH −1 (Ω) = ∇∆−1 ·, ∇∆−1 · 2 . Thereby ∆−1 is the inverse of the negative Laplacian −∆ with zero Dirichlet boundary conditions, cf. Appendix A.4. := k·kH − 1(Ω) := h·, ·iH −1 (Ω) −1 This is a nonstandard space, i.e., H∂ (Ω) is a subspace of R ∗ 1 the space ψ ∈ H (Ω) : Ω ψ dx = 0 . The norm and the inner product are deﬁned as for H −1 (Ω) above, with the only diﬀerence, that ∆−1 is the inverse of the negative Laplacian −∆ with zero Neumann boundary conditions, cf. Section 2.1.3 for details. For X a Banach space with a norm k·kX and v : (0, T ) → X we denote C m (0, T ; X) Lp (0, T ; X) With m ≥ 0, 0 < T < ∞: Space of functions from [0, T ] to X, which are m−times continuously diﬀerentiable. It is a Banach space with the norm kvkC m (0,T ;X) = l ∂ v max0≤l≤m sup0≤t≤T ∂tl (t) . X With 1 ≤ p < ∞: Space of functions v → v(t) measurable on (0, T ) for the measure dt (i.e., the scalar functions t → kvkX are dt-measurable). It is a Banach space with the norm 1/p R T < +∞. kvkLp (0,T ;X) = 0 kvkpX dt For a functional J : X → (−∞, +∞] where X is a Banach space we write argmin {J} := {u ∈ X : J(u) = inf X J} l.s.c. (sequen- Lower semicontinuous: J is called l.s.c. if for every sequence tially) (un ) converging to u we have lim inf n→∞ J(un ) ≥ J(u). xii About Functions For a function f : Ω ⊂ Rd → R and a sequence of functions (f n )n∈N belonging to a Banach space X we have f n → f in X The sequence (f n ) converges strongly to f in X. f n ⇀ f in X The sequence (f n ) converges weakly to f in X. ∗ The sequence (f n ) converges to f in the weak∗ topology of f n ⇀ f in X X. kf kX The norm of f in X; for speciﬁc norm deﬁnitions compare the notations in “Function Spaces and Norms”. supp {f } For a measurable function f : Ω ⊂ Rd → R, let (wi )i∈I be the family of all open subsets such that wi ⊆ Ω and for each i ∈ I, f = 0 a.e. on wi . Then supp (the support of f ) is S deﬁned by supp {f } = Ω \ i wi . Df Distributional derivative of f , cf. Appendix A.1. ∇f Gradient of f . P ∂f . ∇·f Divergence of f , i.e., ∇ · f = di=1 ∂x Pd ∂i2 f ∆f Laplacian operator, i.e., ∆f = i=1 ∂x2 . vt fΩ i Time derivative of a function v : (0, T )R → X for t > 0. 1 Mean value of f over Ω, i.e., fΩ = |Ω| Ω f dx. Let H be a real separable Hilbert space. For a function ψ : H → R ∪ {+∞} we write ψ ∗ (·) The Legendre-Fenchel transform, i.e., convex conjugate ψ ∗ : H∗ → R ∪ {+∞} is deﬁned by ψ ∗ (u) = supv∈H{hv, ui − ψ(v)}. Miscellaneous Notation Let A, B and R be bounded and open sets in Rd . Ω ⊂ Rd An open and bounded set with Lipschitz boundary. A ֒→ B A is continuously embedded into B. A ֒→֒→ B A is compactly embedded into B. T ∈ L(H) T is a bounded linear operator in a Hilbert space H. T∗ The adjoint operator of T in H, i.e., hT ∗ u, vi = hu, T vi, where h·, ·i denotes the inner product in H. kT k The operator norm of T . |·| Euclidean norm in Rd . V∗ The topological dual for a topological vector space V . Hd d-dimensional Hausdorﬀ measure. xiii s>0 1 sign(s) Sign function, i.e., sign(s) = 0 s=0 −1 s < 0. χR Characteristic function of a bounded and open set R, i.e., ( 1 x∈R χR (x) = +∞ otherwise. 1R Indicator(function of a bounded and open set R, i.e., 1 x∈R 1R (x) = 0 otherwise. Where needed, notations and deﬁnitions might be extended in the introductory part of a new chapter or section. xiv Chapter 1 Introduction My Ph.D. thesis is concerned with modern techniques which use partial diﬀerential equations (PDEs) for image processing. I am especially interested in the analysis and numerical solution of higher-order ﬂows such as the Cahn-Hilliard equation, and singular ﬂows such as total variation minimizations, appearing in image inpainting (interpolation) tasks as well as concrete real world applications. Now, inpainting is an artistic word for virtual image restoration or image interpolation, whereby missing parts of damaged images are ﬁlled in, based on the information obtained from the surrounding areas. Virtual image restoration is an important challenge in our modern computerized society: From the reconstruction of crucial information in satellite images of our earth to the renovation of digital photographs and ancient artwork, virtual image restoration is ubiquitous. Inpainting methods based on higher-order ﬂows, i.e., PDEs of third-or fourth differential order, stirred growing interest in the image processing community in recent years. This is because of the high-quality visual results produced by these methods, which are superior to the ones produced via second-order PDEs. Current research in this area mainly concentrates on three issues. The ﬁrst one, being the development of simple but eﬀective higher-order inpainting models, is also the ﬁrst big topic of this thesis. Therein I aim to replace curvature driven models, like Euler’s elastica inpainting, by higher-order, diﬀusion-like models such as Cahn-Hilliard- and TV-H−1 inpainting. The diﬀusion-like inpainting approaches have the advantage that their numerical solution is in general easier and faster, while preserving the good visual results produced by the curvature driven approaches. 1 The second issue is a derivation of rigorous analytic properties and geometric interpretation of these equations. This is a challenging task in its own right, since higher-order equations are very new and little is known about them. Often higher-order equations do not possess a maximum or comparison principle, hence their analysis, like proofs of existence, uniqueness and convergence of solutions to these equations, is very involved and has not been established yet for all cases. A large part of my thesis provides answers to some of these open questions. In particular I shall present results for the Cahn-Hilliard equation, for a fourth-order total variation ﬂow, and for a nonlocal higher-order diﬀusion equation. The third issue is the eﬀective and fast numerical solution of inpainting approaches, and constitutes the third main topic of this work. Although higher-order inpainting approaches produce very good visual results, in general they suﬀer from high computational complexity. This means that the computational eﬀort for producing an inpainted image from one of these methods can be big. This high complexity makes such inpainting approaches less attractive for interactive image manipulation and thus less popular in real-world applications. Hence the focus is on the development of fast and eﬃcient numerical algorithms for their solution. Finally, I am also interested in applications of these inpainting methods in real life. A particular example for this is the recently found Neidhart frescoes in Vienna: Advanced mathematical inpainting tools can restorate the frescoes digitally. The virtually restored frescoes then serve as a virtual template for museums artists who work on the physical restoration of the frescoes. Another real world application is the restoration of satellite images of roads in urban areas in the United States: Here the problem is that the roads are partly covered by trees or buildings and the goal is to remove these obstacles and receive a picture of just the road. For this task PDEs turned out to be the method of choice, not only due to the high quality visual results but also due to the fact that this enables automated restoration of whole databases of roads. In other words, combined with a-priori segmentation algorithms for the trees (providing an initial condition), PDEs inpaint the road images unsupervised, i.e., no user action is required. In the present chapter, in Section 1.1, I start the discussion with some general remarks about the importance of digital image processing methods. Section 1.2 shall explain the term digital image for a general audience. Then in Section 1.3 the task 2 1.1 The Role of Image Processing in our Modern Society of image inpainting is explained in more detail and an overview of existing work in this area is given, in particular in the range of PDE- and energy-based approaches in Section 1.3.1. Further the use of third- and fourth-order partial diﬀerential equations instead of second-order ﬂows in image inpainting is motivated in Section 1.3.2, and an overview of existing numerical methods for higher-order inpainting approaches is given in Section 1.3.3. 1.1 The Role of Image Processing in our Modern Society In our modern society we encounter digital images in a lot of diﬀerent situations: from everyday life, where analogue cameras have long been replaced by digital ones, to their professional use in medicine, earth sciences, arts, and security applications. The images produced in these situations usually have to be organized and possibly postprocessed. The organization and processing of digital images is known under the name of image processing or computer vision. We often have to deal with the processing of images, e.g., the restoration of images corrupted by noise, blur, or intentional scratching. The idea behind image processing is to provide methods that improve the quality of these images by postprocessing them. Examples of damaged images come from medical imaging tools such as brain imaging with MRI (Magnetic Resonance Imaging), PET (Positron Emission Tomography) imaging of inner organs like the human heart, and X-ray imaging of our skeleton. These imaging tools usually produce noisy or incomplete image data. Other examples include satellite images of our earth, which are often blurred. Further, digital image restoration is used in art preservation and restoration, where digital photographs are taken from ancient artwork and are digitally restorated and stored. As such they can serve as templates for restorators and can be kept in a databank for preservation. In sum, digital image restoration provides eﬀective tools to recover or complete lost image information. Keywords in this context are image denoising, image deblurring, image decomposition, image inpainting, and image synthesizing. Another branch of image processing is object recognition and tracking. In these applications one is interested in certain objects in an image that one wants to extract and/or follow, e.g., in a video stream. Keywords here are image segmentation, object recognition, object tracking, and remote sensing. 3 1.2 What is a Digital Image? Finally, there is also the important issue of organizing image data in an eﬃcient way. To do so we have to think about the classiﬁcation of images, e.g., depending on their contents, and an optimal way of storing them in terms of minimizing the storage place. For a more complete introduction to digital image processing we refer to [AK06]. Considering this huge – but by no means complete – amount of image processing applications listed above and the fact that there are still problems in this area which have not been completely and satisfactorily solved, it is not surprising that this is a very active and broad ﬁeld of research. From engineers to computer scientists and mathematicians, a large group of people have been and are still working in this area. In the following we focus on the ﬁrst branch of applications, namely image restoration and in particular on so-called image inpainting techniques, cf. Section 1.3. 1.2 What is a Digital Image? In order to appreciate the following theory and the image inpainting applications, we ﬁrst need to understand what a digital image really is. Roughly speaking a digital image is obtained from an analogue image (representing the continuous world) by sampling and quantization. Basically this means that the digital camera superimposes a regular grid on an analogue image and assigns a value, e.g., the mean brightness in this ﬁeld, to each grid element, cf. [AK06]. In the terminology of digital images these grid elements are called pixels. The image content is then described by grayvalues or colour values prescribed in each pixel. The grayvalues are scalar values ranging between 0 (black) and 255 (white). The colour values are vector values, e.g., (r, g, b), where each channel r, g and b represents the red, green, and blue component of the colour and ranges, as the grayvalues, from 0 to 255. The mathematical representation of a digital image is a so-called image function u deﬁned on a two dimensional (in general rectangular) image domain, the grid. This function is either scalar valued in the case of a grayvalue image, or vector valued in the case of a colour image. Here the function value u(x, y) denotes the grayvalue, i.e., colourvalue, of the image in the pixel (x, y) of the image domain. Figure 1.1 visualizes the connection between the digital image and its image function for the case of a grayvalue image. 4 1.3 Image Inpainting Figure 1.1: Digital image versus image function: On the very left a zoom into a digital photograph where the image pixels (small squares) are clearly visible is shown; in the middle the grayvalues of the red selection in the digital photograph are displayed in matrix form; on the very right the image function of the digital photograph is shown where the grayvalue u(x, y) is plotted as the height over the x, y− plane. Typical sizes of digital images range from 2000 × 2000 pixels in images taken with a simple digital camera, to 10000 × 10000 pixels in images taken with high-resolution cameras used by professional photographers. The size of images in medical imaging applications depends on the task at hand. PET for example produces three dimensional image data, where a full-length body scan has a typical size of 175 × 175 × 500 pixels. Now, since the image function is a mathematical object we can treat it as such and apply mathematical operations to it. These mathematical operations are summarized by the term image processing techniques, and range from statistical methods, morphological operations, to solving a partial diﬀerential equation for the image function, cf. Section 1.1. We are especially interested in the last, i.e., PDE- and variational methods used in imaging and in image inpainting in particular. 1.3 Image Inpainting An important task in image processing is the process of ﬁlling in missing parts of damaged images based on the information obtained from the surrounding areas. It is essentially a type of interpolation and is called inpainting. Let f represent some given image deﬁned on an image domain Ω. Loosely speaking, the problem is to reconstruct the original image u in the (damaged) domain D ⊂ Ω, called inpainting domain or a hole/gap (cf. Figure 1.2). 5 1.3 Image Inpainting Figure 1.2: The inpainting task. Figure from [CS05] The term inpainting was invented by art restoration workers, cf. [Em76, Wa85], and ﬁrst appeared in the framework of digital restoration in the work of Bertalmio et al. [BSCB00]. Therein the authors design a discrete partial diﬀerential equation, which intends to imitate the restoration work of museum artists. Their method shall be explained in more detail in the subsequent section. Applications Applications of digital image inpainting are automatic scratch removal in old photographs and ﬁlms [BSCB00, CS01a, KMFR95b], digital restoration of ancient paintings for conservation purposes [BFMS08], text erasing, like the removal of dates, subtitles, or publicity from a photograph [BSCB00, BBCSV01, CS01a, CS01c], special eﬀects like object disappearance [BSCB00, CS01c], disocclusion [NMS93, MM98], spatial/temporal zooming and super-resolution [BBCSV01, CS01a, Ma00, MG01, TYW01], error concealment [WZ98], lossy perceptual image coding [CS01a], removal of the laser dazzling eﬀect [CCB03], and sinogram inpainting in X-ray imaging [GZYXC06], only to name a few. The beginnings of digital image inpainting The history of digital image inpainting has its beginning in the works of engineers and computer scientists. Their methods were based on statistical and algorithmic 6 1.3 Image Inpainting approaches in the context of image interpolation [AKR97, KMFR95a, KMFR95b], image replacement [IP97, WL00], error concealment [JCL94, KS93], and image coding [Ca96, FL96, RF95]. In [KMFR95b], for example, the authors present a method for video restoration. Their algorithm uses intact information from earlier and later frames to restore the current one and is therefore not applicable to still images. In interpolation approaches for “perceptually motivated” image coding [Ca96, FL96, RF95] the underlying image model is based on the concept of “raw primal sketch” [Ma82]. More precisely this method assumes that the image consists of mainly homogeneous regions, separated by discontinuities, i.e., edges. The coded information then just consists of the geometric structure of the discontinuities and the amplitudes at the edges. Some of these coding techniques already used PDEs for this task, see e.g., [Ca88, Ca96, CT94]. Initiated by the pioneering works of [NMS93, MM98, CMS98, BSCB00], and [CS01a] also the mathematics community got involved in image restoration, using partial differential equations and variational methods for this task. Their approach and some of their methods shall be presented in the following section. 1.3.1 Energy-Based and PDE Methods for Image Inpainting In this section I present the general energy-based, and PDE approach used in image inpainting. After a derivation of both methods in the context of inverse problems and prior image models, I give an overview of the most important contributions within this area. To keep to chronological order, I start with the energy-based approach, also called variational approach, or image prior model. Energy-based methods Energy-based methods can be best explained from the point of view of inverse problems. In a wide range of image processing tasks one encounters the situation that the observed image f is corrupted, e.g., by noise or blur. The goal is to recover the original image u from the observed datum f . In mathematical terms this means that one has to solve an inverse problem T u = f , where T models the process through which the image u went before observation. In the case of an operator T with unbounded inverse, this problem is ill-posed. In such cases one modiﬁes the problem by introducing some additional a-priori information on u, usually in terms of a regularizing term involving, e.g., the total variation of u. This results in a minimization problem for the ﬁdelity T u − f plus 7 1.3 Image Inpainting the a-priori information modeled by the regularizing term R(u). In the terminology of prior image models, the regularizing term is the so-called prior image model, and the ﬁdelity term is the data model. The concept of image prior models has been introduced by Mumford, cf. [Mu94]. For a general overview on this topic see also [AK06]. Inpainting approaches can also be formulated within this framework. More precisely let Ω ⊂ R2 be an open and bounded domain with Lipschitz boundary, and let B1 , B2 be two Banach spaces with B2 ⊆ B1 , f ∈ B1 denoting the given image, and D ⊂ Ω the missing domain. A general variational approach in image inpainting is formulated mathematically as a minimization problem for a regularized cost functional J : B2 → R, J(u) = R(u) + 1 kλ(f − u)k2B1 → min , u∈B2 2 where R : B2 → R and λ(x) = ( λ0 0 Ω\D D, (1.1) (1.2) is the indicator function of Ω \ D multiplied by a constant λ0 ≫ 1. This constant is the tuning parameter of the approach. As before R(u) denotes the regularizing term and represents a certain a-priori information from the image u, i.e., it determines in which space the restored image lies in. In the context of image inpainting, i.e., in the setting of (1.1)-(1.2), it plays the main role of ﬁlling in the image content into the missing domain D, e.g., by diﬀusion and/or transport. The ﬁdelity term kλ(f − u)k2B1 of the inpainting approach forces the minimizer u to stay close to the given image f outside of the inpainting domain (how close is dependent on the size of λ0 ). In this case the operator T from the general approach equals the indicator function of Ω \ D. In general we have B2 ⊂ B1 , which signiﬁes the smoothing eﬀect of the regularizing term on the minimizer u ∈ B2 (Ω). Note that the variational approach (1.1)-(1.2) acts on the whole image domain Ω (global inpainting model), instead of posing the problem on the missing domain D only. This has the advantage of simultaneous noise removal in the whole image and makes the approach independent of the number and shape of the holes in the image. In this global model the boundary condition for D is superimposed by the ﬁdelity term. Before the development of inpainting algorithms one has to understand what an image really is. In the framework of image prior models this knowledge is encoded in the regularizing term R(u). As a consequence diﬀerent image prior models result in diﬀerent 8 1.3 Image Inpainting inpainting methods. As pointed out in [CS01a], the challenge of inpainting lies in the fact that image functions are complex and mostly lie outside of usual Sobolev spaces. Natural images for example are modeled by Mumford as distributions, cf. [Mu94]. Texture images contain oscillations and are modeled by Markov random ﬁelds, see e.g., [GG84, Br98], or by functions in negative Sobolev spaces, see e.g., [OSV03, LV08]. Most nontexture images are modeled in the space of functions of bounded variation ([ROF92, CL97]), and in the Mumford-Shah object-boundary model, cf. [MS89]. Note also that despite its similarity to usual image enhancement methods such as denoising or deblurring, inpainting is very diﬀerent from these approaches. This is because the missing regions are usually large, i.e., larger than the type of noise treated by common image enhancement algorithms. Additionally, in image enhancement the pixels contain both noise and original image information whereas in inpainting there is no signiﬁcant information inside the missing domain. Hence reasonable energy-based approaches in denoising do not necessarily make sense for inpainting. An example for this discrepancy between inpainting approaches and existing image enhancement methods is given in the work of Chan and Shen [CS05]. Therein the authors pointed out that the extension of existing texture modeling approaches in denoising, deblurring and decomposition to inpainting, is not straightforward. In fact the authors showed that the Meyer model [Me01] modiﬁed for inpainting, where the ﬁdelity term modeled in Meyer’s norm only acts outside of the missing domain, is not able to reconstruct interesting texture information inside of the gap: For every minimizer pair (u, g) (where g represents the texture in the image) of the modiﬁed Meyer model, it follows that g is identically zero inside the gap D. PDE methods To segue into the PDE-based approach for image inpainting, we ﬁrst go back to the general variational model in (1.1)-(1.2). Under certain regularity assumptions on a minimizer u of the functional J, the minimizer fulﬁlls a so-called optimality condition on (1.1), i.e., the corresponding Euler-Lagrange equation. In other words, for a minimizer u the ﬁrst variation, i.e., the Fréchet derivative of J, has to be zero. In the case B1 = L2 (Ω), in mathematical terms this reads − ∇R(u) + λ(f − u) = 0 in Ω, 9 (1.3) 1.3 Image Inpainting which is a partial diﬀerential equation with certain boundary conditions on ∂Ω. Here ∇R denotes the Fréchet derivative of R over B1 = L2 (Ω), or more general an element from the subdiﬀerential of R(u). The dynamic version of (1.3) is the so-called steepest- descent or gradient ﬂow approach. More precisely, a minimizer u of (1.1) is embedded in an evolution process. We denote it by u(·, t). At time t = 0, u(·, t = 0) = f ∈ B1 is the original image. It is then transformed through a process that is characterized by ut = −∇R(u) + λ(f − u) in Ω. (1.4) Given a variational formulation (1.1)-(1.2), the steepest-descent approach is used to numerically compute a minimizer of J, whereby (1.4) is iteratively solved until one is close enough to a minimizer of J. In other situations we will encounter equations that do not come from variational principles, such as CDD inpainting [CS01c], Cahn-Hilliard-, and TV-H−1 inpainting in Section 2.1 and 2.2. Then the inpainting approach is directly given as an evolutionary PDE, i.e., ut = F (x, u, Du, D2 u, . . .) + λ(f − u), (1.5) where F : Ω × R × R2 × R4 × . . . → R, and belongs to the class of PDE-based inpainting approaches. State of the art Depending on the choice of the regularizing term R and the Banach spaces B1 , B2 , i.e., the ﬂow F (x, u, Du, . . .), various inpainting approaches have been developed. These methods can be divided into two categories: texture inpainting that is mainly based on synthesizing the texture and ﬁlling it in, and nontexture (or geometric/structure) inpainting that concentrates on the recovery of the geometric part of the image inside the missing domain. In the following we shall only concentrate on nontexture images. In fact the usual variational/PDE approach in inpainting uses local PDEs (in contrast to nonlocal PDEs cf. Section 3.2), which smooth out every statistical ﬂuctuation, i.e., do not see a global pattern such as texture in an image. In [CS01a] the authors call this kind of image restoration low-level inpainting since it does not take into account global features, like patterns and textures. For now, let us start with the presentation of existing nontexture inpainting models. 10 1.3 Image Inpainting Pioneering works The terminology of digital inpainting ﬁrst appeared in the work of Bertalmio et al. [BSCB00]. Their model is based on observations about the work of museum artists, who restorate old paintings. Their approach follows the principle of prolongating the image intensity in the direction of the level lines (sets of image points with constant grayvalue) arriving at the hole. This results in solving a discrete approximation of the PDE ut = ∇⊥ u · ∇∆u, (1.6) solved within the hole D extended by a small strip around its boundary. This extension of the computational domain about the strip serves as the intact source of the image. It is implemented in order to fetch the image intensity and the direction of the level lines which are to be continued. Equation (1.6) is a transport equation for the image smoothness modeled by ∆u along the level lines of the image. Here ∇⊥ u is the per- pendicular gradient of the image function u, i.e., it is equal to (−uy , ux ). To avoid the crossing of level lines, the authors additionally apply intermediate steps of anisotropic diﬀusion, which may result in the solution of a PDE like ut = ∇⊥ u · ∇∆u + ν∇ · (g(|∇u|)∇u), where g(s) deﬁnes the diﬀusivity coeﬃcient and ν > 0 a small parameter. In [BBS01] the authors interpret a solution of the latter equation as a direct solution of the NavierStokes equation for an incompressible ﬂuid, where the image intensity function plays the role of the stream function whose level lines deﬁne the stream lines of the ﬂow. Note that the advantage of this viewpoint is that one can exploit a rich and well-developed history of ﬂuid problems, both analytically and numerically. Also note that Bertalmio et al.’s model actually is a third-order nonlinear partial diﬀerential equation. In the next section we shall see why higher-order PDEs are needed to solve the inpainting task satisfactorily. In a subsequent work of Ballester et al. [BBCSV01] the authors adapt the ideas of [BSCB00] about the simultaneous graylevel- and gradient-continuation to deﬁne a formal variational approach to the inpainting problem. Their variational approach is solved via its steepest descent, which leads to a set of two coupled second-order PDEs, one for the graylevels and one for the gradient orientations. 11 1.3 Image Inpainting An axiomatic approach and elastica curves Chronologically earlier Caselles, Morel and Sbert in [CMS98] and Masnou and Morel in [MM98] initiated the variational/PDE approach for image interpolation. In [CMS98] the authors show that any operator that interpolates continuous data given on a set of curves can be computed as a viscosity solution (cf. [Ev98]) of a degenerate elliptic PDE. This equation is derived via an axiomatic approach, in which the basic interpolation model, i.e., the PDE, results from a series of assumptions about the image function and the interpolation process. We note that this inpainting approach is only able to continue smooth image contents and cannot be used for the continuation of edges. The approach of Masnou and Morel [MM98] belongs to the class of variational approaches and is based on Nitzberg et al.’s work on segmentation [NMS93]. In [NMS93] Nitzberg et al. presented a variational technique for removing occlusions of objects with the goal of image segmentation. Therein the basic idea is to connect T-junctions at the occluding boundaries of objects with Euler elastica minimizing curves. A curve is said to be Eulers elastica if it is the equilibrium curve of the Euler elastica energy Z E(γ) = (a + bκ2 ) ds, γ where ds denotes the arc length element, κ(s) the scalar curvature, and a, b two positive constants. These curves have been originally obtained by Euler in 1744, cf. [Lo27], and were ﬁrst introduced in computer vision by Mumford in [Mu94]. The basic principle of the elastica curves approach is to prolongate edges by minimizing their length and curvature, In [Mu94, NMS93] it is based on a-priori edge detection. Hence, this approach is only applicable to highly segmented images with few T-junctions and is not applicable to natural images. Moreover, edges alone are not reliable information since they are sensitive to noise. In [MM98] Masnou and Morel extend Mumford’s idea of length and curvature minimization from edges to all the level lines of the image function. Their approach is based on the global minimization of a discrete version of a constrained Euler elastica energy for all level lines. This level line approach has the additional advantage that it is contrast invariant; this is diﬀerent from the edge-approach of Nitzberg et al. [NMS93] which depends on the diﬀerence of grayvalues. The discrete version of the Euler elastica energy is connected to the human vision approach of Gestalt theory, in particular Kanizsa’s amodal completion theory [Ka96]. Gestalt theory tries to explain how the human visual system understands partially occluded objects. This gave the 12 1.3 Image Inpainting approach in [MM98] its name disocclusion instead of image inpainting. Details of the theoretical justiﬁcation of the model in [MM98] and the algorithm itself were much later published by Masnou [Ma02]. Note that the Euler elastica energy was used for inpainting later by Chan and Shen in a functionalized form, cf. [CKS02] and later remarks within this section. Anisotropic diffusion: total variation inpainting and CDD Another varia- tional inpainting approach constitutes the work of Chan and Shen in [CS01a]. Their approach is chronologically in between the two works of Bertalmio et al. i.e., [BSCB00, BBCSV01]. The motivation was to create a scheme which is motivated by existing denoising/segmentation methods and is mathematically easier to understand and to analyze. Their approach is based on the most famous model in image processing, the total R variation (TV) model, where R(u) = |Du| (Ω) ≈ Ω |∇u| dx denotes the total variation of u, B1 = L2 (Ω) and B2 = BV (Ω) the space of functions of bounded variation, cf. also [CS01d, CS01a, RO94, ROF92]. It results in the action of anisotropic diﬀusion inside the inpainting domain, which preserves edges and diﬀuses homogeneous regions and small oscillations like noise. More precisely the corresponding steepest descent equation reads ut = ∇ · ∇u |∇u| + λ(f − u). The disadvantage of the total variation approach in inpainting is that the level lines are interpolated linearly, cf. Section 1.3.2. This means that the direction of the level lines is not preserved, since they are connected by a straight line across the missing domain. A straight line connection might still be pleasant for small holes, but very unpleasant in the presence of larger gaps, even for simple images. Another consequence of the linear interpolation is that level lines might not be connected across large distances, cf. Section 1.3.2. A solution for this is the use of higher-order PDEs such as the ﬁrst works of Bertalmio et al. [BSCB00], Ballester et al. [BBCSV01], and the elastica approach of Masnou and Morel [MM98], and also some PDE/variational approaches proposed later on. Within this context, the authors in [CS01c] proposed a new TV-based inpainting method. In their model the conductivity coeﬃcient of the anisotropic diﬀusion depends on the curvature of level lines and it is possible to connect the level lines across large 13 1.3 Image Inpainting distances. This new approach is a third-order diﬀusion equation and is called inpainting with Curvature Driven Diﬀusions (CDD). The CDD equation reads g(κ) ut = ∇ · ∇u + λ(f − u), |∇u| where g : B → [0, +∞) is a continuous function, which penalizes large curvatures, and encourages diﬀusion when the curvature is small. Here B is an admissible class of functions for which the curvature κ is well deﬁned, e.g., B = C 2 (Ω). It is of similar type as other diﬀusion driven models in imaging such as the Perona-Malik equation [PM90, MS95], and like the latter does not (in general) follow a variational principle. To give a more precise motivation for the CDD inpainting model, let us recall that a problem of the usual total variation model is that the diﬀusion strength only depends on the contrast or strength of the level lines. In other words the anisotropic diﬀusion of the total variation model diﬀuses with conductivity coeﬃcient 1/|∇u|. Hence the diﬀusion strength does not depend on geometric properties of the level line, given by its curvature. In the CDD model the conductivity coeﬃcient is therefore changed to g(|κ|)/|∇u|, where g annihilates large curvatures and stabilizes small ones. Interestingly enough CDD performs completely orthogonally to the transport equation of Bertalmio et al. [BSCB00]. Bertalmio et al.’s equation transports the smoothness along the level lines, whereas the CDD equation diﬀuses image pixel information perpendicularly to the level lines. Euler’s elastica inpainting This observation gave Chan, Kang, and Shen the idea to combine both methods, which resulted in the Euler’s elastica inpainting model, cf. [CKS02, CS01b]. Their approach is based on the earlier work of Masnou and Morel [MM98], with the diﬀerence that the new approach poses a functionalized model. This means that instead of an elastica curve model for the level lines of the image, they rewrote the elastica energy in terms of the image function u. Then the regularizing term R ∇u reads R(u) = Ω (a + b(∇ · ( |∇u| ))2 )|∇u| dx with positive weights a and b, B1 = L2 (Ω), and B2 = BV (Ω). In fact in [CKS02] the authors veriﬁed that the Euler elastica inpainting model combines both transportation processes [BSCB00] and [CS01c]. They also presented a very useful geometric interpretation for all three models. We shall discuss this issue in a little more detail in Section 2.4, where I compare this geometric 14 1.3 Image Inpainting interpretation with the newly proposed higher-order inpainting schemes from Section 2.1-2.3. Active contour models Other examples to be mentioned for (1.1) are the active contour model based on Mumford and Shah’s segmentation [MS89, CS01a, TYW01, ES02], and its high-order correction the Mumford-Shah-Euler image model [ES02]. The latter improves the former by replacing the straight-line curve model by the elastica energy. The Mumford and Shah image model reads Z γ |∇u|2 dx + αH1 (Γ), R(u, Γ) = 2 Ω\Γ (1.7) where Γ denotes the edge collection and H1 the one dimensional Hausdorﬀ measure (generalization of the length for regular curves). The corresponding inpainting approach minimizes the Mumford Shah image model plus the usual L2 ﬁdelity on Ω \ D. The idea to consider this model for inpainting goes back to Chan and Shen [CS01a] as an alternative to TV inpainting, and to Tsai et al. [TYW01]. The Mumford-Shah-Euler image model diﬀers from (1.7) in the replacement of the straight line model by Euler’s elastica curve model γ R(u, Γ) = 2 Z Ω\Γ 2 |∇u| dx + Z (a + bκ2 ) ds, Γ where κ denotes the curvature of a level line inside the edge collection Γ, and a and b are positive constants as before. . . . and more More recently Bertozzi, Esedoglu, and Gillette [BEG07a, BEG07b] proposed a modiﬁed Cahn-Hilliard equation for the inpainting of binary images (also cf. Section 2.1), and in a separate work Grossauer and Scherzer [GS03] proposed a model based on the complex Ginzburg-Landau energy. A generalization of Cahn-Hilliard inpainting for grayvalue images, called TV-H−1 inpainting was proposed in [BHS08], also cf. Section 2.2. To ﬁnish this overview let us just list some more inpainting approaches in the literature: total variation wavelet inpainting [CSZ06], fast image inpainting based on coherence transport [BM07], landmark based inpainting [KCS02], inpainting via correspondence map [DSC03], texture inpainting with nonlocal PDEs [GO07], simultaneous 15 1.3 Image Inpainting structure and texture inpainting [BVSO03], cartoon and texture inpainting via morphological component analysis [ESQD05]. For a very good introduction to image inpainting I also refer to [CS05]. The scope of the present work are inpainting methods, which use third- and fourthorder PDEs to ﬁll in missing image contents into gaps in the image domain. In the following section I shall, once again, motivate the choice of higher-order ﬂows for image inpainting. 1.3.2 Second- Versus Higher-Order Approaches In this section I want to emphasize the diﬀerence between second- and higher-order models in inpainting. Now, second-order variational inpainting methods (where the order of the method is determined by the derivatives of highest order in the corresponding Euler-Lagrange equation), like TV inpainting, have drawbacks when it comes to the connection of edges over large distances (Connectivity Principle, cf. Figure 1.3) and the smooth propagation of level lines into the damaged domain (Curvature Preservation, cf. Figure 1.4). This is due to the penalization of the length of the level lines within the minimizing process with a second-order regularizer, thus connecting level lines from the boundary of the inpainting domain via the shortest distance (linear interpolation). R The regularizing term R(u) ≈ Ω |∇u| dx in the TV inpainting approach for example can be interpreted via the coarea formula which gives Z ∞ Z length(Γλ ) dλ, min |∇u| dx ⇐⇒ min u Γλ Ω −∞ where Γλ = {x ∈ Ω : u(x) = λ} is the level line for the grayvalue λ. If we consider on the other hand the regularizing term in the Euler elastica inpainting approach the coarea formula reads 2 ! ∇u a+b ∇· min ) |∇u| dx u |∇u| Ω Z ∞ a length(Γλ ) + b curvature2 (Γλ ) dλ. ⇐⇒ min Z Γλ (1.8) −∞ Hence, not only the length of the level lines but also their curvature is penalized (the penalization of each depends on the ratio b/a). This results in a smooth continuation of level lines over the inpainting domain also over large distances, compare Figure 1.3 and 1.4. 16 1.3 Image Inpainting The performance of higher-order inpainting methods, such as Euler elastica inpainting, can also be interpreted via the second boundary condition, necessary for the well-posedness of the corresponding Euler-Lagrange equation of fourth order. As an example, Bertozzi et al. showed in [BEG07b] that the Cahn-Hilliard inpainting model in fact favors the continuation of the image gradient into the inpainting domain. More precisely, the authors proved that in the limit λ0 → ∞ a stationary solution of the Cahn-Hilliard inpainting equation fulﬁlls u=f on ∂D ∇u = ∇f on ∂D, for a given image f regular enough (f ∈ C 2 ). This means that not only the grayvalues of the image are speciﬁed on the boundary of the inpainting domain but also the gradient of the image function, namely the direction of the level lines are given. Figure 1.3: Two examples of Euler elastica inpainting compared with TV inpainting. In the case of large aspect ratios the TV inpainting fails to comply with the Connectivity Principle. Figure from [CS05]. In an attempt to fulﬁll both the connectivity principle and the curvature preservation a number of third and fourth-order diﬀusions has been suggested for image inpainting, most of which have already been discussed in the previous section. Let us recall the main higher-order approaches within this group in the following short summary. The ﬁrst work connecting image inpainting to a third-order PDE (partial 17 1.3 Image Inpainting Figure 1.4: An example of Euler elastica inpainting compared with TV inpainting. Despite the presence of high curvature TV inpainting truncates the circle inside the inpainting domain (linear interpolation of level lines). Depending on the weights a and b Eulers elastica inpainting returns a smoothly restored object, taking the curvature of the circle into account (Curvature Preservation). Figure from [CKS02]. diﬀerential equation) is the transport process of Bertalmio et al. [BSCB00]. A variational third-order approach to image inpainting is CDD (Curvature Driven Diﬀusion) [CS01c]. Although solving the problem of connecting level lines over large distances (connectivity principle), the level lines are still interpolated linearly and thus curvature is not preserved. The drawbacks of the third-order inpainting models [BSCB00, CS01c] have driven Chan, Kang and Shen [CKS02] to a reinvestigation of the earlier proposal of Masnou and Morel [MM98] on image interpolation based on the Euler elastica energy, compare (1.8). The fourth-order elastica inpainting PDE combines CDD [CS01c] and the transport process of Bertalmio et al. [BSCB00] and is as such able to fulﬁll both the connectivity principle and curvature preservation. Other recently proposed higher-order inpainting algorithms are inpainting with the Cahn-Hilliard equation (cf. [BEG07a, BEG07b] and Section 2.1), TV-H−1 inpainting (cf. [BHS08] and Section 2.2), inpainting with LCIS (cf. [SB09] and Section 2.3), and combinations of second and higher-order methods, e.g. [LT06]. 18 1.3 Image Inpainting 1.3.3 Numerical Solution of Higher-Order Inpainting Approaches As correctly remarked in [CS05], the main challenge in digital inpainting is the fast and eﬃcient numerical solution of energy- and PDE-based approaches. Especially highquality, and thus higher-order, approaches, such as Bertalmio’s inpainting approach [BSCB00], inpainting with CDD [CS01c], Euler’s elastica inpainting [CKS02], and the recently proposed Cahn-Hilliard inpainting - and TV-H−1 inpainting - model [BEG07a, BEG07b, BHS08], despite their superior visual results, demand sophisticated numerical algorithms in order to provide acceptable computation times for large images. In this presentation I will concentrate on numerical methods for higher-order approaches only, since they are the ones we are particularly interested in within this work. Numerical methods for higher-order equations The numerical solution of higher- order equations, like thin ﬁlms, phase ﬁeld models, surface diﬀusion equations, and much more, occupied a big part of research in numerical analysis in the last decades. In [DD71] the authors propose a semi-implicit ﬁnite diﬀerence scheme for the solution of second-order parabolic equations. In order to suppress unstable modes a diﬀusion term was added and subtracted to the numerical scheme, i.e., added implicitly and subtracted explicitly in time. Smereka picked up their idea and used it to solve the fourth-order surface diﬀusion equation, cf. [Sm03]. The same idea was rediscovered by Glasner and applied to a phase ﬁeld approach for the Hele-Shaw interface model, cf. [Gl03]. Besides the ﬁnite diﬀerence approximations, there also exist a lot of ﬁnite element algorithms for fourth-order equations. Barrett, Blowey, and Garcke published a series of papers on the solution of various Cahn-Hilliard equations, cf. [BB98, BB99a, BB99b]. For the sharp interface limit of Cahn-Hilliard, i.e., the Hele-Shaw model, Feng and Prohl analyzed ﬁnite element methods in [FP04, FP05]. Finite element methods for thin ﬁlm equations have been studied, for instance, in [GR00, BGLR02]. In inpainting, eﬃcient numerical schemes for higher-order approaches are still a mostly open issue. Most existing numerical schemes for their solution are iterative by nature. PDE based approaches are approximately solved via ﬁxed-point or timestepping schemes. For energy-based methods the minimizer is usually computed iteratively via the corresponding steepest descent equation. 19 1.3 Image Inpainting Iterative methods for curvature driven approaches For curvature driven inpainting approaches, such as [CS01c] and [CKS02], explicit time stepping schemes have been used until recently. Solving a fourth-order nonlinear evolution equation explicitly in time may restrict the time steps to an order (∆x)4 of the spatial grid size ∆x. Hence this may result in the need of a huge amount of iterations until the inpainted image is obtained. The performance of these schemes could be accelerated by, e.g., the method of Marquina and Osher [MO00]. But still the CPU time required is not appropriate for large images. This computational complexity makes such inpainting approaches less attractive for interactive image manipulation and thus less popular in real-life applications. A very recent advance in the design of fast numerical solvers for these inpainting approaches was made in [BC08], where the authors propose a nonlinear multigrid approach for inpainting with CDD [CS01c]. In order to get a non-singular smoother, i.e., ﬁxed-point algorithm, for the multigrid approach, the authors modify the CDD equation by changing the deﬁnition of the curvature-dependent diﬀusivity coeﬃcient g. Numerical results conﬁrm that the modiﬁed method retains the desirable properties of the original CDD inpainting model while accelerating its numerical computation tremendously. In comparison with the explicit time marching of [CS01c] the multigrid method in [BC08] is four orders of magnitude faster. A matter of future research will be the construction of multigrid solvers for other higher-order inpainting approaches, such as Euler’s elastica inpainting [CKS02], and the recently proposed inpainting methods [BEG07b, BEG07a, GS03, BHS08]. In this regard, the crucial point will be to construct an appropriate non-singular smoother for the respective multigrid algorithm. Numerical methods for diffusion-like inpainting models A fast noniterative method for image inpainting was proposed by Bornemann and März in [BM07]. Their method aims to preserve the visually qualitative results of Bertalmio et al. while performing its numerical solution with a computational speed comparable with the one of Telea’s method [Te04]. In their numerical experiments Bornemann and März showed that their method is in fact at least one order of magnitude faster than Bertalmio’s method. One motivation for the proposal of the Cahn-Hilliard equation for inpainting was its superior computational performance in comparison with curvature driven methods. 20 1.3 Image Inpainting As pointed out in [BEG07a, BEG07b] Cahn-Hilliard inpainting beats Euler’s elastica inpainting by one order of magnitude in its computational complexity. Therein the authors use a certain kind of semi-implicit solver, called convexity splitting, for its numerical solution. The same method was used for TV-H−1 inpainting [BHS08, SB09] and for inpainting with LCIS [SB09]. In [SB09] the authors prove rigorous estimates for their numerical scheme, among them the unconditional stability of the scheme, cf. also Section 4.1. Despite this unconditional stability, these numerical schemes still converge slowly due to a damping on the iterates resulting from the method. For more details on convexity splitting and its application to higher-order inpainting methods, cf. [SB09] and Section 4.1. Further, for TV-H−1 minimization in the case of denoising and cartoon/texture decomposition, Elliott and Smitheman proposed a ﬁnite element method for its numerical solution, cf. [ES07, ES08]. Their scheme is inspired by a work of Feng and Prohl [FP03], who proposed a ﬁnite element method for the ROF model, i.e., the TV-L2 minimization problem. Therein the authors also proved rigorous mathematical results about the approximation and convergence properties of their scheme. An extension of their approach to TV-H−1 inpainting would be interesting. Note that, however, the diﬀerence of the inpainting approach from denoising and decomposition is that the former does not follow a variational principle and the ﬁdelity term is locally dependent on the spatial position. A dual approach for TV-H−1 minimization was proposed by the author in [Sc09], also compare Section 4.2 of this work. This work generalizes the algorithm of Chambolle [Ch04] and Bect et al. [BBAC04] from an L2 ﬁdelity term to an H −1 ﬁdelity. The main motivation for the work in [Sc09] is that with the proposed algorithm the domain decomposition approach developed in [FS07], also cf. Section 4.3, can be applied to the higher-order total variation case. Being able to apply domain decomposition methods to TV-H−1 minimization can result in a tremendous acceleration of computational speed due to the ability to parallelize the computation, cf. Section 4.3, and in particular Subsection 4.3.7. Another promising approach within the design of fast numerical solvers in image processing is the Bregman split method proposed by Goldstein and Osher in 2008, [GO08]. In [GBO09] the authors consider the application of this method to image 21 1.3 Image Inpainting segmentation and surface reconstruction. The latter application is about the reconstruction of surfaces from unorganized data points, which is used for constructing level set representations in three dimensions. The Bregman split method, originally designed to accelerate the computation of ℓ1 regularized minimization problems in general, is thereby used to solve a minimization problem regularized with total variation, i.e., the ℓ1 norm of the gradient of the image function. The computational speed of the Bregman split method in this case is comparable with the one using graph cuts [CD08, DS05], with the additional advantage that it is also able to compute the isotropic total variation. The nature of this application, i.e., surface reconstruction interpreted as an interpolation problem for the given data points, already suggests its possible eﬀectiveness for image inpainting. This is certainly something worthwhile to be considered in a future project. In the following chapters I will summarize my main research directions and present applications of the methods we have developed. Organization of the Thesis Chapter 2 is concerned with mathematical models consisting of (higher-order) partial diﬀerential equations used for the task of image inpainting. In Section 2.1 a binary inpainting approach based on a modiﬁed Cahn-Hilliard equation is presented. This inpainting method has been proposed by Bertozzi et al. in [BEG07a, BEG07b], where the latter paper contains a rigorous analysis of the modiﬁed Cahn-Hilliard equation. We extend this analysis by providing the proof of existence of a stationary solution for the equation. A generalization of this approach for grayvalue images is proposed in Section 2.2. This new inpainting approach is called TV-H−1 inpainting. I state analytical results and present numerical examples. Both sections, i.e., Section 2.1 and Section 2.2, have been obtained in a joint work with Martin Burger and Lin He, cf. [BHS08]. In Section 2.3 I present a new inpainting approach based on Low Curvature Image Simplifiers (LCIS), ﬁrst proposed in a joint work with Andrea Bertozzi in [SB09]. Finally, Section 2.4 contains an interpretation of TV-H−1 inpainting and inpainting with LCIS in terms of transport and diﬀusion of grayvalues into the missing domain. This interpretation was inspired by [CKS02], where the authors discuss this issue for Euler’s elastica inpainting, and by discussions with Massimo Fornasier. The results 22 1.3 Image Inpainting from [CKS02] are also brieﬂy presented in this section, and are further used as a basis for comparison with our two inpainting models. Chapter 3 is dedicated to theoretical results about higher-order ﬂows arising in image inpainting. In Section 3.1 I discuss the Cahn-Hilliard equation as a model for phase separation and coarsening of binary alloys – the original motivation for the equation. In particular, I present the results we achieved in [BCMS08], in collaboration with Martin Burger, Shun-Jin Chu, and Peter A. Markowich, about ﬁnite-time instabilities of the Cahn-Hilliard equation and their connection to the Willmore functional. Additionally, in Section 3.2 I present asymptotic results for nonlocal higher-order evolution equations. The latter section is joint work with Julio D. Rossi and is contained in [RS09]. In Chapter 4, numerical methods, which have been especially designed for higherorder inpainting approaches, are presented. In Section 4.1 I start with the discussion of unconditionally stable numerical schemes for higher-order inpainting approaches proposed together with Andrea Bertozzi in [SB09]. Then, in Section 4.2 I present a dual solver for TV-H−1 minimization problems, cf. [Sc09]. In particular, I show that this dual solver can be applied to solve the TV-H−1 inpainting approach from Section 2.2. Finally, in Section 4.3 a domain decomposition method, which is applicable for minimizing functionals with total variation constraints, is presented. This is joint work with Massimo Fornasier and is contained in [FS07]. The main motivations for such an approach are given, and the results achieved in [FS07] are discussed. With the help of the dual solver proposed in Section 4.2, I further show how the theory developed in [FS07] can be applied to solve TV-H−1 inpainting in a computationally eﬃcient way. Finally, in Chapter 5 I present two applications of the higher-order inpainting methods discussed in Chapter 2. The ﬁrst is the restoration of ancient frescoes in Section 5.1, which evolved from an ongoing project at the University of Vienna in joint work with Wolfgang Baatz, Massimo Fornasier, and Peter Markowich (cf. also [BFMS08]). The second is the reconstruction of roads in satellite images in Section 5.2 and is based on joint work with Andrea Bertozzi. The Appendix contains mathematical preliminaries necessary for the understanding of the presented work. 23 Chapter 2 Image Inpainting With Higher-Order Equations This chapter is dedicated to the presentation and the analysis of three fourth-order PDEs used for image inpainting. Thereby Section 2.1 and Section 2.2 about CahnHilliard- and TV-H−1 inpainting have been mainly developed in collaboration with Martin Burger and Lin He and appeared in [BHS08]. The idea of inpainting with LCIS in Section 2.3 arose in a joint work with Andrea Bertozzi [SB09]. The last section, Section 2.4, about the inpainting mechanism of transport and diﬀusion is inspired by the work of Chan, Kang and Shen [CKS02] and discussions with Massimo Fornasier. 2.1 Cahn-Hilliard Inpainting The Cahn-Hilliard equation is a nonlinear fourth-order diﬀusion equation originating in material science for modeling phase separation and phase coarsening in binary alloys. A new approach in the class of fourth-order inpainting algorithms is inpainting of binary images using a modiﬁed Cahn-Hilliard equation, as proposed in [BEG07a] by Bertozzi, Esedoglu and Gillette. The inpainted version u of f ∈ L2 (Ω) assumed with any (trivial) extension to the inpainting domain is constructed by following the evolution of 1 ′ ut = ∆ −ǫ∆u + F (u) + λ(f − u) in Ω, ǫ (2.1) where F (u) is a so-called double-well potential, e.g., F (u) = u2 (u − 1)2 , and as before ( λ0 Ω \ D λ(x) = 0 D 24 2.1 Cahn-Hilliard Inpainting is the indicator function of Ω \ D multiplied by a constant λ0 ≫ 1. The two wells of F correspond to values of u that are taken by most of the grayscale values. Choosing a potential with wells at the values 0 (black) and 1 (white), equation (2.1) therefore provides a simple model for the inpainting of binary images. The parameter ǫ determines the steepness of the transition between 0 and 1. The Cahn-Hilliard equation is a relatively simple fourth-order PDE used for this task rather than more complex models involving curvature terms, cf. also Section 3.1 for more details on the equation. In fact the numerical solution of (2.1) was shown to be of at least an order of magnitude faster than competing inpainting models, cf. [BEG07a]. Still the Cahn-Hilliard inpainting model has many of the desirable properties of curvature based inpainting models such as the smooth continuation of level lines into the missing domain. In fact the mainly numerical paper [BEG07a] was followed by a very careful analysis of (2.1) in [BEG07b]. Therein the authors prove that in the limit λ0 → ∞ a stationary solution of (2.1) solves 1 ′ ∆ ǫ∆u − F (u) = 0 ǫ u=f ∇u = ∇f in D on ∂D (2.2) on ∂D, for f regular enough (f ∈ C 2 ). This, once more, supports the claim, that fourth-order methods are superior over second-order methods with respect to a smooth continuation of the image contents into the missing domain. In [BEG07b] the authors further proved global existence of a unique weak solution of the evolution equation (2.1). More precisely the solution u was proven to be an element in C([0, T ]; L2 (Ω)) ∩ L2 (0, T ; V ), where V = φ ∈ H 2 (Ω) | ∂φ/∂ν = 0 on ∂Ω , and ν is the outward pointing normal on ∂Ω. Nevertheless the existence of a solution of the stationary equation 1 ′ ∆ −ǫ∆u + F (u) + λ(f − u) = 0 in Ω, ǫ (2.3) remains unaddressed. The diﬃculty in dealing with the stationary equation is the lack of an energy functional for (2.1), i.e., the modiﬁed Cahn-Hilliard equation (2.1) cannot be represented by a gradient ﬂow of an energy functional over a certain Banach space. This is because the ﬁdelity term λ(f − u) is not symmetric with respect to the H −1 25 2.1 Cahn-Hilliard Inpainting inner product, whereby H −1 is deﬁned in the preﬁx of the thesis. In fact the most evident variational approach would be to minimize the functional Z 1 1 ǫ 2 |∇u| + F (u) dx + kλ(u − f )k2−1 . ǫ 2 Ω 2 (2.4) This minimization problem exhibits the optimality condition 1 0 = −ǫ∆u + F ′ (u) + λ∆−1 (λ(u − f )) , ǫ which splits into 1 0 = −ǫ∆u + F ′ (u) ǫ 1 ′ 0 = −ǫ∆u + F (u) + λ20 ∆−1 (u − f ) ǫ in D in Ω \ D. Hence the minimization of (1.4) translates into a second-order diﬀusion inside the inpainting domain D, whereas a stationary solution of (1.1) fulﬁlls 1 ′ 0 = ∆ −ǫ∆u + F (u) in D ǫ 1 0 = ∆ −ǫ∆u + F ′ (u) + λ0 (f − u) in Ω \ D. ǫ One challenge of our work in [BHS08] is to extend the analysis from [BEG07b] by partial answers to questions concerning the stationary equation (2.3) using alternative methods, namely by ﬁxed point arguments. In [BHS08] we prove Theorem 2.1.1. Equation (2.3) admits a weak solution in H 1 (Ω) provided λ0 ≥ for a positive constant C depending on |Ω|, |D|, and F only. C ǫ3 C ǫ3 in In our numerical examples in [BHS08] we can see that the condition λ0 ≥ Theorem 2.1.1 is naturally fulﬁlled, since in order to obtain good visual results in inpainting approaches λ0 has to be chosen rather large in general, cf. Figure 2.2. Note that the same condition also appears in [BEG07b] where it is needed to prove the global existence of solutions of (2.1). 2.1.1 Existence of a Stationary Solution In this section we prove the existence of a weak solution of the stationary equation (2.3), i.e., we shall verify Theorem 2.1.1. Let Ω ⊂ R2 be a bounded Lipschitz domain and 26 2.1 Cahn-Hilliard Inpainting f ∈ L2 (Ω) given. In order to be able to impose boundary conditions in the equation, we assume f to be constant in a small neighborhood of ∂Ω. This assumption is for technical purposes only and does not inﬂuence the inpainting process as long as the inpainting domain D does not touch the boundary of the image domain Ω. Instead of Neumann boundary data as in the original Cahn-Hilliard inpainting approach (cf. [BEG07b]) we use Dirichlet boundary conditions for our analysis, i.e., we consider 1 ′ ut = ∆ −ǫ∆u + F (u) + λ(f − u) in Ω ǫ (2.5) 1 ′ u = f, −ǫ∆u + F (u) = 0 on ∂Ω. ǫ This change from a Neumann- to a Dirichlet problem makes it easier to deal with the boundary conditions in our proofs but does not have a signiﬁcant impact on the inpainting process as long as we assume that D̄ ⊂ Ω. In Appendix 2.1.3 we nevertheless propose a setting to extend the presented analysis for (2.1) to the originally proposed model with Neumann boundary data. In our new setting we deﬁne a weak solution of equation (2.3) as a function u ∈ H = u ∈ H 1 (Ω), u|∂Ω = f |∂Ω that fulﬁlls 1 ′ hǫ∇u, ∇φi2 + (2.6) F (u), φ − hλ(f − u), φi−1 = 0, ∀φ ∈ H01 (Ω). ǫ 2 Remark 2.1.2. With u ∈ H 1 (Ω) and the compact embedding H 1 (Ω) ֒→֒→ Lq (Ω) for every 1 ≤ q < ∞ and Ω ⊂ R2 the weak formulation is well defined. To see that (2.6) deﬁnes a weak formulation for (2.3) with Dirichlet boundary conditions we integrate by parts in (2.6) and get Z 1 ′ −1 −ǫ∆u + F (u) − ∆ (λ(f − u)) φ dx ǫ Ω Z ∆−1 (λ(f − u)) ∇∆−1 φ · ν dH1 = 0, − ∂Ω ∀φ ∈ H01 (Ω), where H1 denotes the one dimensional Hausdorﬀ measure. Since the above equality holds for all φ ∈ H01 (Ω) it holds in particular for all φ in the subset H01 (Ω) ∩ ∇∆−1 φ · ν = 0 on ∂Ω . This yields ǫ∆u − 1 F ′ (u) + ∆−1 (λ(f − u)) = 0 in Ω ǫ (2.7) −1 ∆ (λ(f − u)) = 0 on ∂Ω. 27 2.1 Cahn-Hilliard Inpainting Assuming suﬃcient regularity on u we can use the deﬁnition of ∆−1 to see that u solves 1 − ǫ∆∆u + ∆F ′ (u) + λ(f − u) = 0 in Ω ǫ ∆−1 (λ(f − u)) = −ǫ∆u + 1 F ′ (u) = 0 on ∂Ω, ǫ Since additionally u|∂Ω = f |∂Ω , the function u solves (2.3) with Dirichlet boundary conditions. For the proof of existence of a solution to (2.6) we follow the following strategy. We consider the ﬁxed point operator A : L2 (Ω) → L2 (Ω) where A(v) = u fulﬁlls for a given v ∈ L2 (Ω) the equation 1 ′ 1 −1 −1 ∆ (u − v) = ǫ∆u − F (u) + ∆ [λ(f − u) + (λ0 − λ)(v − u)] in Ω, τ ǫ (2.8) −1 1 (u − v) − λ(f − u) − (λ0 − λ)(v − u) = 0 on ∂Ω, u = f, ∆ τ where τ > 0 is a parameter. The boundary conditions of A are given by the second equation in (2.8). Note that actually the solution u will be in H 1 (Ω) and hence the boundary condition is well-deﬁned in the trace sense, the operator A into L2 (Ω) is then obtained with further embedding. We deﬁne a weak solution of (2.8) as before by a function u ∈ H = u ∈ H 1 (Ω), u|∂Ω = f |∂Ω that fulﬁlls 1 1 ′ + hǫ∇u, ∇φi2 + (u − v), φ F (u), φ τ ǫ (2.9) −1 2 1 − hλ(f − u) + (λ0 − λ)(v − u), φi−1 = 0 ∀φ ∈ H0 (Ω). A ﬁxed point of the operator A, provided it exists, then solves the stationary equation with Dirichlet boundary conditions as in (2.7). Note that in (2.8) the indicator function λ in the ﬁtting term λ(f − u) + (λ0 − λ)(v − u) = λ0 (v − u) + λ(f − v) only appears in combination with given functions f, v and is not combined with the solution u of the equation. For equation (2.8), i.e., (2.9), we can therefore state a variational formulation. This is, for a given v ∈ L2 (Ω) equation (2.8) is the Euler-Lagrange equation of the minimization problem u∗ = argminu∈H 1 (Ω),u|∂Ω =f |∂Ω Jǫ (u, v) with ǫ J (u, v) = Z Ω 1 1 λ0 ǫ 2 |∇u| + F (u) dx+ ku − vk2−1 + 2 ǫ 2τ 2 28 (2.10) 2 u − λ f − 1 − λ v . λ0 λ0 −1 (2.11) 2.1 Cahn-Hilliard Inpainting We are going to use the variational formulation (2.11) to prove that (2.8) admits a weak solution in H 1 (Ω). This solution is unique under additional conditions. Proposition 2.1.3. Equation (2.8) admits a weak solution in H 1 (Ω) in the sense of (2.9). For τ ≤ Cǫ3 , where C is a positive constant depending on |Ω|, |D|, and F only, the weak solution of (2.8) is unique. Further we prove that the operator A admits a ﬁxed point under certain conditions. Proposition 2.1.4. Set A : L2 (Ω) → L2 (Ω), A(v) = u, where u ∈ H 1 (Ω) is the unique weak solution of (2.8). Then A admits a fixed point û ∈ H 1 (Ω) if τ ≤ Cǫ3 and λ0 ≥ ǫC3 for a positive constant C depending on |Ω|, |D|, and F only. Hence the existence of a stationary solution of (2.1) follows under the condition λ0 ≥ C/ǫ3 . We begin with considering the ﬁxed point equation (2.8), i.e., the minimization problem (2.10). In the following we prove the existence of a unique weak solution of (2.8) by showing the existence of a unique minimizer for (2.11). Proof of Proposition 2.1.3. We want to show that Jǫ (u, v) has a minimizer in H = u ∈ H 1 (Ω), u|∂Ω = f |∂Ω . For this we consider a minimizing sequence un ∈ H of Jǫ (u, v). To see that un is uniformly bounded in H 1 (Ω) we show that Jǫ (u, v) is coercive in H 1 (Ω). With F (u) ≥ C1 u2 − C2 for two positive constants C1 , C2 > 0 and the triangular inequality in the H −1 (Ω) space, we obtain 1 1 C1 C2 ǫ 2 2 2 2 ǫ kuk2 − + kuk−1 − kvk−1 J (u, v) ≥ k∇uk2 + 2 ǫ ǫ 2τ 2 2 ! λ λ0 1 λ 2 + kuk−1 − f + 1− v 2 2 λ0 λ0 −1 λ0 C1 1 ǫ kuk22 + + kuk2−1 − C3 (v, f, λ0 , ǫ, Ω, D). ≥ k∇uk22 + 2 ǫ 4 4τ Therefore a minimizing sequence un is bounded in H 1 (Ω) and it follows that un ⇀ u∗ in H 1 (Ω). To ﬁnish the proof of existence for (2.8) we have to show that Jǫ (u, v) is weakly lower semicontinuous in H 1 (Ω). For this we divide the sequence Jǫ (un , v) of (2.11) in two parts. We denote the ﬁrst term by Z ǫ 1 n n 2 n a = |∇u | + F (u ) dx ǫ Ω 2 | {z } CH(un ) 29 2.1 Cahn-Hilliard Inpainting and the second term by 1 λ0 kun − vk2−1 + b = |2τ {z } |2 n D(un ,v) 2 n u − λ f − 1 − λ v . λ0 λ0 −1 {z } F IT (un ,v) Since H 1 ֒→֒→ L2 it follows un → u∗ in L2 (Ω). Further we know that if bn converges strongly, then lim inf (an + bn ) = lim inf an + lim bn . (2.12) We begin with the consideration of the last term in (2.11). We denote f˜ := (1 − λλ0 )v. We want to show λ λ0 f 2 n ˜2 u − f −→ u∗ − f˜ , −1 −1 or equivalently h∆−1 (un − f˜), un − f˜i2 −→ h∆−1 (u∗ − f˜), u∗ − f˜i2 . For this we consider the absolute diﬀerence of the two terms, |h∆−1 (un − f˜), un − f˜i2 − h∆−1 (u∗ − f˜), u∗ − f˜i2 | = |h∆−1 (un − u∗ ), un − f˜i2 − h∆−1 (u∗ − f˜), un − u∗ i2 | ≤ |hun − u∗ , ∆−1 (un − f˜)i2 | + |h∆−1 (u∗ − f˜), u∗ − un i2 | ≤ kun − u∗ k2 · ∆−1 (un − f˜) + kun − u∗ k2 · ∆−1 (u∗ − f˜) | {z } | {z } 2 2 →0 →0 Since the operator ∆−1 : H −1 (Ω) → H01 (Ω) is linear and continuous it follows that Thus −1 ∆ F ≤ ∆−1 · kF k 2 2 for all F ∈ H −1 (Ω). |h∆−1 (un − f˜), un − f˜i2 − h∆−1 (u∗ − f˜), u∗ − f˜i2 | ≤ kun − u∗ k2 ∆−1 un − f˜ + kun − u∗ k2 ∆−1 u∗ − f˜ | {z } | {z } | {z }2 | {z } | {z } | {z }2 →0 const →0 bounded −→ 0 as n → ∞, 30 const const + 2.1 Cahn-Hilliard Inpainting and we conclude that F IT (un , v) converges strongly to F IT (u∗ , v). With the same argument it follows that D(un , v) converges strongly and consequently that the sequence bn converges strongly in L2 (Ω). Further CH(.) is weakly lower semicontinuous, which follows from the lower semicontinuity of the Dirichlet integral and from the continuity of F by applying Fatou’s Lemma. Hence we obtain Jǫ (u∗ , v) ≤ lim inf Jǫ (un , v). Therefore Jǫ has a minimizer in H 1 , i.e., ∃u∗ with u∗ = argminu∈H 1 (Ω) Jǫ (u, v). We next assert that u∗ fulﬁlls the boundary condition u∗ |∂Ω = f |∂Ω . To see this, note that for an admissible function w ∈ H, un − w ∈ H01 (Ω). Now H01 (Ω) is a closed, linear subspace of H 1 (Ω), and so, by Mazur’s theorem (cf. [Ev98] D.4 for example), is weakly closed. Hence u∗ − w ∈ H01 (Ω) and consequently the trace of u∗ on ∂Ω is equal to f . For simplicity let in the following u = u∗ . To see that the minimizer u is a weak solution of (2.8) we compute the corresponding Euler-Lagrange equation to the minimization problem. For this sake we choose any test function φ ∈ H01 (Ω) and compute the ﬁrst variation of Jǫ , i.e., d ǫ , J (u + δφ, v) dδ δ=0 which has to be zero for a minimizer u. Thus we have 1 ′ λ λ 1 ǫ h∇u, ∇φi2 + v ,φ = 0. F (u), φ 2 + (u − v) + λ0 u − f − 1 − ǫ τ λ0 λ0 −1 Integrating by parts in both terms we get 1 ′ λ λ −1 1 −ǫ∆u + F (u) − ∆ (u − v) + λ0 u − f − 1 − v ,φ ǫ τ λ0 λ0 2 Z Z λ λ 1 (u − v) + λ0 u − f − 1 − v ∇∆−1 φ·ν ds = 0. ∆−1 ∇u·νφ ds+ + τ λ0 λ0 ∂Ω ∂Ω Since φ is an element in H01 (Ω), the ﬁrst boundary integral vanishes. Further, a minimizer u fulﬁlls the boundary condition u = f on the boundary ∂Ω. Hence, we obtain that u fulﬁlls the weak formulation (2.9) of (2.8). For the uniqueness of the minimizer, we need to prove that Jǫ is strictly convex. To do so, we prove that for any u1 , u2 ∈ H 1 (Ω), ǫ ǫ ǫ u1 + u2 J (u1 , v) + J (u2 , v) − 2J , v > 0, (2.13) 2 31 2.1 Cahn-Hilliard Inpainting 2 2 based on an assumption that F (.) satisﬁes F (u1 ) + F (u2 ) − 2F ( u1 +u 2 ) > −C(u1 − u2 ) , for a constant C > 0. An example is when F (u) = 18 (u2 − 1)2 , C = 81 . Denoting u = u1 − u2 , we have u1 + u2 1 ǫ λ0 C Jǫ (u1 , v) + Jǫ (u2 , v) − 2Jǫ , v > kuk21 + + kuk2−1 − kuk22 . 2 4 4τ 4 ǫ By using the inequality kuk22 ≤ kuk1 kuk−1 (2.14) and the Cauchy-Schwarz inequality, for (2.13) to be fulﬁlled, we need s ǫ 1 λ0 C + ≥ , 2 4 4τ 4 ǫ i.e., 1 ǫ λ0 + ≥ C 2. τ 3 Therefore Jǫ (u, v) is strictly convex in u and our minimization problem has a unique minimizer if τ is chosen smaller than Cǫ3 for a constant C depending on |Ω|, |D|, and F only. Because of the convexity of Jǫ in ∇u and u, every weak solution of the EulerLagrange equation (2.8) is in fact a minimizer of Jǫ . This proves the uniqueness of a weak solution of (2.8) provided τ < Cǫ3 . Next we want to prove Proposition 2.1.4, i.e., the existence of a ﬁxed point of (2.8) and with this the existence of a stationary solution of (2.1). To do so we are going to apply Schauder’s ﬁxed point theorem. Proof of Proposition 2.1.4. We consider a solution A(v) = u of (2.8) with v ∈ L2 (Ω) given. In the following we will prove the existence of a ﬁxed point by using Schauder’s ﬁxed point theorem. We start with proving that kA(v)k22 = kuk22 ≤ β kvk22 + α (2.15) for constants β < 1 and α > 0. Having this, we have shown that A is a map from the closed ball K = B(0, M ) = u ∈ L2 (Ω) : kuk2 ≤ M into itself for an appropriate constant M > 0. We conclude the proof with showing the compactness of K and the continuity of the ﬁxed point operator A. From Schauder’s theorem the existence of a ﬁxed point follows. 32 2.1 Cahn-Hilliard Inpainting Let us, for the time being, additionally assume that ∇u and ∆u are bounded in Hence we can multiply (2.8) with −∆u and integrate over Ω to obtain L2 (Ω). − Z ∆u∆ −1 Ω 1 (u − v) − λ(f − u) − (λ0 − λ)(v − u) dx τ 1 = −ǫ h∆u, ∆ui2 + ǫ Z F ′ (u)∆u dx Ω After integration by parts we ﬁnd with the short-hand notation w := that Z 1 (u − v) − λ(f − u) − (λ0 − λ)(v − u) τ 1 ′ −1 −1 uw dx − ∇u · ν(∆ w + F (u)) − u∇(∆ w) · ν dH1 ǫ ∂Ω Ω Z 1 2 F ′′ (u) |∇u|2 dx. = −ǫ k∆uk2 − ǫ Ω Z Now we insert the boundary conditions ∆−1 w = 0, u = f =: f1 and F ′ (u) = F ′ (f ) = f2 on ∂Ω with constants f1 and f2 on the left-hand side, i.e. Z Z Z f2 1 −1 uw dx − F ′′ (u) |∇u|2 dx. ∇u · ν − f1 ∇(∆ w) · ν dH1 = −ǫ k∆uk22 − ǫ ǫ Ω ∂Ω Ω An application of Gauss’ Theorem to the boundary integral implies Z Z Z f2 f2 −1 w dx, ∆u dx + f1 ∇u · ν − f1 ∇(∆ w) · ν dH1 = ǫ ǫ Ω Ω ∂Ω and we get Z Z Z Z 1 f2 2 2 ′′ uw dx = −ǫ k∆uk2 − w dx. F (u) |∇u| dx + ∆u dx + f1 ǫ Ω ǫ Ω Ω Ω By further applying Young’s inequality to the last two terms we get Z Z f2 δ f1 δ 1 2 uw dx ≤ F ′′ (u) |∇u|2 dx + − ǫ k∆uk2 − kwk22 + C(f1 , f2 , |Ω|, ǫ, δ). 2ǫ ǫ 2 Ω Ω Using the identity λ(f − u) + (λ0 − λ)(v − u) = λ(f − v) + λ0 (v − u) in the deﬁnition of w yields Z Z f2 δ f1 δ 1 1 2 F ′′ (u) |∇u|2 dx + − ǫ k∆uk2 − kwk22 u · (u − v) dx ≤ τ 2ǫ ǫ Ω 2 Ω ! Z Z u(v − u) dx + C(f1 , f2 , |Ω|, ǫ, δ). + λ0 u(f − u) dx + D Ω\D 33 2.1 Cahn-Hilliard Inpainting By applying the standard inequality (a + b)2 ≤ 2(a2 + b2 ) to the L2 norm of w = ( τ1 + λ0 )u − ( τ1 + λ0 − λ)v − λf and by using (1 − λ/λ0 ) ≤ 1 in the resulting L2 norm of v we get 2 Z Z 1 1 f2 δ 1 2 2 ′′ u · (u − v) dx ≤ F (u) |∇u| dx + f1 δ − ǫ k∆uk2 − + λ0 kuk22 τ 2ǫ ǫ Ω τ Ω ! 2 Z Z 1 + λ0 kvk22 + λ0 u(v − u) dx u(f − u) dx + + 2f1 δ τ D Ω\D + C(f, f1 , f2 , |Ω|, ǫ, δ, λ0 ). With F ′′ (u) ≥ C1 u2 − C2 for some constants C1 , C2 > 0 and for all u ∈ R, and by further applying the Cauchy-Schwarz inequality to the last two integrals we obtain f2 δ C1 C2 1 − ǫ k∆uk22 − ku |∇u|k22 + k∇uk22 u · (u − v) dx ≤ τ 2ǫ ǫ ǫ Ω " 2 2 Z 1 δ2 1 2 2 + f1 δ + λ0 kuk2 + 2f1 δ + λ0 kvk2 + λ0 − 1 − u2 dx τ τ 2 Ω\D Z Z 1 δ1 u2 dx + v 2 dx + C(f, f1 , f2 , |Ω|, |D|, ǫ, λ0 , δ, δ2 ). −1 + 2 2δ1 D D Z Setting δ2 = 1 and δ1 = 2 we see that f2 δ C1 C2 1 − ǫ k∆uk22 − ku |∇u|k22 + k∇uk22 u · (u − v) dx ≤ τ 2ǫ ǫ ǫ Ω " # 2 2 Z Z 1 1 1 1 + f1 δ u2 dx + v 2 dx + λ0 kuk22 + 2f1 δ + λ0 kvk22 + λ0 − τ τ 2 Ω\D 4 D Z + C(f, f1 , f2 , |Ω|, |D|, ǫ, δ, λ0 ). We follow the argument used in the proof of existence for (2.1) in [BEG07b] by observing the following property: A standard interpolation inequality for ∇u reads k∇uk22 ≤ δ3 k∆uk22 + C3 kuk22 . δ3 (2.16) The domain of integration in the second integral of the equation above can be taken to be smaller than Ω by taking a larger constant C3 . Further we use the L1 version of Poincare’s inequality applied to the function u2 . We recall this inequality in the following theorem. Theorem 2.1.5. (Poincare’s inequality in L1 ). Assume that Ω is a precompact open subset of the n-dimensional Euclidean space Rn having Lipschitz boundary (i.e., Ω is 34 2.1 Cahn-Hilliard Inpainting an open, bounded Lipschitz domain). Then there exists a constant C, depending only on Ω, such that, for every function u in the Sobolev space W 1,1 (Ω), ku − uΩ kL1 (Ω) ≤ Ck∇ukL1 (Ω) , R 1 where uΩ = |Ω| Ω u(y) dy is the average value of u over Ω, with |Ω| denoting the Lebesgue measure of the domain Ω. Then, assuming that u 6= 0 in Ω \ D, we choose the constant the size of D compared to Ω) large enough such that Z Z Z Z 2 2 2 2 u − uΩ dx ≤ C4 u dx ≤ u dx − C4 Ω Ω\D or in other words kuk22 Ω Ω ≤ C4 ∇u2 L1 (Ω) + C4 Z C4 (which depends on 2 ∇u dx, u2 dx. (2.17) Ω\D By Hölder’s inequality we also have 2 C5 α ∇u 1 . ≤ ku |∇u|k22 + L (Ω) 2 2α Putting the last three inequalities (2.16)-(2.18) together we obtain Z C3 C4 C5 C3 C4 C3 C4 α u2 dx + ku |∇u|k22 + . k∇uk22 ≤ δ3 k∆uk22 + 2δ3 δ3 2αδ3 Ω\D (2.18) We now use the last inequality to bound the gradient term in our estimates from above to get Z 1 f2 δ + 2C2 δ3 C2 C3 C4 α C1 u · (u − v) dx ≤ ( − ǫ) k∆uk22 + ( − ) ku |∇u|k22 τ 2ǫ 2δ3 ǫ ǫ Ω 2 1 C2 C3 C4 C4 λ0 + λ0 + − ) kuk22 + (f1 δ τ δ3 ǫ 2 2 ! λ0 1 + kvk22 + C(f, f1 , f2 , |Ω|, |D|, ǫ, δ, λ0 ). + 2f2 δ + λ0 4 τ (2.19) 2ǫ2 −f2 δ With δ3 < 2C2 and α, δ small enough, the ﬁrst two terms can be estimated from above by zero. Applying the Cauchy-Schwarz inequality on the left-hand side and rearranging the terms on both sides of the inequality we conclude that ! 2 1 C4 λ0 C2 C3 C4 1 + − f1 δ − λ0 − kuk22 2τ 2 τ δ3 ǫ 2 ! 1 1 λ0 kvk22 + C(f, f1 , f2 , |Ω|, |D|, ǫ, δ, λ0 ). ≤ + + λ0 + 2f2 δ 4 2τ τ 35 2.1 Cahn-Hilliard Inpainting Choosing δ small enough, C4 large enough, and λ0 ≥ CC4 ǫ13 , the solutions u and v fulﬁll kuk22 ≤ β kvk22 + C, (2.20) with β < 1 and a constant C independent of v. Hence u is bounded in L2 (Ω). To see that our regularity assumptions on u from the beginning of the proof are automatically fulﬁlled we consider (2.19) with appropriate constants δ3 , δ, and α as speciﬁed in the paragraph below (2.19). But now we only estimate the second term on the right side by zero and keep the ﬁrst term. By applying the Cauchy-Schwarz inequality and rearranging the terms as before we obtain ! 2 1 C2 C3 C4 1 f2 δ + 2C2 δ3 C4 λ0 kuk22 + (ǫ − + − f1 δ − λ0 − ) k∆uk22 2τ 2 τ δ3 ǫ 2ǫ 2 ! 1 1 λ0 kvk22 + C(f, f1 , f2 , |Ω|, |D|, ǫ, δ, λ0 ), + + λ0 + 2f2 δ ≤ 4 2τ τ 2 δ3 with the coeﬃcient ǫ − f2 δ+2C > 0 due to our choice of δ3 . Therefore not only the 2ǫ L2 - norm of u is uniformly bounded but also the L2 - norm of ∆u. By the standard interpolation inequality (2.16) the boundedness of u in H 1 (Ω) follows. From the last result we additionally get that the operator A is a compact map since A : L2 (Ω) → H 1 (Ω) ֒→֒→ L2 (Ω). Therefore K is a compact and convex subset of L2 (Ω). It remains to show that the operator A is continuous. Indeed if vk → v in L2 (Ω) then A(vk ) = uk is bounded in H 1 (Ω) for all k = 0, 1, 2, . . .. Thus, we can consider a weakly convergent subsequence ukj ⇀ u in H 1 (Ω). Because H 1 (Ω) ֒→֒→ Lq (Ω), 1 ≤ q < ∞ the sequence ukj converges also strongly to u in Lq (Ω). Hence, a weak solution A(vk ) = uk of (2.8) weakly converges to a weak solution u of 1 1 (−∆−1 )(u − v) = ǫ∆u − F ′ (u) − ∆−1 [λ(f − u) + (λ0 − λ)(v − u)] , τ ǫ where u is the weak limit of A(vk ) as k → ∞. Because the solution of (2.8) is unique provided τ ≤ Cǫ3 (cf. Proposition 2.1.3), u = A(v), and therefore A is continuous. Applying Schauder’s Theorem we have shown that the ﬁxed point operator A admits a ﬁxed point û in L2 (Ω) which fulﬁlls Z 1 ′ ∆−1 (λ(f − û)) ∇∆−1 φ · ν dH1 = 0 F (û), φ − hλ(f − û), φi−1 + hǫ∇û, ∇φi2 + ǫ ∂Ω 2 for all φ ∈ H01 (Ω). Because the solution of (2.8) is an element of H also the ﬁxed point û belongs to H. 36 2.1 Cahn-Hilliard Inpainting Following the arguments from the beginning of this section we conclude with the existence of a stationary solution for (2.1). By modifying the setting and the above proof in an appropriate way one can prove the existence of a stationary solution for (2.1) also under Neumann boundary conditions, i.e., ∇u · ν = ∇∆u · ν = 0, on ∂Ω. A corresponding reformulation of the problem is given in Subsection 2.1.3. 2.1.2 Numerical Results In this section numerical results for the Cahn-Hilliard inpainting approach (2.1) are presented. The numerical scheme used is discussed in detail in Section 4.1, and in particular in Section 4.1.2 (cf. also [SB09, BHS08]). In Figures 2.1-2.2 Cahn-Hilliard inpainting was applied to two diﬀerent binary images. In all of the examples we follow the procedure of [BEG07a], i.e., the inpainted image is computed in a twostep process. In the ﬁrst step Cahn-Hilliard inpainting is solved with a rather large value of ǫ, e.g., ǫ = 0.1, until the numerical scheme is close to steady state. In this step the level lines are continued into the missing domain. In a second step the result of the ﬁrst step is put as an initial condition into the scheme for a small ǫ, e.g., ǫ = 0.01, in order to sharpen the contours of the image contents. The reason for this two step procedure is twofold. First of all in [BEG07b] the authors give numerical evidence that the steady state of the modiﬁed Cahn-Hilliard equation (2.1) is not unique, i.e., it is dependent on the initial condition for the equation. As a consequence, computing the inpainted image by the application of Cahn-Hilliard inpainting with a small ǫ only, might not prolongate the level lines into the missing domain as desired. See also [BEG07b] for a bifurcation diagram based on the numerical computations of the authors. The second reason for solving Cahn-Hilliard inpainting in two steps is that it is computationally less expensive. Solving the above time-marching scheme for, e.g., ǫ = 0.1 is faster than solving it for ǫ = 0.01. This is because of a damping introduced by ǫ into the scheme, cf. Section 4.1.2 for details. 37 2.1 Cahn-Hilliard Inpainting Figure 2.1: Destroyed binary image and the solution of Cahn-Hilliard inpainting with switching ǫ value: u(1200) with ǫ = 0.1, u(2400) with ǫ = 0.01 Figure 2.2: Destroyed binary image and the solution of Cahn-Hilliard inpainting with λ0 = 109 and switching ǫ value: u(800) with ǫ = 0.8, u(1600) with ǫ = 0.01 2.1.3 Neumann Boundary Conditions and the Space H∂−1 (Ω) In this section we want to pose the Cahn-Hilliard inpainting problem with Neumann boundary conditions in a way such that the analysis from Section 2.1.1 can be carried out in a similar way. Namely we consider 1 ut = ∆(−ǫ∆u + F ′ (u)) + λ(f − u) in Ω, ǫ ∂∆u ∂u = =0 on ∂Ω, ∂ν ∂ν For the existence of a stationary solution of this equation we consider again a ﬁxed point approach similar to (2.8) in the case of Dirichlet boundary conditions, i.e., u−v 1 = ∆(−ǫ∆u + F ′ (u)) + λ(f − u) + (λ0 − λ)(v − u) in Ω, τ ǫ (2.21) 1 ′ ∂u ∂ ǫ∆u − ǫ F (u) = =0 on ∂Ω. ∂ν ∂ν To reformulate the above equation in terms of the operator ∆−1 with Neumann boundary conditions we ﬁrst have to introduce the space H∂−1 (Ω) in which the operator ∆−1 is now the inverse of −∆ with Neumann boundary conditions. 38 2.1 Cahn-Hilliard Inpainting Thus we deﬁne the non-standard Hilbert space n o H∂−1 (Ω) = F ∈ H 1 (Ω)∗ | hF, 1i(H 1 )∗ ,H 1 = 0 . Since Ω is bounded we know that 1 ∈ H 1 (Ω), hence H∂−1 (Ω) is well deﬁned. Before we deﬁne a norm and an inner product on H∂−1 (Ω) we have to deﬁne more spaces. Let Z 1 1 ψ dx = 0 , Hφ (Ω) = ψ ∈ H (Ω) : Ω with norm kukH 1 := k∇uk2 and inner product hu, viH 1 := h∇u, ∇vi2 . This is a Hilbert φ φ space and the norms k.k1 and k.kH 1 are equivalent (up to constants) on Hφ1 (Ω). Let φ (Hφ1 (Ω))∗ denote the dual of Hφ1 (Ω). We will use (Hφ1 (Ω))∗ to induce an inner product on H∂−1 (Ω). Given F ∈ (Hφ1 (Ω))∗ with associate u ∈ Hφ1 (Ω) (from the Riesz representation theorem) we have by deﬁnition hF, ψi(H 1 )∗ ,H 1 = hu, ψiH 1 = h∇u, ∇ψi2 φ φ φ ∀ψ ∈ Hφ1 (Ω). Let us now deﬁne a norm and an inner product on H∂−1 (Ω). Definition 2.1.6. o n H∂−1 (Ω) := F ∈ H 1 (Ω)∗ | hF, 1i(H 1 )∗ ,H 1 = 0 kF kH −1 := F | Hφ1 (H 1 )∗ ∂ φ hF1 , F2 iH −1 := h∇u1 , ∇u2 i2 , ∂ where F1 , F2 ∈ H∂−1 (Ω) and where u1 , u2 ∈ Hφ1 (Ω) are the associates of F1 | Hφ1 , F2 | Hφ1 ∈ (Hφ1 (Ω))∗ . At this point it is not entirely obvious that H∂−1 (Ω) is a Hilbert space. That this is the case though is veriﬁed in the following theorem. Theorem 2.1.7. 1. H∂−1 (Ω) is closed in (H 1 (Ω))∗ . 2. The norms k.kH −1 and k.k(H 1 )∗ are equivalent on H∂−1 (Ω). ∂ Theorem 2.1.7 can be easily checked just by the application of the deﬁnitions and the fact that the norms k.k1 and k.kH 1 are equivalent on Hφ1 (Ω). From point 1. of φ Theorem 2.1.7 we have that H∂−1 (Ω) is a Hilbert space w.r.t. the (H 1 (Ω))∗ norm and point 2. tells us that the norms k.kH −1 and k.k(H 1 )∗ are equivalent on H∂−1 (Ω). ∂ 39 2.1 Cahn-Hilliard Inpainting Therefore the norm in Deﬁnition 2.1.6 is well deﬁned and H∂−1 (Ω) is a Hilbert space w.r.t. k.kH −1 . ∂ In the following we want to characterize elements F ∈ H∂−1 (Ω). By the above deﬁnition we have for each F ∈ H∂−1 (Ω) that there exists a unique element u ∈ Hφ1 (Ω) such that hF, ψi(H 1 )∗ ,H 1 = We deﬁne Z Ω ∇u · ∇ψ dx, ∀ψ ∈ Hφ1 (Ω). (2.22) ∆−1 F := u (2.23) the unique solution to (2.22). Now let F ∈ L2 (Ω) and assume u ∈ H 2 (Ω). Set hF, ψi(H 1 )∗ ,H 1 := R Ω Fψ dx. Because L2 (Ω) ⊂ H∂−1 (Ω) an element F is also an element in H∂−1 (Ω). Thus there exists a unique element u ∈ Hφ1 (Ω) such that Z Ω (−∆u − F )ψ dx + Z ∂Ω ∇u · νψ dH1 = 0, ∀ψ ∈ Hφ1 (Ω). Since hF, 1i(H 1 )∗ ,H 1 = 0, we see that hF, ψ + Ki(H 1 )∗ ,H 1 = hF, ψi(H 1 )∗ ,H 1 for all con- stants K ∈ R and therefore (2.22) extends to all ψ ∈ H 1 (Ω). Therefore u ∈ Hφ1 (Ω) is the unique weak solution of the following problem: ( − ∆u − F = 0 in Ω ∇u · ν = 0 on ∂Ω. (2.24) Remark 2.1.8. With the above characterization of elements F ∈ H∂−1 (Ω) and the notation (2.23) for its associates the inner product and the norm in H∂−1 can be written as Z ∇∆−1 F1 · ∇∆−1 F2 dx, ∀F1 , F2 ∈ H∂−1 (Ω), hF1 , F2 iH −1 := ∂ Ω and norm kF kH −1 := ∂ sZ Ω (∇∆−1 F )2 dx. Throughout the rest of this section we will write the short forms h., .i−1 and k.k−1 for the inner product and the norm in H∂−1 (Ω) respectively. Note that this notation is only valid within this section. It is important to notice that in order to rewrite (2.21) in terms of ∆−1 we require the ”right hand side” of the equation, i.e., u−v τ 40 + λ(u − f ) + (λ0 − λ)(u − v) to be an 2.2 TV-H−1 Inpainting element of our new space H∂−1 (Ω) (cf. Deﬁnition 2.1.6). In other words the ”right hand side” has to have zero mean over Ω. Because we cannot guarantee this property for solutions of the ﬁxed point equation (2.21) we are going to modify the right hand side by subtracting its mean. Let 1 1 F + λ0 FΩ2 τ ΩZ 1 (u − v) dx FΩ1 = |Ω| Ω Z λ λ 1 2 (u − f ) + 1 − (u − v) dx, FΩ = |Ω| Ω λ0 λ0 FΩ = and consider instead of (2.21) the equation 1 u − v ′ −1 − λ(f − u) − (λ0 − λ)(v − u) − FΩ in Ω, ǫ∆u − F (u) = ∆ ǫ τ ∂u = 0 on ∂Ω, ∂ν where the second Neumann boundary condition ∂ (ǫ∆u− 1ǫ F ′ (u)) ∂ν = 0 on ∂Ω is included in the deﬁnition of ∆−1 . The functional of the corresponding variational formulation then reads ǫ J (u, v) = Z Ω 1 1 ǫ 2 (u − v) − FΩ1 2 |∇u| + F (u) dx + −1 2 ǫ 2τ 2 λ λ λ0 2 u− f − 1− + v − FΩ . 2 λ0 λ0 −1 With these deﬁnitions the proof for the existence of a stationary solution for the modiﬁed Cahn-Hilliard equation with Neumann boundary conditions can be carried out similarly to the proof in Section 2.1.1. Note that every solution of (2.1) fulﬁlls R R d u dx = dt Ω Ω λ(f − u) dx. This means that for a stationary solution û the integral R C Ω λ(f − u) dx = 0 for every constant C ∈ R (i.e., the “right hand side” has zero mean and therefore FΩ1 = FΩ2 = 0). 2.2 TV-H−1 Inpainting Let Ω ⊂ Rd , for d = 1, 2 be a bounded open set with Lipschitz boundary, and H = L2 (Ω). We recall that for u ∈ L1loc (Ω) Z 1 d u∇ · ϕ dx : ϕ ∈ Cc (Ω) , kϕk∞ ≤ 1 V (u, Ω) := sup Ω 41 (2.25) 2.2 TV-H−1 Inpainting is the variation of u and that u ∈ BV (Ω) (the space of functions of bounded variation, [AFP00, EG92]) if and only if V (u, Ω) < ∞, see [AFP00, Proposition 3.6]. In such a case, |D(u)|(Ω) = V (u, Ω), where |D(u)|(Ω) is the total variation (TV) of the ﬁnite Radon measure Du, the derivative of u in the sense of distributions. For more details about the total variation and the space of functions of bounded variation we refer to Appendix A.5. The minimization of energies with total variation constraints, i.e., n o min kT u − f k2H + 2α|D(u)|(Ω) , u∈BV (Ω) (2.26) for a given f ∈ H, where H is a suitable Hilbert space, e.g., H = L2 (Ω), and T ∈ L(H) is a linear bounded operator in H, traces back to the ﬁrst uses of such a functional model in noise removal in digital images as proposed by Rudin, Osher, and Fatemi [ROF92]. There the operator T is just the identity. Extensions to more general operators T and numerical methods for the minimization of the functional appeared later in several important contributions [CL97, DV97, AV97, Ve01, Ch04]. From these pioneering and very successful results, the scientiﬁc output related to total variation minimization and its applications in signal and image processing increased dramatically in the last decade. For brevity we do not list here all the possible directions and contributions. Now, after the result about a stationary solution for the Cahn-Hilliard inpainting approach (2.1) presented in Section 2.1, the second contribution of [BHS08] is to generalize the latter approach to grayvalue images. This is realized by using subgradients of the TV functional within the ﬂow, which leads to structure inpainting with smooth curvature of level sets. In the present section the results about this new inpainting approach shall be reported. We motivate this new approach by a Γ-convergence result for the Cahn-Hilliard energy. In fact we prove that the sequence of functionals for an appropriate timediscrete Cahn-Hilliard inpainting approach Γ-converges to a functional regularized with the total variation for binary arguments u = 1E , where E is some Borel measurable subset of Ω. This is stated in more detail in the following Theorem. Throughout this section we keep to the notation, and the deﬁnitions from Section 2.1, i.e., Ω ⊂ R2 is a bounded open set with Lipschitz boundary, D ⊂ Ω denotes the inpainting domain, and λ is the indicator function of Ω \ D multiplied by a constant λ0 ≫ 1 as deﬁned in (1.2). 42 2.2 TV-H−1 Inpainting Theorem 2.2.1. Let f, v ∈ L2 (Ω) be two given functions, and τ > 0 a positive parameter. Let further k·k−1 be the norm in H −1 (Ω), defined in more detail in the Appendix A.4, and ǫ > 0 a parameter. Then the sequence of functionals 2 Z λ 1 1 λ0 λ ǫ 2 2 ǫ u− f − 1− |∇u| + F (u) dx + ku − vk−1 + v J (u, v) = ǫ 2τ 2 λ0 λ0 Ω 2 −1 Γ-converges for ǫ → 0 in the topology of L1 (Ω) to where 2 λ λ λ0 1 2 u− f − 1− ku − vk−1 + J(u, v) = T V (u) + v , 2τ 2 λ0 λ0 −1 C |Du| (Ω) 0 T V (u) = +∞ if u = 1E for some Borel measurable subset E ⊂ Ω otherwise, 1E denoting the indicator function of E and C0 = 2 R1p 0 F (s) ds. Remark 2.2.2. Setting v = uk and a minimizer u of the functional Jǫ (u, v) to be u = uk+1 , the minimization of Jǫ can be seen as an iterative approach with stepsize τ to solve (2.1). R Remark 2.2.3. The Γ-convergence of the Ginzburg-Landau energy Ω (ǫ/2 |∇u|2 + 1/ǫF (u)) dx to the total variation of an indicator function for a subset of Ω goes back to Modica and Mortola [MM77a, MM77b]. Their result can also be understood in terms of a sharp interface limit. In other words, the limit of the energy of a phase field model (the Ginzburg-Landau energy), where black (0) and white (1) regions are separated by a transition layer of order ǫ, was shown to be a model that describes the motion of the sharp interface between these regions without transition layer, i.e., ǫ = 0. Similar results have been shown for the Cahn-Hilliard equation directly. Here, the corresponding sharp interface limit is known as the Mullins-Sekerka problem or Hele-Shaw problem, cf. [MS63, Pe89, ABC94, St96]. Now, by extending the validity of the total variation functional T V (u) from functions u = 0 or 1 to functions |u| ≤ 1 we receive an inpainting approach for grayvalue images rather than binary images. We shall call this new inpainting approach TVH−1 inpainting and deﬁne it in the following way: The inpainted image u of f ∈ L2 (Ω), shall evolve via ut = ∆p + λ(f − u), 43 p ∈ ∂T V (u), (2.27) 2.2 TV-H−1 Inpainting with T V (u) = ( |Du| (Ω) +∞ if |u(x)| ≤ 1 a.e. in Ω otherwise. (2.28) Here, ∂T V (u) denotes the subdiﬀerential of the functional T V (u) (cf. Appendix A.2 for the deﬁnition). The L∞ bound in the deﬁnition of the TV functional (2.28) is quite natural as we are only considering digital images u whose grayvalue can be scaled to [−1, 1]. It is further motivated by the Γ-convergence result of Theorem 2.2.1. Remark 2.2.4. Similar to the use of the functional Jǫ from Theorem 2.2.1 as an iterative approach to solve (2.1) (cf. Remark 2.2.2), the functional J from Theorem 2.2.1 might serve as an iterative process to solve (2.27). A similar form of the TV-H−1 inpainting approach already appeared in the context of decomposition and restoration of grayvalue images, see for example [VO03, OSV03, LV08]. Diﬀerent to the situation of TV-H−1 denoising / decomposition the TV-H−1 inpainting approach does not exhibit a variational formulation and hence, its analytical and numerical treatment is diﬀerent. Further, in Bertalmio at al. [BVSO03] an application of the model from [VO03] to image inpainting has been proposed. In contrast to the inpainting approach (2.27) the authors in [BVSO03] only use a diﬀerent form of the TV-H−1 approach for a decomposition of the image into cartoon and texture prior to the inpainting process, which is accomplished with the method presented in [BSCB00]. Using the same methodology as in the proof of Theorem 2.1.1 we obtain the following existence theorem, Theorem 2.2.5. Let f ∈ L2 (Ω). The stationary equation ∆p + λ(f − u) = 0, p ∈ ∂T V (u) (2.29) admits a solution u ∈ BV (Ω). We shall also give a characterization of elements in the subdiﬀerential ∂T V (u) for T V (u) deﬁned as in (2.28), i.e., T V (u) = |Du| (Ω) + χ1 (u), where ( 0 if |u| ≤ 1 a.e. in Ω χ1 (u) = +∞ otherwise. Namely, we shall prove the following theorem. 44 2.2 TV-H−1 Inpainting Theorem 2.2.6. Let p̃ ∈ ∂χ1 (u). An element p ∈ ∂T V (u) with |u(x)| ≤ 1 in Ω, fulfills the following set of equations ∇u p = −∇ · a.e. on supp ({|u| < 1}) |∇u| ∇u p = −∇ · + p̃, p̃ ≤ 0 a.e. on supp ({u = −1}) |∇u| ∇u + p̃, p̃ ≥ 0 a.e. on supp ({u = 1}) . p = −∇ · |∇u| For (2.27) we additionally give error estimates for the inpainting error and stability information in terms of the Bregman distance. Let ftrue be the original image and û a stationary solution of (2.27). In our considerations we use the symmetric Bregman distance deﬁned as (û, ftrue ) = hû − ftrue , p̂ − ξi2 , DTsymm V p̂ ∈ ∂T V (û), ξ ∈ ∂T V (ftrue ). We prove the following result Theorem 2.2.7. Let ftrue fulfill the so-called source condition, namely that there exists ξ ∈ ∂T V (ftrue ) such that ξ = ∆−1 q for a source element q ∈ H −1 (Ω), and û ∈ BV (Ω) be a stationary solution of (2.27) given by û = us + ud , where us is a smooth function and ud is a piecewise constant function. Then the inpainting error reads (û, ftrue ) + DTsymm V λ0 1 kû − ftrue k2−1 ≤ kξk21 + Cλ0 |D|(r−2)/r errinpaint , 2 λ0 with 2 < r < ∞, constant C > 0 and errinpaint := K1 + K2 |D| C (M (us ), β) + 2 R(ud ) , where K1 and K2 are appropriate positive constants, and C is a constant depending on the smoothness bound M (us ) for us and β that is determined from the shape of D. The error region R(ud ) is determined by the level lines of ud . Finally we present numerical results for the proposed TV-H−1 inpainting approach and brieﬂy explain the numerical implementation. Let us continue with the veriﬁcation of the stated results in Theorems 2.2.1-2.2.7. 45 2.2 TV-H−1 Inpainting 2.2.1 Γ-Convergence of the Cahn-Hilliard Energy In the following we want to motivate our new inpainting approach (2.27) by considering the Γ-limit for ǫ → 0 of an appropriate time-discrete Cahn-Hilliard inpainting approach, i.e., the Γ-limit of the functionals from our ﬁxed point approach in (2.11). More precisely we want to prove Theorem 2.2.1. Before starting our discussion let us recall the deﬁnition of Γ-convergence and its impact within the study of optimization problems. For more details on Γ-convergence we refer to [Ma93]. Definition 2.2.8. Let X = (X, d) be a metric space and (Fh ), h ∈ N be family of functions Fh : X → [0, +∞]. We say that (Fh ) Γ-converges to a function F : X → [0, +∞] on X as h → ∞ if ∀x ∈ X we have (i) for every sequence xh with d(xh , x) → 0 we have F (x) ≤ lim inf Fh (xh ); h (ii) there exists a sequence x¯h such that d(x¯h , x) → 0 and F (x) = lim Fh (x¯h ) h (or, equivalently, F (x) ≥ lim suph Fh (x¯h )). Then F is the Γ-limit of (Fh ) in X and we write: F (x) = Γ − limh Fh (x), x ∈ X. The formulation of the Γ-limit for ǫ → 0 is analogous by deﬁning a sequence ǫh with ǫh → 0 as h → ∞. The important property of Γ-convergent sequences of functions Fh is that their minima converge to minima of the Γ-limit F . In fact we have the following theorem Theorem 2.2.9. Let (Fh ) be like in Definition 2.2.8 and additionally equicoercive, that is there exists a compact set K ⊂ X (independent of h) such that inf {Fh (x)} = inf {Fh (x)}. x∈X x∈K If Fh Γ-converges on X to a function F we have min {F (x)} = lim inf {Fh (x)} . x∈X h x∈X After recalling these facts about Γ-convergence we continue this section with the proof of Theorem 2.2.1. 46 2.2 TV-H−1 Inpainting Proof of Theorem 2.2.1. Modica and Mortola have shown in [MM77a, MM77b] that the sequence of Cahn-Hilliard functionals Z ǫ 1 2 CH(u) = |∇u| + F (u) dx ǫ Ω 2 Γ-converges in the topology L1 (Ω) to C |Du| (Ω) if u = 1 for some Borel measurable subset E ⊂ Ω 0 E T V (u) = +∞ otherwise R1p as ǫ → 0, where C0 = 2 0 F (s) ds. (The space BV (Ω) and the total variation |Du| (Ω) are deﬁned in the Appendix in Section A.5.) Now, for a given function v ∈ L2 (Ω) the functional Jǫ from our ﬁxed point approach (2.8), i.e., 2 Z ǫ 1 λ 1 λ λ0 2 2 ǫ u − f − (1 − )v , J (u, v) = |∇u| + F (u) dx + ku − vk−1 + 2 ǫ 2τ {z λ0 λ0 −1 } |2 |Ω {z } | {z } :=CH(u) :=D(u,v) :=F IT (u,v) is the sum of the regularizing term CH(u), the damping term D(u, v) and the ﬁtting term F IT (u, v). We recall the following fact, Theorem 2.2.10. [Ma93, Dal Maso, Proposition 6.21.] Let G : X → R be a continuous function and (Fh ) Γ-converges to F in X, then (Fh + G) Γ-converges to F + G in X. Since the H −1 -norm is continuous in H −1 (Ω) and hence in particular in L1 (Ω), the two terms in Jǫ that are independent from ǫ, i.e., D(u, v) and F IT (u, v), are continuous in L1 (Ω). Together with the Γ-convergence result of Modica and Mortola for the CahnHilliard energy, we have proven that the modiﬁed Cahn-Hilliard functional Jǫ can be seen as a regularized approximation in the sense of Γ-convergence of the TV-functional J(u, v) = T V (u) + D(u, v) + F IT (u, v), for functions u ∈ BV (Ω) with u(x) = 1E for a Borel measurable subset E ⊂ Ω. In fact we have gone from a smooth transition layer between 0 and 1 in the Cahn-Hilliard inpainting approach (depending on the size of ǫ) to a sharp interface limit in which the image function now jumps from 0 to 1. This sharp interface limit motivates the extension of J(u, v) to grayvalue functions such that |u| ≤ 1 on Ω and hence leads us from the Cahn-Hilliard inpainting approach for binary images to a generalization for grayvalue images, namely our so-called TVH−1 inpainting method (2.27)-(2.28). 47 2.2 TV-H−1 Inpainting 2.2.2 Existence of a Stationary Solution Our strategy for proving the existence of a stationary solution for TV-H−1 inpainting (2.27) is similar to our existence proof for a stationary solution of the modiﬁed CahnHilliard equation (2.1) in Section 2.1.1. Similarly as in our analysis for (2.1) in Section 2.1.1, we consider equation (2.27) with Dirichlet boundary conditions, namely ut = ∆p + λ(f − u) in Ω u=f on ∂Ω, for p ∈ ∂T V (u). Now let f ∈ L2 (Ω), |f | ≤ 1 be the given grayvalue image. For v ∈ Lr (Ω), 1 < r < 2, we consider the minimization problem u∗ = arg min u∈BV (Ω) J(u, v), with functionals J(u, v) := T V (u) + 1 λ0 λ λ ||u − v||2−1 + ||u − f − (1 − )v||2−1 , λ0 λ0 |2τ {z } |2 {z } D(u,v) (2.30) F IT (u,v) with T V (u) deﬁned as in (2.28). Note that Lr (Ω) can be continuously embedded in H −1 (Ω). Hence the functionals in (2.30) are well deﬁned. First we will show that for a given v ∈ Lr (Ω) the functional J(., v) attains a unique minimizer u∗ ∈ BV (Ω) with |u∗ (x)| ≤ 1 a.e. in Ω. Proposition 2.2.11. Let f ∈ L2 (Ω) be given with |f (x)| ≤ 1 a.e. in Ω and v ∈ Lr (Ω). Then the functional J(., v) has a unique minimizer u∗ ∈ BV (Ω) with |u∗ (x)| ≤ 1 a.e. in Ω. Proof. Let (un )n∈N be a minimizing sequence for J(u, v), i.e., n→∞ J(un , v) → inf u∈BV (Ω) J(u, v). Then un ∈ BV (Ω) and |un (x)| ≤ 1 in Ω (because otherwise T V (un ) would not be ﬁnite). Therefore |Dun | (Ω) ≤ M, for an M ≥ 0 and for all n ≥ 1, 48 2.2 TV-H−1 Inpainting and, because of the uniform boundedness of |u(x)| for every point x ∈ Ω, kun kLp (Ω) ≤ M̃ , for an M̃ ≥ 0, ∀n ≥ 1, and 1 ≤ p ≤ ∞. Thus un is uniformly bounded in Lp (Ω) and in particular in L1 (Ω). Together with the boundedness of |Dun | (Ω), the sequence un is also bounded in BV (Ω) and there exists a subsequence, still denoted un , and a u ∈ BV (Ω) such that un ⇀ u weakly in Lp (Ω), 1 ≤ p ≤ ∞ and weakly∗ in BV (Ω). Because L2 (Ω) ⊂ L2 (R2 ) ⊂ H −1 (Ω) (by zero extensions of functions on Ω to R2 ) un ⇀ u also weakly in H −1 (Ω). Because |Du| (Ω) is lower semicontinuous in BV (Ω) and by the lower semicontinuity of the H −1 norm we get J(u, v) = T V (u) + D(u, v) + F IT (u, v) ≤ lim inf n→∞ (T V (un ) + D(un , v) + F IT (un , v)) = lim inf n→∞ J(un , v). So u is a minimizer of J(u, v) over BV (Ω). To prove the uniqueness of the minimizer we (similarly as in the proof of Theorem 2.1.3) show that J is strictly convex. Namely we prove that for all u1 , u2 ∈ BV (Ω), u1 6= u2 u1 + u2 , v > 0. J(u1 , v) + J(u2 , v) − 2J 2 We have J(u1 , v) + J(u2 , v) − 2J u1 + u2 ,v 2 2 ! u + u 1 2 ku1 k2−1 + ku2 k2−1 − 2 = 2 −1 u1 + u2 +T V (u1 ) + T V (u2 ) − 2T V 2 1 λ0 + ≥ ku1 − u2 k2−1 > 0. 4τ 4 λ0 1 + 2τ 2 This ﬁnishes the proof. Next we shall prove the existence of stationary solution for (2.27). For this end we consider the corresponding Euler-Lagrange equation to (2.30), i.e., −1 u − v + p − ∆−1 (λ(f − u) + (λ0 − λ)(v − u)) = 0, ∆ τ 49 2.2 TV-H−1 Inpainting with weak formulation 1 + hp, φi2 − hλ(f − u) + (λ0 − λ)(v − u), φi−1 = 0 ∀φ ∈ H01 (Ω). (u − v), φ τ −1 A ﬁxed point of the above equation, i.e., a solution u = v, is then a stationary solution for (2.27). Thus, to prove the existence of a stationary solution of (2.27), i.e., to prove Theorem 2.2.5, as before we are going to use a ﬁxed point argument. Let A : Lr (Ω) → Lr (Ω), 1 < r < 2, be the operator which maps a given v ∈ Lr (Ω) to A(v) = u under the condition that A(v) = u is the minimizer of the functional J(., v) deﬁned in (2.30). The choice of the ﬁxed point operator A over Lr (Ω) was made in order to obtain the necessary compactness and continuity properties for the application of Schauder’s theorem. Since here the treatment of the boundary conditions is similar to that in Section 2.1.1, we will leave this part of the analysis in the upcoming proof to the reader and just carry out the proof without explicitly taking care of the boundary. Proof of Theorem 2.2.5. Let A : Lr (Ω) → Lr (Ω), 1 < r < 2, be the operator that maps a given v ∈ Lr (Ω) to A(v) = u, where u is the unique minimizer of the functional J(., v) deﬁned in (2.30). Existence and uniqueness u follow from Proposition 2.2.11. Since u minimizes J(., v) we have u ∈ L∞ (Ω) hence u ∈ Lr (Ω). Additionally we have J(u, v) ≤ J(0, v), i.e., 1 λ0 λ λ ||u − v||2−1 + ||u − f − (1 − )v||2−1 + T V (u) 2τ 2 λ0 λ0 1 λ0 λ λ |Ω| ≤ ||v||2−1 + || f + (1 − )v||2−1 ≤ + λ0 (|Ω| + |D|). 2τ 2 λ0 λ0 2τ Here the last inequality was obtained since Lr (Ω) ֒→ H −1 (Ω) and hence ||v||−1 ≤ C ′ and ||λv||−1 ≤ C for a C > 0. (In fact, since H 1 (Ω) ֒→ Lr (Ω) for all 1 ≤ r′ < ∞ it follows from duality that Lr (Ω) ֒→ H −1 (Ω) for, 1 < r < ∞.) By the last estimate we obtain u ∈ BV (Ω). Since BV (Ω) ֒→֒→ Lr (Ω) compactly for 1 ≤ r < 2 and Ω ⊂ R2 (cf. Theorem A.5.7), the operator A maps Lr (Ω) → BV (Ω) ֒→֒→ Lr (Ω), i.e., A : Lr (Ω) → K, where K is a compact subset of Lr (Ω). Thus, for v ∈ B(0, 1) (where B(0, 1) denotes the ball in L∞ (Ω) with center 0 and radius 1), the operator A : B(0, 1) → B(0, 1) ∩ K = K̃, where K̃ is a compact and convex subset of Lr (Ω). Next we have to show that A is continuous in Lr (Ω). Let (vk )k≥0 be a sequence which converges to v in Lr (Ω). Then uk = A(vk ) solves ∆pk = u k − vk − (λ(f − uk ) + (λ0 − λ)(vk − uk )) , τ 50 2.2 TV-H−1 Inpainting where pk ∈ ∂T V (uk ). Thus uk is uniformly bounded in BV (Ω) ∩ L∞ (Ω) (and hence in Lr (Ω)) and, since the right-hand side of the above equation is uniformly bounded in Lr (Ω), also ∆pk is bounded in Lr (Ω). Thus there exists a subsequence pkl such that ∆pkl ⇀ ∆p in Lr (Ω) and a subsequence ukl that converges weakly ∗ to a u in BV (Ω) ∩ L∞ (Ω). Since BV (Ω) ֒→֒→ Lr (Ω) we have ukl → u strongly in Lr (Ω). Therefore the limit u solves ∆p = u−v − (λ(f − u) + (λ0 − λ)(v − u)) . τ (2.31) If we additionally apply Poincare’s inequality to ∆pk we conclude that k∇pk − (∇pk )Ω kLr (Ω) ≤ C k∇ · (∇pk − (∇pk )Ω )kLr (Ω) , R 1 where (∇pk )Ω = |Ω| Ω ∇pk dx. In addition, since pk ∈ ∂T V (uk ), it follows that (pk )Ω = 0 and kpk kBV ∗ (Ω) ≤ 1. Thus (∇pk )Ω < ∞ and pk is uniformly bounded in W 1,r (Ω). Thus there exists a subsequence pkl such that pkl ⇀ p in W 1,r (Ω). In ′ addition Lr (Ω) ֒→ BV ∗ (Ω) for 2 < r′ < ∞ (this follows again from Theorem A.5.7 2r (Rellich-Kondrachov by a duality argument) and W 1,r (Ω) ֒→֒→ Lq (Ω) for 1 ≤ q < 2−r 2r Compactness Theorem, cf. Theorem A.3.1 in Appendix A.3). By choosing 2 < q < 2−r we therefore have W 1,r (Ω) ֒→֒→ BV ∗ (Ω). Thus pkl → p strongly in BV ∗ (Ω). Hence the element p in (2.31) is an element in ∂T V (u). Because the minimizer of (2.30) is unique, u = A(v), and therefore A is continuous in Lr (Ω). The existence of a stationary solution follows from Schauder’s ﬁxed point theorem. 2.2.3 Characterization of Solutions Finally we want to compute elements p ∈ ∂T V (u). In particular we shall prove Theorem 2.2.6. Like in [BRH07] the model for the regularizing functional is the sum of a standard regularizer plus the indicator function of the L∞ constraint. In particular, we have T V (u) = |Du| (Ω) + χ1 (u), where |Du| (Ω) is the total variation of Du and ( 0 if |u| ≤ 1 a.e. in Ω χ1 (u) = +∞ otherwise. (2.32) We want to compute the subgradients of T V by assuming ∂T V (u) = ∂ |Du| (Ω) + ∂χ1 (u). This means that we can separately compute the subgradients of χ1 . To guar- antee that the splitting above is allowed we have to consider a regularized functional of 51 2.2 TV-H−1 Inpainting R q the total variation, like Ω |∇u|2 + δ dx. This is suﬃcient because both |D.| (Ω) and χ1 are convex and |D.| (Ω) is continuous (compare [ET76, Proposition 5.6., pp. 26]). The subgradient ∂ |Du| (Ω) is already well described, as, for instance, in [AK06, Ve01]. We will just brieﬂy recall its characterization. Thereby we do not insist on the details of the rigorous derivation of these conditions, and we limit ourself to mentioning the main facts. It is well known [Ve01, Proposition 4.1] that p ∈ ∂|Du|(Ω) implies ∇u in Ω p = −∇ · |∇u| ∇u ·ν =0 on ∂Ω. |∇u| The previous conditions do not fully characterize p ∈ ∂|Du|(Ω), additional conditions would be required [AK06, Ve01], but the latter are, unfortunately, hardly numerically implementable. Since we anyway consider a regularized version of |Du| (Ω), the subdiﬀerential becomes a gradient which reads p = −∇ · ∇u p |∇u|2 + δ ! ∇u ·ν =0 p |∇u|2 + δ in Ω on ∂Ω. The subgradient of χ1 is computed like in the following Lemma. Lemma 2.2.12. Let χ1 : Lr (Ω) → R ∪ {∞} be defined by (2.32), and let 1 ≤ r ≤ ∞. ∗ r , is a subgradient p ∈ ∂χ1 (u) for u ∈ Lr (Ω) with Then p ∈ Lr (Ω), for r∗ = r−1 χ1 (u) = 0, if and only if p=0 a.e. on supp({|u| < 1}) p≤0 a.e. on supp({u = −1}) p≥0 a.e. on supp({u = 1}). Proof. Let p ∈ ∂χ1 (u). Then we can choose v = u + ǫw for w being any bounded function supported in {|u| < 1 − α} for arbitrary 0 < α < 1. If ǫ is suﬃciently small we have |v| ≤ 1. Hence Z wp dx. 0 ≥ hv − u, pi2 = ǫ {|u|<1−α} 52 2.2 TV-H−1 Inpainting Since we can choose ǫ either positive or negative, we obtain Z wp dx = 0. {|u|<1−α} Because 0 < α < 1 and w are arbitrary we conclude that p = 0 on the support of {|u| < 1}. If we choose v = u + w with w an arbitrary bounded function with 0 ≤ w ≤ 1 on supp({−1 ≤ u ≤ 0}) w = 0 on supp({0 < u ≤ 1}), then v is still between −1 and 1 and Z Z wp dx + 0 ≥ hv − u, pi2 = {u=−1} wp dx = Z wp dx. {u=−1} {u=1} Because w is arbitrary and positive on {u = −1} it follows that p ≤ 0 a.e. on {u = −1}. If we choose now v = u + w with w is an arbitrary bounded function with w = 0 on supp({−1 ≤ u ≤ 0}) −1 ≤ w ≤ 0 on supp({0 < u ≤ 1}), then v is still between −1 and 1 and Z Z wp dx + 0 ≥ hv − u, pi2 = {u=−1} wp dx = Z wp dx. {u=1} {u=1} Analogously to before, since w is arbitrary and negative on {u = 1} it follows that p ≥ 0 a.e. on {u = 1}. On the other hand assume that p=0 a.e. on supp({|u| < 1}) p≤0 a.e. on supp({u = −1}) p≥0 a.e. on supp({u = 1}). We need to verify the subgradient property hv − u, pi2 ≤ χ1 (v) − χ1 (u) = χ1 (v) for all v ∈ Lr (Ω) only for χ1 (v) = 0, since it is trivial for χ1 (v) = ∞. So let v ∈ Lr (Ω) be a function between −1 and 1 almost everywhere on Ω. Then with p as above we obtain Z Z p(v − u) dx p(v − u) dx + hv − u, pi2 = {u=1} {u=−1} Z Z p(v − 1) dx. p(v + 1) dx + = {u=1} {u=−1} Since −1 ≤ v ≤ 1 the ﬁrst and the second term are always ≤ 0 since p ≤ 0 for {u = −1} and p ≥ 0 for {u = 1} respectively. Therefore hv − u, pi2 ≤ 0 and we are done. 53 2.2 TV-H−1 Inpainting 2.2.4 Error Estimation and Stability Analysis With the Bregman Distance In the following analysis we want to present estimates for both the error we actually make in inpainting an image with our TV-H−1 approach (2.27) (see (2.40)) and for the stability of solutions for this problem (see (2.41)) in terms of the Bregman distance. This section is motivated by the error analysis for variational models in image restoration with Bregman iterations in [BRH07], and the error estimates for inpainting models developed in [CK06]. In [BRH07] the authors consider among other things the general optimality condition p + λ0 A∗ (Au − fdam ) = 0, (2.33) where p ∈ ∂R(u) for a regularizing term R, A is a bounded linear operator and A∗ its adjoint. Now the error that is to be estimated depends on the form of smoothing of the image contained in (2.33). Considering this equation one realizes that smoothing consists of two steps. The ﬁrst is created by the operator A which depends on the image restoration task at hand, and actually smooths the subgradient p. The second smoothing step is the one which is directly implied by the regularizing term, i.e., its subgradient p, and depends on the relationship between the primal variable u and the dual variable p. A condition that represents this dual smoothing property of functions, i.e., subgradients, is the so-called source condition. Letting ftrue be the original image, the source condition for ftrue reads There exists ξ ∈ ∂R(ftrue ) such that ξ = A∗ q for a source element q ∈ D(A∗ ), (2.34) where D(A∗ ) is the domain of the operator A∗ . It can be shown (cf. [BO05]) that this is equivalent to requiring from ftrue to be a minimizer of R(u) + λ0 kAu − fdam k22 2 for arbitrary fdam ∈ D(A∗ ) and λ0 ∈ R. Now, the source condition has a direct consequence for the Bregman distance, which gives rise to its use for subsequent error analysis. To be more precise, the Bregman distance is deﬁned as p (v, u) = R(v) − R(u) − hv − u, pi2 , DR p ∈ ∂R(u). Then, if ftrue fulﬁlls the source condition with a particular subgradient ξ, we obtain ξ DR (u, ftrue ) = R(u) − R(ftrue ) − hu − ftrue , ξi2 = R(u) − R(ftrue ) − hq, Au − Aftrue i2 , 54 2.2 TV-H−1 Inpainting and thus the Bregman distance can be related to both the error in the regularization functional (R(u) − R(ftrue )) and the output error (Au − Aftrue ). For the sake of symmetry properties in the sequel we shall consider the symmetric Bregman distance, which is deﬁned as p2 p1 symm (u1 , u2 ) = hu1 − u2 , p1 − p2 i2 , (u2 , u1 ) + DR (u1 , u2 ) = DR DR pi ∈ ∂R(ui ). Additionally to this error analysis we shall get a control on the inpainting error |u − ftrue | inside the inpainting domain D by means of estimates from [CK06]. Therein the authors analyzed the inpainting process by understanding how the regularizer continues level lines into the missing domain. The inpainting error was then determined by means of the deﬁnition of an error region, smoothness bounds on the level lines, and quantities taking into the account the shape of the inpainting domain. In the following we are going to implement both strategies, i.e., [BRH07, CK06], in order to prove Theorem 2.2.7. Proof of Theorem 2.2.7. Let fdam ∈ L2 (Ω) be the given damaged image with inpainting domain D ⊂ Ω and ftrue the original image. We consider the stationary equation to (2.27), i.e., − ∆p + λ(u − fdam ) = 0, p ∈ ∂T V (u), (2.35) where T V (u) is deﬁned as in (2.28). More precisely T V (u) is interpreted as a functional over L2 (Ω), |Du| (Ω) if u ∈ BV (Ω), kuk ∞ ≤ 1 L T V (u) = +∞ otherwise. In the subsequent we want to characterize the error we make by solving (2.35) for u, i.e., how large do we expect the distance between the restored image u and the original image ftrue to be. Now, let ∆−1 be the inverse operator to −∆ with zero Dirichlet boundary conditions as before. In our case the operator A in (2.33) is the embedding operator from H01 (Ω) into H −1 (Ω) and stands in front of the whole term A(u − fdam ), cf. (2.35). The adjoint operator is A∗ = ∆−1 which maps H −1 (Ω) into H01 (Ω). We assume that the given image fdam coincides with ftrue outside of the inpainting domain, i.e., fdam = ftrue in Ω \ D fdam = 0 in D. 55 (2.36) 2.2 TV-H−1 Inpainting Further we assume that ftrue satisﬁes the source condition (2.34), i.e., There exists ξ ∈ ∂T V (ftrue ) such that ξ = ∆−1 q for a source element q ∈ H −1 (Ω). (2.37) For the following analysis we ﬁrst rewrite (2.35). For û, a solution of (2.35), we get p̂ + λ0 ∆−1 (û − ftrue ) = ∆−1 [(λ0 − λ)(û − ftrue )] , p̂ ∈ ∂T V (û). Here we replaced fdam by ftrue using assumption (2.36). By adding a ξ ∈ ∂T V (ftrue ) from (2.37) to the above equation we obtain λ −1 −1 (û − ftrue ) p̂ − ξ + λ0 ∆ (û − ftrue ) = −ξ + λ0 ∆ 1− λ0 Taking the duality product with û − ftrue (which is just the inner product in L2 (Ω) in our case) we get (û, ftrue ) + λ0 kû − ftrue k2−1 = ∇ξ, ∇∆−1 (û − ftrue ) 2 DTsymm V λ , (û − ftrue ), û − ftrue 1− + λ0 λ0 −1 where DTsymm (û, ftrue ) = hû − ftrue , p̂ − ξi2 , V p̂ ∈ ∂T V (û), ξ ∈ ∂T V (ftrue ). An application of Young’s inequality yields DTsymm (û, ftrue ) V 2 λ 1 λ0 2 2 kû − ftrue k−1 ≤ kξk1 + λ0 1 − + (2.38) (û − ftrue ) 2 λ0 λ0 −1 For the last term we obtain λ 1 − λ v = sup v φ, 1 − λ0 λ0 φ,kφk−1 =1 −1 −1 λ −1 = sup − ∆ φ, 1 − v λ0 φ,kφk−1 =1 2 λ = sup − 1− ∆−1 φ, v λ0 φ,kφk−1 =1 2 λ −1 ≤Hölder kvk2 · sup 1 − λ0 ∆ φ . φ,kφk−1 =1 2 56 2.2 TV-H−1 Inpainting With ∆−1 : H −1 → H 1 ֒→ Lr , 2 < r < ∞ we get Z Ω λ 1− λ0 ∆ −1 2 Z dx = φ D =choose i.e. q= 2r |D| r−2 r ∆ −1 1 2 φ dx ≤Hölder |D| q′ · Z ∆ −1 Ω 2q φ 1 q 2 r−2 r−2 · ∆−1 φr ≤H 1 ֒→Lr C|D| r kφk2−1 = C|D| r , 2 2 1 − λ v ≤ C|D| r−2 r kvk . 2 λ0 −1 (2.39) Applying (2.39) to (2.38) we see that (û, ftrue ) + DTsymm V λ0 1 kû − ftrue k2−1 ≤ kξk21 + Cλ0 |D|(r−2)/r kû − ftrue k22 . 2 λ0 To estimate the last term we use some error estimates for TV inpainting computed in [CK06]. First we have Z Z 2 2 (û − ftrue ) dx + (û − ftrue )2 dx. kû − ftrue k2 = D Ω\D Since û − ftrue is uniformly bounded in Ω (this follows from the L∞ bound in the deﬁnition of T V (u)) we estimate the ﬁrst term by a positive constant K1 and the second term by the L1 norm over D. We obtain Z |û − ftrue | dx. kû − ftrue k22 ≤ K1 + K2 D Now let û ∈ BV (Ω) be given by û = us + ud , where us is a smooth function and ud is a piecewise constant function. Following the error analysis in [CK06, Theorem 8.] for functions û ∈ BV (Ω) we have kû − ftrue k22 ≤ K1 + K2 err(D) ≤ K1 + K2 |D| C (M (us ), β) + 2 R(ud ) , where M (us ) is the smoothness bound for us , β is determined from the shape of D, and the error region R(ud ) is deﬁned from the level lines of ud . Note that in general the error region from higher-order inpainting models including the TV seminorm is smaller than that from TV-L2 inpainting (cf. [CK06, Section 3.2.]). Finally we end up with DTsymm (û, ftrue ) + V 1 λ0 kû − ftrue k2−1 ≤ kξk21 + Cλ0 |D|(r−2)/r errinpaint , 2 λ0 57 (2.40) 2.2 TV-H−1 Inpainting with errinpaint := K1 + K2 |D| C (M (us ), β) + 2 R(ud ) . The ﬁrst term in (2.40) depends on the regularizer T V , and the second term on the size of the inpainting domain D. Remark 2.2.13. From inequality (2.40) we derive an optimal scaling for λ0 , i.e., a scaling which minimizes the inpainting error. It reads λ20 |D| r−2 r ∼1 and therefore λ0 ∼ |D|− r−2 2r . p In two space dimensions r can be chosen arbitrarily big, which gives λ0 ∼ 1/ |D| as the optimal order for λ0 . Stability estimates for (2.35) can also be derived with an analogous technique. For ui being the solution of (2.35) with fdam = fi (again assuming that fi = ftrue in Ω \ D), the estimate DJsymm (u1 , u2 ) λ0 λ0 ku1 − u2 k2−1 ≤ + 2 2 Z D (f1 − f2 )2 dx (2.41) holds. 2.2.5 Numerical Results In this section numerical results for the TV-H−1 inpainting approach (2.27) are presented. The numerical scheme used is discussed in detail in Section 4.1, and in particular in Section 4.1.3 (see also [SB09, BHS08]). In Figures 2.3–2.6 examples for the application of TV-H−1 inpainting to grayvalue images are shown. q In all examples the total variation |∇u| is approximated by its regularized version |∇u|2 + δ with δ = 0.01 and the time stepsize τ is chosen to be equal to 1. In Figure 2.4 a comparison of the TV-H−1 inpainting result with the re- sult obtained by the second-order TV-L2 inpainting model for a crop of the image in Figure 2.3 is presented. The superiority of the fourth-order TV-H−1 inpainting model to the second-order model with respect to the desired continuation of edges into the missing domain is clearly visible. Other examples which support this claim are presented in Figures 2.5 and 2.6 where the line is connected by the TV-H−1 inpainting 58 2.2 TV-H−1 Inpainting model but clearly split by the second-order TV-L2 model. It would be interesting to strengthen this numerical observation with a rigorous result as it was done in [BEG07b] for Cahn-Hilliard inpainting, cf. (2.2). The author considers this as another important contribution of future research. Figure 2.3: TV-H−1 inpainting: u(1000) with λ0 = 103 Figure 2.4: (l.) u(1000) with TV-H−1 inpainting, (r.) u(5000) with TV-L2 inpainting Figure 2.5: TV-H−1 inpainting compared to TV-L2 inpainting: u(5000) with λ0 = 10 59 2.3 Inpainting with LCIS Figure 2.6: TV-H−1 inpainting compared to TV-L2 inpainting: u(5000) with λ0 = 10 2.3 Inpainting with LCIS Another higher-order inpainting model proposed in [SB09] is inpainting with LCIS (low curvature image simpliﬁers). This approach is motivated by two famous second-order nonlinear PDEs in image processing, the work of Rudin, Osher and Fatemi [ROF92] and Perona & Malik [PM90]. These methods are based on a nonlinear version of the heat equation ut = ∇ · (g(|∇u|)∇u), in which g is small in regions of sharp gradients. LCIS represent a fourth-order relative of these nonlinear second-order approaches. They have been proposed in [TT99] and later used by Bertozzi and Greer in [BG04] for the denoising of piecewise linear signals. Related fourth-order equations, combining diﬀusion and convection, have been studied by the latter authors in [GB04a, GB04b]. In [SB09] we consider LCIS for image inpainting. With f ∈ L2 (Ω) our inpainted image u evolves in time as ut = −∇ · (g(∆u)∇∆u) + λ(f − u), with thresholding function g(s) = 1 1+s2 and λ is deﬁned as the indicator function of Ω \ D multiplied by a constant λ0 ≫ 1, as in the previous two sections. Note that with 60 2.3 Inpainting with LCIS g(∆u)∇∆u = ∇(arctan(∆u)) the above equation can be rewritten as ut = −∆(arctan(∆u)) + λ(f − u). (2.42) Using the second derivative of the arctangent as a regularizing term, spikes in the initial signal are preserved versus step functions are diﬀused (cf. [BG04]). This means that solutions of this equation are piecewise linear functions and not piecewise constant functions as the ones produced by the TV model [ROF92]. Introducing a regularizing parameter 0 < δ ≤ 1 in G′ (y) = arctan(y/δ) we are able to control the smoothing eﬀect. Namely, the smaller δ the less diﬀusive the equation will be. In [BG04] the authors prove regularity of solutions of (2.42) for the case λ(x) ≡ λ0 in all of Ω. Namely they show that for smooth initial data u(., t = 0) = u0 and smooth data f a unique smooth solution exists, globally in time in one space dimension and locally in time in two dimensions. For numerical purposes it is suﬃcient to have wellposedness and certain regularity properties of (2.42) on a ﬁnite time interval. Although the diﬀerence between the equation in [BG04] and our inpainting equation (2.42) is that the ﬁdelity term is discontinuous, the results in [BG04] suggest its validity also in this case. Nevertheless a rigorous analysis of the LCIS inpainting equation is still open for future research. 2.3.1 Numerical Results In this section numerical results for the LCIS inpainting approach (2.42) are presented. The numerical scheme used is discussed in detail in Section 4.1, and in particular in Section 4.1.4 (cf. also [SB09]). For the comparison with TV-H−1 inpainting we apply (2.42) to the same image as in Section 2.2.5. This example is presented in Figure 2.7. In Figure 2.8 the LCIS inpainting result is compared with TV-H−1 - and TV-L2 inpainting, for a small part in the given image. Again the result of this comparison indicates the continuation of the gradient of the image function into the inpainting domain. A rigorous proof of this observation is a matter of future research. 61 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison Figure 2.7: LCIS inpainting u(500) with δ = 0.1 and λ0 = 102 . Figure 2.8: (l.) u(1000) with LCIS inpainting, (m.) u(1000) with TV-H−1 inpainting, (r.) u(5000) with TV-L2 inpainting 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison In [CKS02] the authors gave an interpretation of Euler’s elastica inpainting in terms of the mechanisms of transport and diﬀusion. More precisely, they derived the optimality condition for elastica minimizing curves, and compared it with the transport equation of Bertalmio et al. [BSCB00] and the CDD inpainting approach of Chan and Shen [CS01c]. Thereby the optimality condition for elastica minimizing curves, or preferably the corresponding ﬂux ﬁeld, shows a natural decomposition into its normal and tangent ﬁeld, cf. Theorem 2.4.1. In this section we want to make a similar analysis for TV-H−1 inpainting and inpainting with LCIS, presented in Section 2.2 and Section 2.3 respectively. To do so, let us brieﬂy recall the results from [CKS02]. 62 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison Euler’s Elastica Inpainting For the following statement we assume that the image u is smooth enough and that the curvature κ is well deﬁned as κ=∇· ∇u |∇u| . Then the authors of [CKS02] proved the following theorem. Theorem 2.4.1. [CKS02, Theorem 5.1.] Let φ ∈ C 1 (R, (0, ∞)) be given and Z φ(κ)|∇u| dx. R(u) = Ω Then the gradient descent time marching is given by ∂u(x, t) ~ (x, t), =∇·V ∂t x ∈ Ω, t > 0, with the boundary conditions along ∂Ω ∂u = 0, ∂ν ∂(φ′ (κ)|∇u|) = 0, ∂ν ~ is given by where ν denotes the outward pointing normal on ∂Ω. The flux field V ~ = φ(κ)~n − V ~t ∂(φ′ (κ)|∇u|) . |∇u| ∂~t Here ~n is the ascending normal field ∇u/|∇u|, and ~t is the tangent field (whose exact orientation does not matter due to the parity of ~t in the expression). In particular, for the elastica inpainting model with Z R(u) = (a + bκ2 )|∇u| dx Ω Theorem 2.4.1 gives the following. Corollary 2.4.2. [CKS02, Corollary 5.3.] For the elastica inpainting model, the gradient descent is given by ∂u(x, t) ~ (x, t) − λ(u(x) − u0 (x)), =∇·V ∂t x ∈ Ω, t > 0, with the boundary conditions as in Theorem 2.4.1 and ~ = (a + bκ2 )~n − 2b ∂(κ|∇u|) ~t. V |∇u| ∂~t 63 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison To see the connection to the inpainting models of Bertalmio et al. [BSCB00] and Chan and Shen [CS01c], let us recall them brieﬂy. These two inpainting models have been already presented in the introduction of this work, i.e., in Section 1.3.1. Bertalmio’s approach is based on solving the following transport equation inside of the inpainting domain D, ∂u = ∇⊥ u · ∇L(u), ∂t where ∇⊥ = (−uy , ux ) = |∇u|~t, and L(u) can be any smoothness measure of the image u. For their numerical experiments the authors in [BSCB00] chose L(u) = ∆u, the Laplacian of u. For the equilibrium state the equation reads ~t · ∇L(u) = 0, i.e., ∂L(u) = 0, ∂~t which means that the smoothness measure remains constant along any completed level line. In other words, assuming available boundary data, boundary smoothness gets transported along the level lines into the missing domain. Orthogonal to this idea of smoothness transport along level lines, Chan and Shen proposed the CDD inpainting model in [CS01c], i.e., ∂u g(κ) =∇·( ∇u), ∂t |∇u| where g : B → [0, +∞) is a continuous function with g(0) = 0 and g(±∞) = +∞, and B equals, e.g., C 2 (Ω). The function g penalizes large curvatures, and encourages diﬀusion when the curvature is small. This model diﬀuses the image contents across the level lines (since ∇u/|∇u| is the normal vector to the level lines!), which is completely orthogonal to the behavior of the Bertalmio et al. approach. Now, in [CKS02] the authors showed that the Euler elastica inpainting model uniﬁes ~ for the inpainting energy these two mechanisms. Theorem 2.4.1 says that the ﬂux V consists of two components, the normal part ~nEuler = φ(κ) · ~n, V and the tangential part ′ ~tEuler = − 1 ∂(φ (κ)|∇u|) · ~t. V |∇u| ∂~t We immediately see that the normal ﬂux corresponds to the ﬂux of the CDD inpainting equation with g(κ) = φ(κ). By rewriting the tangential component, we see that this 64 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison corresponds to the Bertalmio ﬂux with a special smoothness measure. In fact it turns out that the tangential component of the Euler’s elastica ﬂux corresponds to the scheme of Bertalmio with smoothness measure LEuler = φ −1 ∂(φ′ (κ)|∇u|) . |∇u|2 ∂~t This measure can be further rewritten in a way which makes its connection to the Laplacian visible. In the case φ(s) = |s| we get LEuler = φ ±1 [∇ × ∇u](~n, ~t), |∇u|2 which closely resembles Bertalmio’s choice of the Laplacian, which can written as LBertalmio (u) = ∆u = tr(∇ × ∇u) = [∇ × ∇u](~n, ~n) + [∇ × ∇u](~t, ~t). TV-H−1 Inpainting Now we want to do a similar analysis for TV-H−1 inpainting. The gradient descent of the TV-H−1 inpainting regularizer is given by ∇u ∇u ∂u(x, t) = −∆ ∇ · = ∇ · −∇ ∇ · , ∂t |∇u| |∇u| with ﬂux ﬁeld ∇u T V −H −1 ~ . V = −∇ ∇ · |∇u| Now to write this ﬂux as the sum of its normal and tangential component we project it both onto the normal and the tangential direction and use the fact that the sum of these projections gives the identity, i.e., ~n ⊗ ~n + ~t ⊗ ~t = Id, where ⊗ denotes the tensor product. With this the steepest descent for TV-H−1 inpainting reads ∂u(x, t) ∂t = ∇ · (−∇κ) = ∇ · −∇κ(~t ⊗ ~t + ~n ⊗ ~n) ∂κ ~ ∂κ ~n . = ∇· − t+ ∂~n ∂~t 65 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison Now we want to compare the normal and the tangential component with the components of the Euler’s elastica inpainting approach and with the approaches of CDD and Bertalmio et al respectively. Let us start with the normal part of the TV-H−1 ﬂow, which reads ∂κ T V −H −1 ~ ∇ · Vn = ∇ · − ~n . ∂~n At the same time the normal part in Euler’s elastica is ~nEuler = ∇ · (φ(κ)~n), ∇·V or more speciﬁcally ~nEuler = ∇ · ((a + bκ2 )~n), ∇·V and the CDD ﬂux reads ~ CDD = ∇ · (g(κ)~n). ∇·V Note that the strength of the diﬀusion in all three approaches depends on the size of the diﬀusivity constant. This is, in the case of TV-H−1 inpainting D = −∂κ/∂~ n |∇u| , the diﬀusion depends on the change of the curvature across the level lines. The larger the curvature changes across the level lines, the more diﬀusive this approach is. Thereby the sign of the diﬀusion depends on the sign of the covariant derivative (forward or backward diﬀusion!). In particular this means that there is no diﬀusion in areas of the image with homogeneous grayvalue and strong diﬀusion of edge information into the missing domain. ~ . For TV-H−1 inpainting Next we do the same for the tangential component in V this reads ~ T V −H −1 ∇·V t ∂κ ~ = ∇· − t ∂~t ∂κ −1 ⊥ = ∇· ·∇ u ∂~t |∇u| −1 ∂κ ⊥ = ∇ u·∇ , |∇u| ∂~t since ∇⊥ u is divergence-free. This corresponds to the scheme of Bertalmio with smoothness measure −1 ∂κ , |∇u| ∂~t whereas Euler’s elastica tangent component has the smoothness measure −2b ∂(κ|∇u|) LEuler (u) = . |∇u|2 ∂~t LT V −H −1 (u) = 66 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison LCIS Inpainting The steepest descent for the LCIS regularizer reads ∂u(x, t) ~ LCIS = ∇ · (−g(∆u)∇∆u). =∇·V ∂t Taking g(s) = 1/(1 + s2 ), the divergence of the ﬂux changes to ~ LCIS = ∇ · (∇(arctan(∆u))). ∇·V Similarly as for Euler’s elastica inpainting and the TV-H−1 approach, we split the ﬂux of LCIS into its normal and tangential component and compare it with CDD and Bertalmio et al.’s approach respectively. For the normal component we get ~nLCIS = ∇ · − ∂arctan(∆u) ~n , ∇·V ∂~n i.e., the normal ﬂux diﬀuses the grayvalues with diﬀusivity constant D=− ∂arctan(∆u) 1 . ∂~n |∇u| Next we compare the tangential component of LCIS with Bertalmio’s approach. The divergence of the tangential ﬂux of LCIS reads ∂arctan(∆u) LCIS ~t ~t ∇·V = ∇· − ∂~t ∂arctan(∆u) 1 ⊥ = ∇· − ∇ u |∇u| ∂~t ∂arctan(∆u) 1 = ∇⊥ u · ∇ − . |∇u| ∂~t This reassembles Bertalmio’s smoothness transport along the level lines with smoothness measure ∂arctan(∆u) 1 |∇u| ∂~t 1 ∂(∆u) 1 = − 2 |∇u| 1 + (∆u) ∂~t LLCIS (u) = − 67 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison Bertalmio et al.’s transport Cahn and Shen’s CDD Euler’s elastica TV-H−1 LCIS ~n ∇·V L(u) ∂κ ∇ · (− ∂~ n) n~ −2b ∂(κ|∇u|) |∇u|2 ∂~t −1 ∂κ |∇u| ∂~t ∂(∆u) 1 1 − |∇u| . 1+(∆u)2 ∂~t 0 ∇ · (g(κ)~n) ∇ · ((a + bκ2 )~n) ∇ · (− ∂arctan(∆u) ~n) ∂~ n ∆u 0 ~n (which represents the diﬀusive Table 2.1: A comparison of the normal ﬂux ﬁeld V part of the ﬂux), and the smoothness measure L(u), which is transported along the tangential ﬂux, for all discussed inpainting approaches Summary and Numerical Results In Tabular 2.1 we summarize the comparison of the ﬂux ﬁeld of the discussed inpainting approaches. For a better understanding of the derived geometric properties of TV-H−1 inpainting and inpainting with LCIS we present a comparison of those with Euler’s elastica inpainting for the inpainting of a straight line in Figure 2.9. For all three inpainting approaches several intermediate steps of the iteration (evolution) are shown. The processes seem quite similar. First the inpainting domain is ﬁlled in where homogeneous boundary condition are given, i.e., the intensity values are transported into the inpainting domain very fast. At the edges (black/white boundary) the evolution takes a longer time until the edge is continued. Note that at the edge the change of the curvature in normal direction is equal to 0. It becomes nonzero when approaching the boundary of the inpainting domain in between the two edges. 68 2.4 The Inpainting Mechanisms of Transport and Diffusion - A Comparison Solution of TV−H−1 inpainting at iteration=100 Solution of LCIS inpainting at iteration=6 Solution of TV−H−1 inpainting at iteration=900 Solution of LCIS inpainting at iteration=120 Solution of TV−H−1 inpainting at iteration=3000 Solution of LCIS inpainting at iteration=1200 Figure 2.9: A comparison of the evolution of TV-H−1 inpainting, inpainting with LCIS, and Euler’s elastica inpainting at three diﬀerent time steps: (l.) the evolution of TVH−1 inpainting at time steps t = 100, 900, and 3000; (m.) the evolution of LCIS inpainting at time steps t = 6, 120, and 1200; (r.) the evolution of Euler’s elastica inpainting at time steps t = 1200, 3000, and 30000. 69 Chapter 3 Analysis of Higher-Order Equations The study of higher-order PDEs is still very young and therefore their theoretical analysis poses challenging problems. In this chapter we are especially interested in analytic properties of higher-order ﬂows, i.e., PDEs of fourth diﬀerential order, that arise in image inpainting. In Section 3.1 we study the Cahn-Hilliard equation, which we already have discussed in the context of its inpainting application in Section 2.1. In particular we study instabilities of the Cahn-Hilliard equation and their connection to the Willmore functional. This work has been developed in collaboration with Martin Burger, Shun-Yin Chu, and Peter Markowich, and has been published in [BCMS08]. Section 3.2 is dedicated to the analysis of higher-order nonlocal evolution equations. This is based on joint work with Julio Rossi, cf. [RS09]. Nonlocal ﬂows have been recently proposed for inpainting of texture images in [GO07] and are therefore within the scope of the present work. 3.1 Instabilities of the Cahn-Hilliard Equation and the Willmore Functional In this section we are interested in the ﬁnite-time stability of transition solutions of the Cahn-Hilliard equation and its connection to the Willmore functional. We shall present the results from [BCMS08]. Therein we show that the Willmore functional locally decreases or increases in time in the linearly stable or unstable case respectively. 70 3.1 Instabilities in the Cahn-Hilliard Equation This linear analysis explains the behavior near stationary solutions of the Cahn-Hilliard equation. We perform numerical examples in one and two dimensions and show that in the neighborhood of transition solutions local instabilities occur in ﬁnite time. Beside that we show convergence of solutions of the Cahn-Hilliard equation for arbitrary dimension to a stationary state by proving asymptotic decay of the Willmore functional in time. We consider the Neumann problem for the Cahn-Hilliard equation, x ∈ Ω, ut = ∆ −ǫ2 ∆u + F ′ (u) ∂u = ∂ (−ǫ2 ∆u + F ′ (u)) = 0, x ∈ ∂Ω, ∂n ∂n (3.1) with Ω ⊆ Rd a bounded domain and d ≥ 1, though we shall focus primarily on the cases d = 1 and 2. F is a double-well potential with F ′ (u) = 12 (u3 − u) and 0 < ǫ ≪ 1 is a small parameter. Note that the function F ′ is of bistable type. Considering only constant solutions u = c of (3.1), these are classiﬁed in the following way. If F ′′ (u) < 0 then u corresponds to the so-called spinodal interval (|u| < √1 ) 3 and it is an unstable stationary state. Otherwise u corresponds to the metastable intervals (i.e. u ∈ (−1, − √13 ) or u ∈ ( √13 , 1)) and is asymptotically stable, see [Fi00] for a detailed description. The Cahn-Hilliard equation The Cahn-Hilliard equation is a classic model for phase separation and subsequent phase coarsening of binary alloys. There the solution u(x, t) represents the concentration of one of the two metallic components of the alloy. For further information about the physical background of the Cahn-Hilliard equation we refer, for instance, to [CH58, Pe89, PF90, Gu88]. Numerical studies about the behavior of solutions of (3.1) can be found, e.g., in [EF87, FP05, MF05]. Solutions of (3.1) have a time-conserved mean value Z Z u(x, t = 0)dx, for all t > 0. u(x, t)dx = (3.2) Ω Ω A way of deriving equation (3.1) has been suggested by Fife in [Fi00]. One takes the Ginzburg-Landau free energy E[u](t) = Z Ω ǫ2 2 |∇u(x, t)| + F (u(x, t)) dx 2 71 (3.3) 3.1 Instabilities in the Cahn-Hilliard Equation and looks for constrained (mass-conserving) gradient ﬂows of this functional. The Cahn-Hilliard equation (3.1) is obtained by taking the gradient ﬂow of (3.3) in the sense of the H −1 (Ω) inner product, cf. Section 2.1.3. The mass constraint is a consequence of the natural boundary conditions for (3.1). Using the ideas of [Ha88] it is not hard to see that the Cahn-Hilliard equation possesses a global attractor. Zheng proved in [Zh86] convergence to equilibria for solutions of (3.1) in two and three dimensions. In [RH99] the authors proved that solutions of the Cahn-Hilliard equation converge to equilibria in dimensions d = 1, 2 and 3. In one space dimension the equilibria are isolated [NST89] and the global attractor is ﬁnite-dimensional. Further Grinfeld and Novick-Cohen give a full description of the stationary solutions of the viscous Cahn-Hilliard equation in one space dimension, compare [GN95, GN99]. So far equilibria have been determined and their properties studied only in the one dimensional case. It is a major problem characterizing the equilibria in more than one dimension. The reason is that the limit set of the solutions can be large. However, some papers as [WW98a, WW98b, WW98c, FKMW97, CK96] still provide certain special types of equilibrium solutions. A special type of stationary solutions of (3.1) are so-called transition solutions, which continuously connect the two stable equilibria -1 and 1. In one dimension the x so-called kink solution u0 = tanh 2ǫ is such a stationary solution of the Cahn-Hilliard equation. The radially-symmetric analogue in two dimensions are the so-called bubble solutions. In [BK90] asymptotic stability was shown for the kink solution for ﬁxed ǫ = 1, i.e. small perturbations of u0 in some suitable norm will decay to zero in time. Further studies for the one-dimensional case describe the motion of transition layers, compare [ABF91, BH92, BX94, Gr95]. In [AF94] Alikakos and Fusco proved spectral estimates of the linearized fourth-order Cahn-Hilliard operator in two dimensions near bubble solutions. Some of their results are discussed later in Section 3.1.2. In this section we investigate the stability of transition solutions of the Cahn-Hilliard equation in ﬁnite time and its connection to the Willmore functional. The backward second-order diﬀusion (for |u| < √1 ) 3 in the equation gradually ef- fects the solution, which can result in phenomena like local instabilities or oscillating patterns, controlled by the fourth-order term on scales of order ǫ. We are going to show that in the neighborhood of transition solutions small instabilities occur in ﬁnite time. 72 3.1 Instabilities in the Cahn-Hilliard Equation In general it is natural to study stationary solutions of the Cahn-Hilliard equation by analyzing the energy functional (3.3). The energy functional decreases in time since Z d |∇(−ǫ2 ∆u(x, t) + F ′ (u(x, t)))|2 dx = 0. (3.4) E[u](t) + dt Ω Because of this monotonicity the energy functional is not suitable for the study of local (in space and time) behavior of the Cahn-Hilliard equation. Instead we present analytical and numerical evidence that the numerically observed instabilities are connected with the evolution of the Willmore functional. The Willmore functional The Willmore functional of the Cahn-Hilliard equation (3.1) is given by Z 1 1 W [u](t) = (ǫ∆u(x, t) − F ′ (u(x, t)))2 dx, 4ǫ Ω ǫ (3.5) and is used to describe the geometric boundary of two diﬀerent stable states and the movement of curves under anisotropic ﬂows. It has its origin in diﬀerential geometry, where it appears as a phase ﬁeld approximation for solutions of the so-called Willmore problem [Wi93]. The Willmore problem is to ﬁnd a surface Γ in an admissible class R embedded in R3 which minimizes the mean curvature energy Γ H 2 dS under certain constraints on the surface, where H = (κ1 + κ2 )/2 is the mean curvature and κ1 , κ2 are the principal curvatures of Γ. For the analytical and computational modeling of a minimizing surface of the Willmore problem the phase ﬁeld method is considered among other approaches. In [DLRW05] the authors consider solutions of a constrained √ + ǫh with ﬁxed mass and minimization problem for (3.5) of the form uǫ (x) = tanh d(x) 2ǫ ﬁxed energy (3.3) where d is the signed distance function to the zero level set of uǫ and h is an arbitrary function in C 2 (Ω) independent of ǫ. They show that the level sets {uǫ = 0} converge uniformly to a critical point of the Willmore problem as ǫ → 0. Furthermore, the authors of [RS06] considered a modiﬁed De Giorgi conjecture, i.e., they considered functionals Fǫ : L1 (Ω) → R for domains Ω ∈ R2 and R3 with Fǫ [u](t) = E[u](t) + 4W [u](t) if u ∈ L1 (Ω) ∩ W 2,2 (Ω) and Fǫ [u](t) = ∞ if u ∈ L1 (Ω) \ W 2,2 (Ω). They showed that this sequence of functionals applied to characteristic functions χ = 2χE − 1 with E ⊂ Ω 73 3.1 Instabilities in the Cahn-Hilliard Equation Γ− converges in L1 (Ω) as ǫ → 0 to a functional F [χ] given by Z n−1 |H∂E |2 dHn−1 . F [χ](t) = σH (∂E ∩ Ω) + σ ∂E∩Ω Here σ = R1 √ −1 2F (where F is the double well potential), H∂E denotes the mean curvature vector of ∂E and Hn−1 is the n − 1 dimensional Hausdorﬀ measure. This result indicates that possible instabilities of (3.1) disappear for the limit ǫ → 0. We will encounter this observation again in our numerical examples for small values of ǫ. For additional considerations of Γ− limits of this type see [RT07], and especially [Ch96]. In [BCMS08] we use the Willmore functional to detect local instabilities in ﬁnite time of transition solutions of (3.1) for small values of ǫ ≪ 1. As said above, in one √ . In two space dimension they are given by the kink solutions of the form tanh d(x) 2ǫ dimensions we consider their radially-symmetric analogues called bubble solutions. To get an insight into the behavior of the Willmore functional for solutions of (3.1) we study the asymptotic limit of solutions of the Cahn-Hilliard equation for arbitrary space dimension d ≥ 1 by showing asymptotic decay of the Willmore functional in time. Asymptotic behavior In Section 3.1.1 we study the asymptotic limit of solutions of the Cahn-Hilliard equation for arbitrary space dimension d ≥ 1 by showing asymptotic decay of the Willmore functional in time. The main challenge of our proof of convergence is that we avoid the use of the Lojasiewicz inequality as say, in [RH99]. It was shown in [Lo93] that gradient ﬂows in Rd (and even in L2 , compare [Si83]) fulﬁll the Lojasiewicz inequality which implies convergence to equilibrium of solutions of the gradient system. For the application to the Cahn-Hilliard equation it takes a serious eﬀort to prove validity of the Lojasiewicz inequality for gradient ﬂows in H −1 , as proved in [RH99]. We circumvent this diﬃculty and prove the following Theorem. Theorem 3.1.1. Let u be the solution of the Cahn-Hilliard equation with initial data u0 = u0 (x), posed either as a Cauchy problem in Ω = Rd , d ≥ 1, or in a bounded domain Ω with Neumann boundary conditions. We assume that Z F ′ (u)dx = 0 for all t > 0. (3.6) Ω 74 3.1 Instabilities in the Cahn-Hilliard Equation For Ω = Rn further suppose that ( 2 0 ǫ ∆u − F ′ (u0 ) and ∇(ǫ2 ∆u0 − F ′ (u0 )) are spatially exponentially decaying as |x| → ∞. (3.7) Then it follows that lim W [u](t) = 0. t→∞ Remark 3.1.2. • Note that the assumption (3.6) in Theorem 3.1.1 is no restricR tion on F. Since Ω F ′ (u)dx is a constant, we can rewrite the equation as Z 1 2 ′ F ′ (u)dx) ut = ∆(−ǫ ∆u + F (u) − |Ω| Ω = ∆(−ǫ2 ∆u + f˜(u)), with R 1 |Ω| R Ω ˜ Ω f (u)dx F ′ (u)dx. = 0 where f˜(u) is equal to F ′ (u) shifted by the constant In the case of Neumann boundary conditions it further follows from (3.1) that ∂(−ǫ2 ∆u + f˜(u)) ∂u = = 0. ∂n ∂n Thus shifting F ′ (u) by a constant does not change the equation and the assumption (3.6) is reasonable. • Note further that condition (3.7) extends the exponential decay of the quantities therein to solutions u(., t) for arbitrary times t > 0 due to the mass conservation (3.2) of solutions of (3.1). The challenge of proving convergence of the Willmore functional is that it is generally not monotone in time. To overcome this we construct a nonnegative functional balancing the Willmore functional with the energy functional so that the strong decay property of the energy plays the main role in controlling increasing parts appearing in the Willmore functional. Finite-time behavior The next step is to analyze the behavior of the Willmore functional and its connection to the behavior of solutions of (3.1) in ﬁnite time. For this sake we consider the linearized Cahn-Hilliard equation. In fact the behavior of solutions of the nonlinear equation is similar in the neighborhood of stationary solutions to that of the linear equation. Sander and Wanner discussed in [SW00] that solutions of (3.1) which start 75 3.1 Instabilities in the Cahn-Hilliard Equation near a homogeneous equilibrium and within the spinodal interval remain close to the corresponding solution of the linearized equation with high probability (depending on how likely it is to ﬁnd an appropriate initial condition) for an unexpectedly long time. In particular we are interested in instabilities that appear locally in both time and space. To motivate this better let us consider the Cahn-Hilliard equation near a constant equilibrium state ũ on the whole space Ω = Rd . Set u = ũ + δv, δ ∈ R small, then the perturbation v fulﬁlls in ﬁrst approximation vt = −ǫ2 ∆2 v + a∆v in Rd , t > 0 in Rd , v(t = 0) = v0 (x) with a = F ′′ (ũ) = 21 (3ũ2 − 1). The above equation reads after a Fourier transform vˆt (ξ, t) = (−ǫ2 |ξ|4 − a |ξ|2 )v̂, v̂(t = 0) = vˆ0 (ξ), ξ ∈ Rd , t > 0 ξ ∈ Rd . This equation can be explicitly solved: v̂(ξ, t) = exp (− |ξ|2 (ǫ2 |ξ|2 + a)t)vˆ0 (ξ). √ √ If a = F ′′ (ũ) < 0 (this means that ũ lies in the spinodal interval (−1/ 3, 1/ 3)) o n p p then v̂(ξ, t) → ∞ for ξ ∈ |ξ| < |a|/ǫ ∩ supp v̂0 , i.e. frequencies < |a|/ǫ become ampliﬁed if they are present in the initial data. In our case we consider solutions in the neighborhood of transition solutions, i.e. transition solutions perturbed where they assume values from the spinodal interval. Referring again to [SW00] the linear instability explained above is also valid for the nonlinear Cahn-Hilliard equation for a ﬁnite time until the eﬀect of the nonlinearity becomes strong enough to stabilize the solution again. This results in instabilities local in space and time. In Section 3.1.2 spectral estimates for transition solutions tracing back to Alikakos, Bates and Fusco [ABF91, AF94] are presented for one and two dimensions. Further, the important role of the Willmore functional for ﬁnite-time stability/instability analysis for the Cahn-Hilliard equation is motivated. For ǫ ﬁxed we linearize the Willmore functional at a stationary solution of the Cahn-Hilliard equation (3.1) perturbed by an eigenvector of the linearized Cahn-Hilliard operator. We show that the Willmore 76 3.1 Instabilities in the Cahn-Hilliard Equation functional decreases in time for eigenvectors corresponding to a negative eigenvalue and increases in the case of a positive eigenvalue. In other words d W [v](t) ≤ 0 dt d W [v](t) ≥ 0 dt ⇐⇒ λ < 0 ⇐⇒ linearly stable ⇐⇒ λ > 0 ⇐⇒ linearly unstable, where v = ũ + δv0 , with ũ a stationary solution of (3.1) and v0 the eigenvector to the eigenvalue λ of the linearized equation. Roughly speaking this means that linear instabilities – which correspond to positive eigenvalues of the Cahn-Hilliard equation – can be detected by considering the evolution of the Willmore functional. In Section 3.1.3 we perform numerical computations for (3.1) near transition solutions in one and two dimensions. We remark that in the past 20 years numerical approximations of the solutions of the Cahn-Hilliard equation – for purposes diﬀerent from ours – have been studied by many authors, see [FP04, FP05] for further references. We use a semi-implicit approximation in time and ﬁnite elements for the space discretization. We start the computation at t = 0 with a transition solution perturbed within its transition area with values from the spinodal interval of the equation. Motivated by the linear stability analysis of subsection 3.1.2, we discuss stability in terms of the Willmore functional. We say a function u(x, t) shows an unstable behavior at time 0 < t0 < ∞ if d W [u](t0 ) > 0. dt Conversely, we say the function u(x, t) is stable for all times 0 < t < t0 if d W [u](t) ≤ 0. dt 3.1.1 Asymptotic Behavior In this section we present the proof of Theorem 3.1.1 which will be split into several lemmas and propositions. In the following the long-time asymptotic behavior of solutions of the Cahn-Hilliard equation is studied by exploring the Willmore functional. We consider the d-dimensional case of the Cahn-Hilliard equation. All following arguments hold true both for the Neumann boundary problem and the Cauchy problem in Rd with certain conditions on the spatial decay of the solutions. We start by introducing some useful properties of the functionals in our setting of the stationary proﬁle. 77 3.1 Instabilities in the Cahn-Hilliard Equation Lemma 3.1.3. Let u be the solution of the Cahn-Hilliard equation as posed in Theorem 3.1.1. Then for any test function φ = φ(x, t) ∈ C0∞ (Ω × (0, ∞)) we have d dt Z φ( Ω Z ǫ2 |∇u|2 + F (u)) dx + φ|∇(ǫ2 ∆u − F ′ (u))|2 dx 2 Ω Z Z ǫ2 1 2 ∆φ(F ′ (u) − ǫ2 ∆u)2 dx φt ( |∇u| + F (u)) dx + = 2 2 Ω ZΩ 2 −ǫ ∇(∇φ · ∇u) · ∇(ǫ2 ∆u − F ′ (u)) dx. Ω Lemma 3.1.4. Let u be the solution of the Cahn-Hilliard equation as posed in Theorem 3.1.1 then we have Z Z 1 1 d (ǫ∆u − F ′ (u))2 dx + 2ǫ2 |∆(ǫ∆u − F ′ (u))|2 dx dt Ω ǫ ǫ Ω Z 1 1 = 2 F ′′ (u)(ǫ∆u − F ′ (u))∆(ǫ∆u − F ′ (u))dx, ǫ ǫ Ω and d dt Z Z 1 1 (∆(ǫ∆u − F ′ (u)))2 dx + 2ǫ2 (∆2 (ǫ∆u − F ′ (u)))2 dx ǫ ǫ Ω Z Ω 1 1 = 2 F ′′ (u)∆(ǫ∆u − F ′ (u))∆2 (ǫ∆u − F ′ (u))dx. ǫ ǫ Ω The proofs of Lemma 3.1.3 and 3.1.4 are straightforward. Note that both lemmas hold when Ω is a bounded domain as well as when Ω is unbounded, provided that condition (3.7) from Theorem 3.1.1 holds. Proposition 3.1.5. Let f, g ∈ C 1 ([0, ∞)) be nonnegative functions with g ′ (t) ≤ 0 ev′ + erywhere, supp f ⊂ supp g, supt∈supp g fg and supt∈supp g (f g) bounded, where (f ′ )+ (t) = max{f ′ (t), 0}. Under the same assumptions as in Theorem 3.1.1 and for a sufficiently large constant C we have Z Z i 1 ′ ǫ2 C dh 2 f (t)(ǫ∆u − F (u)) dx + 2 g(t)( |∇u|2 + F (u))dx ≤ 0. dt Ω ǫ ǫ Ω 2 In particular, ǫ3 W [u](t) + CE[u](t) ≤ ǫ3 W [u0 ] + CE[u0 ]. Proof. Consider the functional Z Z 1 ǫ2 C f (t)(ǫ∆u − F ′ (u))2 dx + 2 U [u](t) = g(t)( |∇u|2 + F (u))dx. ǫ ǫ Ω 2 Ω 78 3.1 Instabilities in the Cahn-Hilliard Equation R R In the following we use the short form := Ω . By using the identity in Lemma 3.1.4 for the ﬁrst term in U [u](t) and 3.1.3 for the second term we derive Z Z 2 1 ′ C ′ ǫ d ′ 2 U [u](t) = f (t) (ǫ∆u − F (u)) dx + 2 g (t) |∇u|2 + F (u)dx dt ǫ ǫ 2 h Z 1 1 + f (t) 2 F ′′ (u)(ǫ∆u − F ′ (u))∆(ǫ∆u − F ′ (u))dx ǫ ǫ Z Z i C 1 − 2ǫ2 |∆(ǫ∆u − F ′ (u))|2 dx − 2 g(t) |∇(−ǫ2 ∆u + F ′ (u))|2 dx. (3.1) ǫ ǫ For the case Ω = Rd we need the following Lemma to deal with the last term in (3.1). Lemma 3.1.6 (modiﬁed Poincare inequality on Rd ). Let a > 0, C1 > 0, C2 > 0 be fixed constants. Then there exists a positive constant C0 = C0 (a, C1 , C2 ) such that for any functions f in Z 1 d f dx = 0, |f (x)| ≤ C1 e−a|x| |f |L2 , Va,C1 ,C2 = {f ∈ H (R ) : Rd ′ |f (x)| ≤ C2 e−a|x| |f |L2 for x ∈ Rd }, we have |f |L2 (Rd ) ≤ C0 |∇f |L2 (Rd ) . Taking into account condition (3.7) for the spatial decay of the involved quantities and by using Lemma 3.1.6 for Ω = Rd , the right side of (3.1) can be bounded by Z Cg(t) 1 ′ + (f (t)) − (ǫ∆u − F ′ (u))2 dx C0 ǫ Z 1 1 ′′ + 2f (t) sup |F (u)| |ǫ∆u − F ′ (u)| · |∆(ǫ∆u − F ′ (u))|dx ǫ ǫ |u|≤2 Z 1 C − 2f (t)ǫ2 |∆(ǫ∆u − F ′ (u))|2 dx + 2 g ′ (t)E[u](t). ǫ ǫ This term is non-positive when C is chosen to be large enough such that Cg(t) (−2f (t)ǫ2 ) ≤ 0, f (t)2 sup |F ′′ (u)|2 − (f ′ (t))+ − C 0 |u|≤2 that is, choosing C > C0 f (t)(sup|u|≤2 |F ′′ (u)|)2 2ǫ2 g(t) + C0 (f ′ (t))+ g(t) for t ∈ supp g. Remark 3.1.7. Note that the Poincare inequality on Rd is not valid in general. Conn+2 b sider for example the function ha (x) = a 2b xe−a|x| with a > 0 and some fixed b > 0. R 2 R ′2 2 Then ha dx = O(1) and ha dx = O(a b ) as a → 0+. Therefore 1 |h′a |L2 = O(a b ) → 0 |ha |L2 79 3.1 Instabilities in the Cahn-Hilliard Equation as a tends to zero, which contradicts the Poincare inequality. Finally we arrive at the proof of the main result of this section, Theorem 3.1.1. Proof of Theorem 3.1.1. Step 1: Let 1+t 0 ≤ t ≤ 1, 2 , f (t) = 2 − t, 1 ≤ t ≤ 2, 0, t≥2 and g(t) = ( 1, t ≤ 2, . 0, t ≥ 3 Let C be a ﬁxed constant chosen as in Proposition 3.1.5. The functional C g(t)E[u](t) ǫ2 is decreasing in time and U [u](t = 1) ≤ U [u](t = 0). That is, U [u](t) ≡ 4ǫf (t)W [u](t) + C 1 C E[u](t = 1) ≤ W [u](t = 0) + 3 E[u](t = 0). 3 4ǫ 2 4ǫ Step 2: For each n ∈ N, setting f (t − n + 1) and g(t − n + 1) as in Proposition 3.1.5, we can write down the same inequality as in the previous step but for n − 1 ≤ t ≤ n and obtain C 1 C W [u](t = n) + 3 E[u](t = n) ≤ W [u](t = n − 1) + 3 E[u](t = n − 1). (3.2) 4ǫ 2 4ǫ Set αn = W [u](t = n). Then (3.2) can be rewritten as Z n Z 1 C αn ≤ αn−1 + 3 |∇(−ǫ2 ∆u + F ′ (u))|2 dxdt 2 4ǫ n−1 Ω W [u](t = 1) + where we used the decay property (3.4) of the energy functional. Step 3: We want to show that αn tends to zero as n tends to inﬁnity. By an iterative argument we get n 1 αn ≤ α0 2 Z Z C 1 n−1 1 |∇(−ǫ2 ∆u + F ′ (u))|2 dxdt+ + 3 4ǫ 2 0 Ω n−2 Z 2 Z 1 |∇(−ǫ2 ∆u + F ′ (u))|2 dxdt 2 1 Ω Z Z 1 n−1 |∇(−ǫ2 ∆u + F ′ (u))|2 dxdt + ··· + 2 n−2 Ω Z n Z |∇(−ǫ2 ∆u + F ′ (u))|2 dxdt . + n−1 Ω 80 3.1 Instabilities in the Cahn-Hilliard Equation Using a standard fact from calculus formulated in the following Lemma 3.1.8, it follows that αn converges to 0 for n → ∞. P Lemma 3.1.8. Let (an ), (bn ) are two nonnegative sequences such that their sums n an P and n bn are convergent. Then lim n→∞ n X ai bn−i = 0. i=0 We conclude our proof in step 4. Step 4: It remains to prove that for any sequence (tn ) tending to inﬁnity, W [u](tn ) converges to zero. To do so it suﬃces to prove that for any ﬁxed integer q > 0, (W [u](t = nq ))n converges to zero. Repeating the computations of Step 3 for the inequality (3.2) for all rational values of t in between n − 1 and n the proof is similar and we omit the details here. Remark 3.1.9. Note that the proof of Theorem 3.1.1 also provides trivially for a decay of ut in H −2 , namely kut kH −2 = ≤ ∆(−ǫ2 ∆u + F ′ (u)) −2 H 2 ′ C −ǫ ∆u + F (u) L2 → 0 as t → ∞. So we have shown that the Willmore functional asymptotically decreases to zero under the assumptions of Theorem 3.1.1. This additionally proves that ut → 0 for t → ∞ in H −2 (Ω) for every ǫ > 0 and arbitrary dimension d. 3.1.2 Linear Stability / Instability In this section we consider the short-time behavior of solutions of the Cahn-Hilliard equation. We relate local-in-time instabilities of solutions with the Willmore functional by comparing the eigenvalues of the linearized operator of the equation with the evolution of the Willmore functional. We begin with a short discussion of spectral estimates and conclude with presenting the new result. For the one dimensional case Alikakos, Bates and Fusco showed in [ABF91] that there is exactly one unstable eigenvalue of the linearized Cahn-Hilliard eigenvalue problem. For simplicity let D = [0, 1]. They consider the problem linearized at an invariant 81 3.1 Instabilities in the Cahn-Hilliard Equation manifold M formed by the translation of a self-similar solution uξǫ (x) = uǫ (x − ξ) ∈ M with parameter ξ: ( − ǫ2 H ′′′′ + (F ′′ (uξǫ )H ′ )′ = λ(ǫ)H, 0 < x < 1, H = H ′′ = 0 x = 0, 1. (3.1) The ﬁrst eigenvalue is simple and exponentially small for small ǫ > 0 0< λξ1 (ǫ) e− 2νδǫ ξ (uξ (0))2 xx =O , =O ǫ3 ǫ7 (3.2) where δξ is a small positive constant given in the proof of (3.2) in [ABF91] and ν is a generic constant, see [CGS84]. The remaining spectrum is bounded from above by λξi (ǫ) ≤ −C < 0, i = 2, 3, · · · with C is positive and independent of ǫ, ξ. Both results are contained in [ABF91]. In two dimensions Alikakos and Fusco [AF94] proved that there is a two-dimensional invariant manifold with exponentially small eigenvalues where the solutions asymptotically develop droplets on the boundary with a speed which is exponentially small. These superslow solutions are called bubble solutions and correspond to an approximate spherical interface drifting slowly towards the boundary, without changing its shape. Solutions like that are typical in the ﬁnal stages of evolution of (3.1) for general initial conditions. Further they showed that the dimension of eigenspaces of superslow eigenvalues of the linearized Cahn-Hilliard equation on D ⊆ Rd is at most d for d ≥ 2. For simplicity of explanation we consider the eigenvalue problem of the linearized fourth-order Cahn-Hilliard operator in D ⊆ R2 . The results in higher dimen- sions are analogous to this case. Let U (η) be the unique increasing bounded solution of U ′′ − F (U ) = 0 on R, and V (η) a bounded function that satisﬁes the orthogonality condition Z ∞ f ′′ (U (η))U̇ 2 (η)V (η)dη = 0, −∞ where f (u) = F ′ (u). We consider a one-parameter family of functions uξǫ (x) represented by uξǫ (x) = x−ρ 2 U ( x−ρ ǫ ) + ǫV ( ǫ ) + O(ǫ ), |y − ρ| ≤ λ, qǫ (x), |y − ρ| > λ where y = |x − ξ|, ρ > 0 and qǫ (x) is an arbitrary function with f ′ (qǫ (x)) ≥ c > 0. The function uξǫ (x) represents a bubble with center ξ ∈ D and radius ρ. 82 3.1 Instabilities in the Cahn-Hilliard Equation The eigenvalue problem for the Cahn-Hilliard operator linearized in uξǫ (x), i.e., Lξ = ∆(−ǫ2 ∆ + F ′′ (uξǫ )), is given by Lξ (φ) = λφ, x ∈ D ⊆ R2 (3.3) with Neumann boundary conditions ∂φ ∂ = (−ǫ2 ∆φ + F ′′ (uξǫ )φ) = 0, ∂n ∂n x ∈ ∂D. In [AF94] Alikakos and Fusco stated the following result. Theorem 3.1.10. Let λǫ1 ≥ λǫ2 ≥ λǫ3 ≥ · · · be the eigenvalues of (3.3). Let ρ > 0, δ > 0 be fixed. Then there exists an ǫ0 > 0 and constants c, C, C ′ > 0, independent of ǫ, such that for 0 < ǫ < ǫ0 and ξ ∈ D with d(ξ, ∂D) > δ, the following estimates hold true: c c Ce− ǫ ≥ λǫ1 ≥ λǫ2 ≥ −Ce− ǫ , λǫ3 ≤ −C ′ ǫ. The ﬁrst two eigenvalues λǫ1 , λǫ2 are superslow and the others are negative. Now we present the connection between linear stability properties of the CahnHilliard equation and the Willmore functional. In the following we provide a linear stability analysis around an equilibrium state u0 satisfying −ǫ2 ∆u0 + F ′ (u0 ) = 0. More precisely, we look for a solution of the form u(x, t) = u0 (x) + δv(x, t) + O(δ 2 ) for suﬃciently small 0 < δ ≪ 1 and some perturbation v(x, t). Due to mass conservation we assume that v has mean zero for all times Z v(x, t)dx = 0 ∀t > 0. Ω We obtain the ﬁrst-order evolution with respect to δ via the linearized equation vt = ∆(−ǫ2 ∆v + F ′′ (u0 )v) := ∆L0 v. (3.4) We now compute the asymptotic expansion of the Willmore functional as δ → 0. It can be expanded as W [u] = W [u0 ] + δW ′ [u0 ]v + 83 δ 2 ′′ W [u0 ](v, v) + O(δ 3 ), 2 (3.5) 3.1 Instabilities in the Cahn-Hilliard Equation where the ﬁrst and second-order derivatives are taken as variations Z ′ W [u0 ]v = (−ǫ2 ∆u0 + F ′ (u0 ))(−ǫ2 ∆v + F ′′ (u0 )v)dx and ′′ W [u0 ](v, w) = Z (L0 v)(L0 w)dx + Since u0 is a stationary solution, we have ′ Z (−ǫ2 ∆u0 + F ′ (u0 ))F ′′′ (u0 )vwdx. ′′ W [u0 ]v = 0 and W [u0 ](v, v) = Z (L0 v)2 dx. Now let v0 be an eigenfunction of the linearized fourth-order Cahn-Hilliard operator, i.e. there is λ 6= 0 such that with Neumann boundary condition ∆(L0 v0 ) = λv0 , ∂v0 ∂n |∂Ω = 0 and R Ω v0 = 0. Note that λ is real, since v0 solves a symmetric eigenvalue problem in the scalar product of H −1 , deﬁned here as R the dual of H 1 (Ω) ∩ u : Ω u dx = 0 , cf. also Section 2.1.3. The standard linear stability analysis yields that the perturbation of u0 by δv0 is linearly stable for λ < 0 and unstable for λ > 0. These two cases can be translated directly into the local-in-time behavior of the Willmore functional, whose time derivative at time t = 0 is given by d W [u(t)]|t=0 = δW ′ [u0 ]vt |t=0 + δ 2 W ′′ [u0 ](v, vt )|t=0 + O(δ 3 ) dt Z 2 = δ (L0 v0 )(L0 vt )dx + O(δ 3 ) Z 2 = δ (L0 v0 )(L0 ∆L0 v0 )dx + O(δ 3 ) Z 2 = δ (L0 v0 )(L0 (λv0 ))dx + O(δ 3 ) Z = λδ 2 (L0 v0 )2 dx + O(δ 3 ). This means that to leading order, the time derivative of W [u] has the same sign as λ, i.e., the Willmore functional is locally increasing in time in the unstable case, and locally decreasing in the stable case. 84 3.1 Instabilities in the Cahn-Hilliard Equation 3.1.3 Nonlinear Stability / Instability We expect the behavior of solutions of the nonlinear equation (3.1) to be dominated by that of the linear equation in the neighborhood of stationary solutions. Therefore numerical examples may very well give us a good idea about the behavior of solutions and their connection to the Willmore functional even for the nonlinear case. In the following a semi-implicit ﬁnite element discretization for the Cahn-Hilliard equation is brieﬂy described and numerical examples are discussed. Numerical Discretization To discretize a fourth-order equation with boundary conditions as above it is often convenient to write it as a system of two diﬀerential algebraic equations of second order. In our case of the Cahn-Hilliard equation this results in the following system ut = ∆v v = −ǫ2 ∆u + F ′ (u), with Neumann boundary conditions ∂u ∂v = = 0, ∂n ∂n x ∈ ∂Ω. The following issues have to be taken into consideration. • Explicit schemes for fourth-order equations restrict time steps to be of order O((∆x)4 ), where ∆x is the spatial grid size. • Fully implicit schemes can be unconditionally stable. The disadvantage is the high computational eﬀort for solving nonlinear equations. • Semi-implicit schemes are a compromise between explicit and implicit discretiza- tion. Brieﬂy, semi-implicit means that the equation is split into a convex and a concave part and discretized implicitly and explicitly respectively, see ([BE92, Ey98]). Therefore the restriction on the step sizes is less severe and we do not have to solve nonlinear equations. 85 3.1 Instabilities in the Cahn-Hilliard Equation For these reasons we use the following semi-implicit approximation u(t, x) − u(t − ∆t, x) = ∆v(t, x) ∆t v(t, x) = −ǫ2 ∆u(t, x) + F ′ (u(t − ∆t, x)) + F ′′ (u(t − ∆t, x)) · (u(t, x) − u(t − ∆t, x)). Note that F ′ is Taylor-expanded at the solution of the previous time step u(t − ∆t). For the space discretization we use linear ﬁnite elements on an equidistant grid in one dimension and on a rectangular grid in two dimensions. Numerical Examples In the following examples we consider the solution of the Cahn-Hilliard equation in one and two dimensions for diﬀerent initial states in a neighborhood of a transition solution. In the one dimensional case the so-called kink solution is given by u0 (x) = x tanh ( 2ǫ ). As a ﬁrst approach in the one dimensional analysis we take as initial value u(x, t = 0) = u0 (x) + p(x). The function p denotes a particular kind of zero-mean perturbation, namely ( x ) a · sin (f π Cǫ p(x) = 0 x ∈ (−C · ǫ, C · ǫ) otherwise, with amplitude a > 0, frequency f > 0 and support (−C · ǫ, C · ǫ) with C > 0. The amplitude a is chosen so that the values of u within the support of the perturbation lie in the spinodal interval of the equation (which is the back diﬀusion interval). Varying the parameter ǫ and the support, the amplitude and the frequency of p, we want to observe how the solutions evolve in time. The behavior of the solutions is further compared with the evolution of the corresponding energy functional and the Willmore functional. We begin with a ﬁxed ǫ = 0.1. For the ﬁrst two examples in Figure 3.1 and 3.2 a ﬁxed step size in space and time discretization was used. As spatial step size we took ∆x = 0.05 · ǫ and for the time steps ∆t = 103 · ǫ. The amplitude and the frequency of the perturbation are also ﬁxed to 1. The diﬀerence between the two examples is the support of the perturbation and dependent on this is the convergence process. In the ﬁrst example an unstable state occurs. This means that over a certain time interval the perturbation locally grows. The second example is stable in the numerical computations. 86 3.1 Instabilities in the Cahn-Hilliard Equation (a) t=1 (b) t=500 (c) t=1000 (d) t=1999 (e) Energy functional (f) Willmore functional Figure 3.1: (a)-(d): Evolution of the solution in time for ǫ = 0.1 and a zero-mean perturbation supported on (−15ǫ, 15ǫ) with a = 1, f = 1 and with corresponding energy functional (e) and Willmore functional (f) In the ﬁrst case the supporting interval for the perturbation is (−15ǫ, 15ǫ) (Figure 3.1) and we begin with an initial state having two peaks on both sides of zero. When time proceeds the peaks grow in the beginning, resulting in an unstable transient state. After this unstable state the solution converges to the kink solution. In the case of the 87 3.1 Instabilities in the Cahn-Hilliard Equation (a) t=1 (b) t=50 (c) t=100 (d) t=499 (e) Energy functional (f) Willmore functional Figure 3.2: (a)-(d): Evolution of the solution in time for ǫ = 0.1 and a zero-mean perturbation supported on (−3ǫ, 3ǫ) with a = 1, f = 1 and with corresponding energy functional (e) and Willmore functional (f) supporting interval (−3ǫ, 3ǫ) in Figure 3.2 the solution converges uniformly to the kink solution without transitional state. Comparing the time evolution of our ﬁrst two examples with the corresponding energy functionals and Willmore functionals, we can easily see diﬀerences in the graphs of functionals. We can see that in the unstable case the energy functional almost has 88 3.1 Instabilities in the Cahn-Hilliard Equation a saddle point, in the stable case it is rapidly decreasing. Also instabilities seem to correspond to peaks in the graph of the Willmore functional over time. The Willmore functional increases over a ﬁnite time interval in contrast to the stable case where it decreases for all times t. (a) t=1 (b) t=30 (c) t=130 (d) t=1999 (e) Energy functional (f) Willmore functional Figure 3.3: (a)-(d): Evolution of the shifted solution in time for ǫ = 0.1 and a zeromean perturbation supported on (−15ǫ, 15ǫ) with a = 1, f = 1 and with corresponding energy functional (e) and Willmore functional (f) Another interesting phenomenon can be seen by starting with modiﬁed versions of 89 3.1 Instabilities in the Cahn-Hilliard Equation the perturbation p. For example we could shift the sinusoidal perturbation to the left or to the right of zero to start with an asymmetric initial state. The example in Figure 3.3 is a result of a shift of the perturbation used in the example of Figure 3.1. The shift of the perturbation leads to a shift of the kink solution asymptotically in time as a consequence of conservation of mass. Considering again the Willmore functional, an increase in time, visibly caused by a transitional instability, occurs. −0.5 time of instability amplitude of instability −1 log(time)/log(amplitude) −1.5 −2 −2.5 −3 −3.5 −4 −4.5 −5 1.2 (a) 1.25 1.3 1.35 1.4 1.45 1.5 range of perturbation 1.55 1.6 1.65 1.7 (b) Figure 3.4: Length of time interval and maximal amplitude of the instability for different amplitudes of the perturbation (a) and diﬀerent supporting intervals of the perturbation (b). ǫ = 0.1 ﬁxed. Considering the behavior of the perturbed solution in several numerical tests we can further make claims on how the perturbation has to look like so that an unstable state occurs. For a ﬁxed ǫ ≪ 1 we can see that the length of the supporting interval is most relevant. Extending the supporting interval of the perturbation brings with it an extension of the time interval and the size of the instability as a consequence, compare Figure 3.4 right diagram. If the supporting interval is too small, no unstable state can be seen, compare Figure 3.2. Also the amplitude and the frequency of the perturbation have an inﬂuence on the occurrence and size of the instability. With growing amplitude of the perturbation, the time interval and the size of the instability change, compare Figure 3.4, left diagram. If the amplitude exceeds a certain threshold the solution will not converge to the kink solution anymore. Furthermore the higher the frequency of the perturbation, the faster the solution converges to the kink solution. Therefore the higher the frequency of the perturbation, the smaller the time interval 90 3.1 Instabilities in the Cahn-Hilliard Equation of the instability. Because of the important role of the support of the perturbation, it seems that perturbations with high frequency bring no additional information for the study of the time-local instability. Point in time of maximal amplitude of the instability 4 2 0 log(time) −2 −4 −6 −8 −10 −12 0 0.1 0.2 0.3 0.4 0.5 ε 0.6 0.7 0.8 0.9 1 Figure 3.5: Length of the time interval of the instability for diﬀerent values of ǫ for a perturbation supported on (−15ǫ, 15ǫ) with a = 1, f = 1. By changing the parameter ǫ one can see that with decreasing ǫ the time interval of the instability decreases. In Figure 3.5 the point in time of the maximal amplitude of the instability is shown for diﬀerent ǫ. The maximal amplitude of the instability stays approximately the same. In the two-dimensional case the analogue of the kink solutions are the so-called bubble solutions. In Figure 3.6 the evolution of a solution of the two dimensional CahnHilliard equation near a bubble solution over a ﬁnite time interval is shown. In this example we used equidistant space and time discretization ∆x = ∆y = ǫ and ∆t = ǫ4 . Here the time stepsize is chosen rather small in order not to miss the instability which appears for a very short time period only. As initial value we take a radial-symmetric bubble solution perturbed by a sine wave in x1 –direction. For a better comparison with the one-dimensional case a vertical cut in x2 = 0.5 of the solution is shown. Again the solution exhibits a local growth of amplitude before converging uniformly to the stationary solution. Especially near x1 = 0.5 the solution initially tends away from the bubble solution. As predicted, this phenomenon causes an increase of the Willmore functional in a short time interval before it decays to 0. 91 3.1 Instabilities in the Cahn-Hilliard Equation (a) t=21 (b) t=600 (c) t=21 (d) t=600 (e) Bubble solution (f) Willmore functional Figure 3.6: Surface plot (a)-(b) and vertical cut (c)-(d) of the solution for ǫ = 0.1 perturbed with a sinusoidal wave in x–direction. (e) shows the bubble solution and (f) the Willmore functional 3.1.4 Consequences We considered local instability and asymptotic behavior of the Cahn-Hilliard equation in the neighborhood of certain transition solutions. We found that studying instabilities of the Cahn-Hilliard equation in ﬁnite time is closely connected to studying the monotonicity behavior of the Willmore functional. We found that the Willmore 92 3.2 Nonlocal Higher-Order Evolution Equations functional is a good monitoring quantity to study structures of instability patterns of the solutions. We found the Willmore functional to be monotonically decreasing if the solution converges to the equilibrium state without transitional instability and having maxima when local-time instabilities occur. Therefore the Willmore functional could be used for the mathematical and numerical analysis of the Cahn-Hilliard equation, e.g., by providing a more eﬃcient tool for determining stability/instability of solutions. Further it can be useful in applications of the Cahn-Hilliard equation as in its inpainting application presented in Section 2.1. The Willmore functional could thereby serve as an indicator reporting how far away from the steady state the solution is. 3.2 Nonlocal Higher-Order Evolution Equations In this section we study the asymptotic behavior of solutions to the nonlocal operator ut (x, t) = (−1)n−1 (J ∗ Id − 1)n (u(x, t)), x ∈ Rd which is the nonlocal analogue to the higher-order local evolution equation vt = (−1)n−1 (∆)n v. We prove that the solutions of the nonlocal problem converge to the solution of the higher-order problem with right hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to inﬁnity. This section traces the contents of [RS09]. Our main concern in the present section is the asymptotic behavior of solutions of a nonlocal diﬀusion operator of higher order in the whole Rd , d ≥ 1. We consider the following nonlocal evolution problem: ut (x, t) =(−1)n−1 (J ∗ Id − 1)n (u(x, t)) ! n X n =(−1)n−1 (−1)n−k (J∗)k (u) (x, t), k k=0 u(x, 0) =u0 (x), for x ∈ Rd , t > 0 and arbitrary n ≥ 1. Here (J ∗ u)(x, t) = the usual convolution of J and u and (J∗)k (u) R Rd (3.1) J(x − y)u(y, t) dy is denotes the convolution with J iterated k times. Further, let J ∈ C(Rd , R) be a nonnegative, radially invariant function with R 1 d Rd J(x) dx = 1 and u0 ∈ L (R ) denote the initial condition for (3.1). 93 3.2 Nonlocal Higher-Order Evolution Equations Nonlocal problems like (3.1) have been recently widely used to model diﬀusion processes, see [Fi03]. The solution u(x, t) of (3.1) is thereby interpreted as the density of a single population at the point x at time t and J(x − y) is the probability of “jumping” from location y to location x. The convolution (J ∗ u)(x) is then the rate at which individuals arrive to position x from all other positions, while −u(x, t) = R − Rd J(y − x)u(x, t) dy is the rate at which they leave position x to reach any other position. Their evolution is described by (3.1) for the case n = 1. For more references concerning the use of nonlocal evolution problems and their stationary counterparts for modeling diﬀusion processes we quote for instance [BCC05, BFRW97, Co06, Co07, CDM08, CD05, CD07, Zh04], devoted to traveling front type solutions. Further nonlocal equations like (3.1) also found applications in image processing. The main advantage of nonlocal operators in image processing is the ability to process structures like edges (local image features), but also textures (nonlocal image features), within the same framework. In [BCM05] a nonlocal ﬁlter, referred to as nonlocal means, was suggested for image denoising. A variational understanding of this ﬁlter was ﬁrst presented in [KOJ05] as a nonconvex functional and later in [GO07] as a convex quadratic functional. In the latter reference the authors investigated the functional Z 1 |u(x) − u(y)|2 w(x, y) dxdy, E(u) = 2 Ω×Ω where the weight function w(x, y) ∈ Ω × Ω, Ω ⊂ R2 open and bounded, is positive and symmetric, i.e. w(x, y) = w(y, x). The proposed ﬂow for minimizing the energy E(u) was then deﬁned as Z ut (x) = (u(y) − u(x)) w(x, y) dy, Ω u(x, 0) = u0 (x), x ∈ Ω, (3.2) taking the given (noisy) image u0 as the initial condition. With w(x, y) = J(x − y) equation (3.2) has the same structure as the nonlocal equation (3.1) for n = 1. We call equation (3.1) a nonlocal diﬀusion equation of order n. Thereby the diﬀusion of the density u at a point x and time t does not only depend on u and its derivatives at the point (x, t) (local behavior), but on all the values of u in a ﬁxed neighborhood of x through the convolution term J ∗ u (nonlocal behavior). In our problem (3.1) the application of the nonlocal operator J ∗ Id − 1 on the density u is iterated n times. 94 3.2 Nonlocal Higher-Order Evolution Equations This can be seen as a nonlocal generalization of higher-order equations of the form vt (x, t) = −An (−∆) αn 2 v(x, t), (3.3) with A and α are positive constants speciﬁed later in this section. Note that when α = 2 (3.3) is just vt (x, t) = −An (−∆)n v(x, t). Higher-order diﬀusions of this type appear in various applications. The Cahn-Hilliard equation, for instance, is a fourth-order reaction diﬀusion equation which models phase separation and coarsening of binary alloys, see [Fi00] for more details and references. A modiﬁed Cahn-Hilliard equation was further proposed in [BEG07a, BEG07b] for inpainting, i.e., image interpolation of binary images. Another fourth-order example is the Kuramoto-Sivashinsky equation (cf. e.g. [Mi86]), used in the study of spatiotemporal chaos (cf. [CH93]). In both equations a linear fourth-order diﬀusion as in (3.3) for n = α = 2 is involved. Nonlocal higher-order problems have been, for instance, proposed as models for periodic phase separation. Here the nonlocal character of the problem is associated with long-range interactions of “particles” in the system. An example is the nonlocal Cahn-Hilliard equation (cf. e.g. [Ha04, NO95, OK86]). Problem (3.1) was studied in great detail for the case n = 1 only. In fact the Cauchy problem is considered in [CCR06, IR08], while the “Neumann” boundary condition for the same problem is treated in [AMRT08, CERW07, CERW08]. See also [IR07] for the appearance of convective terms and [CCEM07, CER05] for other interesting features in related nonlocal problems. In this section we consider (3.1) as a model for higher-order nonlocal evolution for arbitrary n ≥ 1 and extend the existing analysis for the Cauchy problem within this general setting. For this model, we ﬁrst prove existence and uniqueness of a solution, but our main aim is to study the asymptotic behavior as t → ∞ of solutions to (3.1). Moreover, we prove that solutions to (3.1) converge to the solution to (3.3) when the problem is rescaled in an appropriate way. Now, let us proceed with the precise description of our main results. Statement of the results. For a function f we denote by fˆ the Fourier transform of f and by fˇ the inverse Fourier transform of f . Our hypotheses on the convolution kernel J that we will assume throughout the paper are: 95 3.2 Nonlocal Higher-Order Evolution Equations The kernel J ∈ C(Rd , R) is a nonnegative, radial function with total mass equalling R one, Rd J(x) dx = 1. This means that J is a radial density probability, which implies ˆ ˆ = 1. Moreover, we assume that that its Fourier transform satisfies |J(ξ)| ≤ 1 with J(0) ˆ = 1 − A |ξ|α + o(|ξ|α ) J(ξ) for ξ → 0, (3.4) for some A > 0 and α > 0. Remark 3.2.1. Note that assumption (3.4) on the behaviour of Jˆ near the origin plays a central role in the following analysis of the nonlocal problem. In fact, the decay rate as t goes to infinity of solutions of this nonlocal problem is determined by (3.4). Under these conditions on J we have the following results. First, we show existence and uniqueness of a solution Theorem 3.2.2. Let u0 ∈ L1 (Rd ) such that uˆ0 ∈ L1 (Rd ). There exists a unique solution u ∈ C 0 ([0, ∞); L1 (R)d ) of (3.1) that, in Fourier variables, is given by the n−1 ˆ n explicit formula, û(ξ, t) = e(−1) (J(ξ)−1) t uˆ0 (ξ). Next, we deal with the asymptotic behavior as t → ∞. Theorem 3.2.3. Let u be a solution of (3.1) with u0 , uˆ0 ∈ L1 (Rd ). Then the asymptotic behavior of u(x, t) is given by d lim t αn max |u(x, t) − v(x, t)| = 0, t→+∞ x αn where v is the solution of vt (x, t) = −An (−∆) 2 v(x, t) with the initial condition v(x, 0) = u0 (x) and A and α as in (3.4). Moreover, we have that there exists a constant C > 0 such that d ku(., t)kL∞ (Rd ) ≤ C t− αn and the asymptotic profile is given by d 1 lim max t αn u(yt αn , t) − ku0 kL1 (Rd ) GA (y) = 0, t→+∞ y αn n where GA (y) satisfies GˆA (ξ) = e−A |ξ| . To prove this result we use ideas from [CCR06] using Fourier variables, but with a nontrivial extra reﬁnement that is due to the fact that we are dealing with higher-order problems. In fact, we have to take extra care and distinguish between even and odd n. Next, we show that solutions to nonlocal problems like (3.1), rescaled in an appropriate way, converge to a solution to the problem (3.3) as the scaling parameter tends to zero. 96 3.2 Nonlocal Higher-Order Evolution Equations Theorem 3.2.4. Let uε be the unique solution to n (u ) (x, t) = (−1)n−1 (Jε ∗ Id − 1)) (u (x, t)), ε t ε εαn u(x, 0) = u0 (x), (3.5) where Jε (s) = ε−d J( sε ). Then, for every T > 0, we have lim kuε − vkL∞ (Rd ×(0,T )) = 0, ε→0 where v is the solution to the local problem vt (x, t) = −An (−∆) initial condition v(x, 0) = u0 (x). αn 2 v(x, t) with the same The previous proofs of convergence of nonlocal problems to their local counterparts were performed in physical space with subtle arguments, see [CERW08, IR07]. For our proof of this result, we just look at Fourier variables, simplifying the previous analysis. Note that this approach is valid only when linear problems in the whole space are considered. The veriﬁcation of our results is organized as follows: in Section 3.2.1 we prove existence and uniqueness of a solution; in Section 3.2.2 we deal with the asymptotic behavior and, ﬁnally, in Section 3.2.3 we approximate the local higher-order problem by nonlocal ones. 3.2.1 Existence and Uniqueness To prove existence and uniqueness of solutions we make use of the Fourier transform. Proof of Theorem 3.2.2. We have ut (x, t) = (−1)n−1 ! n X n (−1)n−k (J∗)k (u) (x, t). k k=0 Applying the Fourier transform to this equation we obtain n X n n−1 k ˆ ût (ξ, t) = (−1) (−1)n−k (J(ξ)) û(ξ, t) k k=0 n−1 = (−1) ˆ − 1)n û(ξ, t). (J(ξ) Hence û(ξ, t) = e(−1) n−1 (J(ξ)−1) nt ˆ nt ˆ (−1)n−1 (J(ξ)−1) uˆ0 (ξ). Since uˆ0 (ξ) ∈ L1 (Rd ) and e is continuous and bounded, û(·, t) ∈ L1 (Rd ) and the result follows by taking the inverse Fourier transform. 97 3.2 Nonlocal Higher-Order Evolution Equations Now we prove a lemma concerning the fundamental solution of (3.1). Lemma 3.2.5. The fundamental solution w of (3.1), that is the solution of the equation with initial condition u0 = δ0 , can be decomposed as w(x, t) = e−t δ0 (x) + v(x, t), (3.6) with v(x, t) smooth. Moreover, if u is a solution of (3.1) it can be written as Z w(x − z, t)u0 (z) dz. u(x, t) = (w ∗ u0 )(x, t) = Rd Proof. By the previous result we have ˆ − 1)n ŵ(ξ, t). ŵt (ξ, t) = (−1)n−1 (J(ξ) Hence, as the initial data veriﬁes wˆ0 = δˆ0 = 1, we get n n−1 ˆ n−1 ˆ n ŵ(ξ, t) = e(−1) (J(ξ)−1) t = e−t + e−t e[(−1) (J(ξ)−1) +1]t − 1 . The ﬁrst part of the lemma follows applying the inverse Fourier transform. To ﬁnish the proof we just observe that w ∗ u0 is a solution of (3.1) with (w ∗ u0 )(x, 0) = u0 (x). 3.2.2 Asymptotic Behavior Next we prove the ﬁrst part of Theorem 3.2.3. Theorem 3.2.6. Let u be a solution of (3.1) with u0 , uˆ0 ∈ L1 (Rd ). Then, the asymptotic behavior of u(x, t) is given by d lim t αn max |u(x, t) − v(x, t)| = 0, t→+∞ x where v is the solution of vt (x, t) = −An (−∆) u0 (x). αn 2 v(x, t), with initial condition v(x, 0) = Proof. As in the previous section, we have, using Fourier variables, ˆ − 1)n û(ξ, t). ût (ξ, t) = (−1)n−1 (J(ξ) Hence û(ξ, t) = e(−1) n−1 (J(ξ)−1) nt ˆ 98 uˆ0 (ξ). 3.2 Nonlocal Higher-Order Evolution Equations αn On the other hand, let v(x, t) be a solution of vt (x, t) = −An (−∆) 2 v(x, t), with the same initial data v(x, 0) = u0 (x). Solutions of this equation are understood in the sense that αn n v̂(ξ, t) = e−A |ξ| t uˆ0 (ξ). Hence in Fourier variables Z Z nt ˆ (−1)n−1 (J(ξ)−1) −An |ξ|αn t |û − v̂| (ξ, t) dξ = −e )uˆ0 (ξ) dξ (e d d R R Z αn nt n ˆ (−1)n−1 (J(ξ)−1) ≤ − e−A |ξ| t uˆ0 (ξ) dξ e |ξ|≥r(t) Z nt ˆ (−1)n−1 (J(ξ)−1) −An |ξ|αn t + e − e u ˆ (ξ) dξ 0 |ξ|<r(t) = I + II, where I and II denote the ﬁrst and the second integral respectively, and r(t) a nonnegative function in t. To get a bound for I we decompose it in two parts, Z Z nt ˆ −An |ξ|αn t (−1)n−1 (J(ξ)−1) I≤ uˆ0 (ξ) dξ + uˆ0 (ξ) dξ e e |ξ|≥r(t) |ξ|≥r(t) = I1 + I2 . First we consider I1 . Setting η = ξt1/(αn) and writing I1 in the new variable η we get, Z αn d n I1 ≤ kuˆ0 kL∞ (Rd ) e−A |η| t− αn dη, 1 |η|≥r(t)t αn and hence t d αn I1 ≤ kuˆ0 kL∞ (Rd ) Z 1 |η|≥r(t)t αn e−A n |η|αn t→∞ dη −→ 0 provided that we impose 1 t→∞ r(t)t αn −→ ∞. (3.7) To deal with I2 we have to use diﬀerent arguments for n even and n odd. Let us begin with the easier case of an even n. - n even - Using our hypotheses on J we get I2 ≤ Ce−t , t→∞ with r(t) −→ 0 and therefore d d t→∞ t αn I2 ≤ Ce−t t αn −→ 0. 99 3.2 Nonlocal Higher-Order Evolution Equations Now consider the case when n is odd. - n odd - From our hypotheses on J we have that Jˆ satisﬁes ˆ ≤ 1 − A |ξ|α + |ξ|α h(ξ), J(ξ) where h is bounded and h(ξ) → 0 as ξ → 0. Hence there exists D > 0 and a constant a such that ˆ ≤ 1 − D |ξ|α , for |ξ| ≤ a. J(ξ) ˆ Moreover, because J(ξ) ≤ 1 and J is a radial function, there exists a δ > 0 such that ˆ ≤ 1 − δ, J(ξ) for |ξ| ≥ a. Therefore I2 can be bounded by Z Z nt nt ˆ ˆ (−1)n−1 (J(ξ)−1) (−1)n−1 (J(ξ)−1) I2 ≤ uˆ0 (ξ) dξ + uˆ0 (ξ) dξ e e a≥|ξ|≥r(t) |ξ|≥a Z αn n n e−D |ξ| t dξ + Ce−δ t . ≤ kuˆ0 kL∞ (Rd ) a≥|ξ|≥r(t) Changing variables as before, η = ξt1/(αn) , we get Z αn d d n n αn e−D |η| dη + Ct αn e−δ t t I2 ≤ kuˆ0 kL∞ (Rd ) 1 1 at αn ≥|η|≥r(t)t αn Z αn d n n e−D |η| dη + Ct αn e−δ t → 0, ≤ kuˆ0 kL∞ (Rd ) 1 |η|≥r(t)t αn as t → ∞ if (3.7) holds. It remains only to estimate II. We proceed as follows Z n +An |ξ|αn ] ˆ −An |ξ|αn t t[(−1)n−1 (J(ξ)−1) II = e − 1 |uˆ0 (ξ)| dξ. e |ξ|<r(t) Applying the binomial formula and taking into account the two diﬀerent cases when n is even and odd we can conclude that Z αn d d n e−A |ξ| t t(|ξ|αn h(ξ) + K(|ξ|αk h(ξ)k )) dξ, t αn II ≤ Ct αn |ξ|<r(t) where K(|ξ|αk h(ξ)k ) is a polynomial in |ξ|α and h(ξ) with 0 < k ≤ n and provided that we impose t(r(t))αn h(r(t)) → 0 as t → ∞. (3.8) 100 3.2 Nonlocal Higher-Order Evolution Equations In this case we have Z d t αn II ≤ C 1 |η|<r(t)t αn e−A n |η|αn (|η|αn h(η/t1/(αn) ) + K(|η|αk h(η/t1/(αn) )k ) 1 t(αk)/(αn) ) dη. To show the convergence of II to zero we use dominated convergence. Because of our assumption on h we know that h(η/t1/(αn) ) → 0 as t → ∞ (note that clearly also h(η/t1/(αn) )k converges to zero for every k > 0). Further the integrand is dominated αn n by khkL∞ (Rd ) e−A |η| |η|αn , which belongs to L1 (Rd ). Combining this with our previous results we have that Z d d |û − v̂| (ξ, t) dξ ≤ t αn (I + II) → 0 as t → ∞, (3.9) t αn Rd provided we can ﬁnd a r(t) → 0 as t → ∞ which fulﬁlls both conditions (3.7) and (3.8). This is done in Lemma 3.2.7, which is postponed to just after the present proof. To conclude we only have to observe that the convergence of u − v in L∞ follows from the convergence of the Fourier transforms û(·, t) − v̂(·, t) → 0 in L1 . Indeed, from (3.9) we obtain Z d d |û − v̂| (ξ, t) dξ → 0, t → ∞, t αn max |u(x, t) − v(x, t)| ≤ t αn x Rd which completes the proof of the theorem. The following Lemma shows that there exists a function r(t) satisfying (3.7) and (3.8), as required in the proof of the previous theorem. Lemma 3.2.7. Given a real valued function h ∈ C(R) such that h(ρ) → 0 as ρ → 0 with h(ρ) > 0 for small ρ, there exists a function r with r(t) → 0 as t → ∞ which for α > 0 satisfies 1 lim r(t)t αn = ∞ t→∞ and lim t(r(t))αn h(r(t)) = 0. t→∞ Proof. For ﬁxed t large enough, we choose r(t) as a solution of 1 1 r(h(r)) 2αn = t− αn . (3.10) This equation deﬁnes a function r = r(t) which, by continuity arguments goes to zero as t tends to inﬁnity, satisfying also the additional asymptotic conditions in the lemma. Indeed, if there exists tn → ∞ with no solution of (3.10) for r ∈ (0, δ) then h(r) ≡ 0 in (0, δ), which is a contradiction to our assumption that h(r) > 0 for r small. 101 3.2 Nonlocal Higher-Order Evolution Equations As a consequence of Theorem 3.2.6, we obtain the following corollary which completes the results gathered in Theorem 3.2.3. Corollary 3.2.8. The asymptotic behavior of solutions of (3.1) is given by ku(., t)kL∞ (Rd ) ≤ C d t αn . Moreover, the asymptotic profile is given by d 1 lim max t αn u(yt αn , t) − ku0 kL1 (Rd ) GA (y) = 0, t→+∞ y αn n where GA (y) satisfies GˆA (ξ) = e−A |ξ| . Proof. From Theorem 3.2.6 we obtain that the asymptotic behavior is the same as the one for solutions of the evolution given by a power n of the fractional Laplacian. It is easy to check that the asymptotic behavior is in fact the one described in the statement of the corollary. In Fourier variables we have 1 lim v̂(ηt− αn , t) = lim e−A n |η|αn t→∞ −An |η|αn t→∞ =e −An |η|αn =e Therefore 1 uˆ0 (ηt− αn ) uˆ0 (0) ku0 kL1 (Rd ) . d 1 lim max t αn v(yt αn , t) − ku0 kL1 (Rd ) GA (y) = 0, t→+∞ y αn n where GA (y) satisﬁes GˆA (ξ) = e−A |ξ| . With similar arguments as in the proof of Theorem 3.2.6 one can prove that also the asymptotic behavior of the derivatives of solutions u of (3.1) is the same as the one for derivatives of solutions v of the evolution of a power n of the fractional Laplacian, assuming suﬃcient regularity of the solutions u of (3.1). Theorem 3.2.9. Let u be a solution of (3.1) with u0 ∈ W k,1 (Rd ), k ≤ αn and uˆ0 ∈ L1 (Rd ). Then, the asymptotic behavior of Dk u(x, t) is given by d+k lim t αn max Dk u(x, t) − Dk v(x, t) = 0, t→+∞ x where v is the solution of vt (x, t) = −An (−∆) u0 (x). 102 αn 2 v(x, t) with initial condition v(x, 0) = 3.2 Nonlocal Higher-Order Evolution Equations Proof. We begin again by transforming our problem for u and v into a problem for the corresponding Fourier transforms û and v̂. For this we consider k k v(ξ, t))∨ max Dk u(x, t) − Dk v(x, t) = max (D\ u(ξ, t))∨ − (D\ x ξ Z Z \ \ k k ≤ |ξ|k |û(ξ, t) − v̂(ξ, t)| dξ. D u(ξ, t) − D v(ξ, t) dξ = Rd Rd R Showing Rd |ξ|k |û(ξ, t) − v̂(ξ, t)| dξ → 0 as t → ∞ works analogue to the proof of Theorem 3.2.6. The additional term |ξ|k is always dominated by the exponential terms. 3.2.3 Scaling the Kernel In this section we show that the problem vt (x, t) = −An (−∆) αn 2 v(x, t) can be approx- imated by nonlocal problems like (3.1) when rescaled in an appropriate way. Proof of Theorem 3.2.4. The proof uses once more the explicit formula for the solutions in Fourier variables. We have, arguing exactly as before, uˆε (ξ, t) = e(−1) c ε (ξ)−1) n−1 (J εαn n t uˆ0 (ξ). and v̂(ξ, t) = e−A n |ξ|αn t uˆ0 (ξ). ˆ Now, we just observe that Jbε (ξ) = J(εξ) and therefore we obtain n ˆ (−1)n−1 (J(εξ)−1) n |ξ|αn t t −A αn ε |uˆε − v̂| (ξ, t) dξ = (e −e )uˆ0 (ξ) dξ Rd Rd Z n ˆ (−1)n−1 (J(εξ)−1) n |ξ|αn t t −A dξ e εαn −e ≤ kuˆ0 kL∞ (Rd ) Z Z |ξ|≥r(ε) + Z |ξ|<r(ε) ! n ˆ (−1)n−1 (J(εξ)−1) n |ξ|αn t t −A e dξ . εαn −e For t ∈ [0, T ] we can proceed as in the proof of Theorem 3.2.3 (Section 3.2.2) to obtain that Z |uˆε − v̂| (ξ, t) dξ → 0, ε → 0. max |uε (x, t) − v(x, t)| ≤ x Rd I leave the details to the reader. 103 Chapter 4 Numerical Solution of Higher-Order Inpainting Approaches One main challenge in inpainting with higher-order ﬂows is their eﬀective numerical implementation. For example the straightforward discretization of a fourth-order evolution equation may result in a strong restriction on the time step ∆t, i.e., ∆t ≤ O(∆x)4 where ∆x denotes the mesh size. Therefore high-order equations and possible nonconvex ﬂows require elaborate discretization schemes to guarantee stability and a fast convergence of the algorithm. For an overview of numerical methods for higher-order inpainting approaches see Section 1.3.3. In this chapter we present numerical algorithms that approximate solutions of the three inpainting approaches presented in Sections 2.1-2.3. Our ﬁrst approach are semiimplicit solvers, presented in Section 4.1. This section mainly follows the lines of [SB09] and is joint work with Andrea Bertozzi. We show the application of these schemes to Cahn-Hilliard inpainting (2.1), TV-H−1 inpainting (2.27), and inpainting with LCIS (2.42), and present rigorous estimates verifying, e.g., their unconditional stability in the sense that the numerical solution is uniformly bounded on a ﬁnite time interval. In Section 4.2 we present an alternative method to solve TV-H−1 inpainting. This method is based on a dual solver for TV minimization, introduced by Chambolle in [Ch04] for an L2 constraint, and generalized by the author in [Sc09] for an H −1 constraint. Section 4.2 is based on ideas of [Sc09] with emphasis on the inpainting application. In the last 104 4.1 Unconditionally Stable Solvers section we present a domain decomposition approach for TV minimization. This work is mainly based on results obtained in joint work with Massimo Fornasier in [FS07]. Domain decomposition methods are able to decrease the computational complexity of a numerical scheme by means of parallel computation. In particular for the higher-order TV-H−1 inpainting approach, the potential consequences with respect to computation time are of major interest. Numerical results for both TV-L2 - and TV-H−1 minimization are presented. 4.1 Unconditionally Stable Solvers In this section we discuss an eﬃcient semi-implicit approach presented in Eyre [Ey98] (also cf. citeVLR) called convexity splitting for its application to higher-order inpainting approaches. We consider the following problem: Let J ∈ C 2 (RN , R) be a smooth func- tional from RN into R, where N is the dimension of the data space. Let Ω be the spatial domain of the data space. Find u ∈ RN such that ( ut = −∇J(u) in Ω, u(., t = 0) = u0 in Ω, (4.1) with initial condition u0 ∈ RN . The basic idea of convexity splitting is to split the functional J into a convex and a concave part. In the semi implicit scheme the convex part is treated implicitly and the concave one explicitly in time. Under additional assumptions on (4.1) this discretization approach is unconditionally stable, of order 2 in time, and relatively easy to apply to a large range of variational problems. Moreover we shall see that the idea of convexity splitting can be applied to more general evolution equations, and in particular to those that do not follow a variational principle. The main focus of this section is to illustrate the application of the convexity splitting idea to the three fourth-order inpainting approaches (2.1), (2.27) and (2.42) presented in Sections 2.1-2.3. We show that this idea results in an unconditionally stable ﬁnite diﬀerence scheme and allows us to (approximately) compute strong solutions of the continuous problem. Moreover, we prove consistency of these schemes, and convergence to the exact solution under possible additional restrictions on the latter. For Cahn-Hilliard inpainting and TV-H−1 inpainting the developed convexity splitting schemes can be 105 4.1 Unconditionally Stable Solvers proven to be of order 1 in time. For inpainting with LCIS the numerical scheme is of order 2 in time. Notation In this section we discuss the numerical solution of evolutionary diﬀerential equations. Therefore we have to distinguish between the exact solution u of the continuous equation and the approximate solution U of the corresponding time-discrete numerical scheme. We write capital Uk for the solution of the time-discrete equation at time k∆t and small uk = u(k∆t) for a solution of the continuous inpainting equation at time k∆t with time step size ∆t. Let ek denote the discretization error given by ek = uk − Uk . In Subsection 4.1.1 u and Uk are vectors in RN , where N denotes the dimension of the data. In all other parts of this section u and Uk are assumed to be elements in L2 (Ω). Let J ∈ C 2 (H, R) denote a functional from a suitable Hilbert space H to R, and ∇J(u) its ﬁrst variation with respect to u. In the discrete setting H = RN . Finally, in the discrete setting, h., .i denotes the inner product on R with corresponding norm kuk2 = hu, ui. In the continuous setting we stick to the notation speciﬁed in the preﬁx of the thesis. 4.1.1 The Convexity Splitting Idea Convexity splitting was originally proposed to solve energy minimizing equations. Nevertheless we will see that it can also be applied to more general PDEs, such as (2.1) and (2.27), which do not fulﬁll a variational principle. Convexity splitting methods, although known under diﬀerent names, have a long tradition in several parts of numerical analysis. For example Barrett, Blowley, and Garcke used the idea of convexity splitting in [BB97, Equation (3.42)] to approximate the solution of a phase separation model with ﬁnite elements. In [ES02, Equation (5.4)] a ﬁnite diﬀerence scheme for second-order parabolic equations is presented which also uses the convexity splitting idea. A discussion on convexity splitting in the context of more general optimization problems can be found in [YR03, Chapter 2.]. First we would like to introduce the notion of gradient ﬂows and the application of convexity splitting methods in this context. To do so we follow the work of Eyre 106 4.1 Unconditionally Stable Solvers [Ey98]. We consider equation (4.1). If J satisﬁes (i) J(u) ≥ 0, ∀u ∈ RN (4.2) (ii) J(u) → ∞ as kuk → ∞ (4.3) (iii) hJ(∇J)(u)u, ui ≥ λ ∈ R, ∀u ∈ RN (4.4) then we refer to (4.1) as a gradient system and to its solutions as gradient flows. Here J(∇J)(u) denotes the Jacobian of ∇J in u. All gradient systems fulﬁll the dissipation property, i.e., dJ(u) = − k∇J(u)k2 dt and therefore J(u(t)) ≤ J(u0 ) for all t ≥ 0. If J(u) is convex, i.e., λ > 0 in (4.2), then the system has a single equilibrium. In this case unconditionally stable and uniquely solvable numerical schemes exist (cf. [SH94]). If J(u) is not convex, i.e., λ < 0, multiple minimizers may exist and the gradient ﬂow can possibly expand in u(t). The stability of an explicit gradient descent algorithm, i.e., Uk+1 = Uk − ∆t∇J(Uk ), in this case may require extremely small time steps, depending of course on the functional J. Therefore the development of stable and eﬃcient discretizations for non-convex functionals J is highly desirable. The basic idea of convexity splitting is to write the functional J as J(u) = Jc (u) − Je (u), (4.5) where Jo ∈ C 2 (RN , R) and Jo (u) is strictly convex for all u ∈ RN , o ∈ {c, e}. (4.6) The semi-implicit discretization of (4.1) is then given by Uk+1 − Uk = −∆t (∇Jc (Uk+1 ) − ∇Je (Uk )) , (4.7) where U0 = u0 . Remark 4.1.1. We want to anticipate that the setting of Eyre, and hence the subsequent presentation of convexity splitting, is a purely discrete one. Nevertheless it actually holds in a more general framework, i.e., for more general gradient flows. In the case of an L2 gradient flow for example, the Jacobian J of the discrete functional J just has to be replaced by the second variation of the continuous functional J in L2 (Ω). 107 4.1 Unconditionally Stable Solvers In the following we will show that convexity splitting can be applied to the inpainting approaches (2.1), (2.27), and (2.42) and produces unconditionally gradient stable or unconditionally stable numerical schemes. Let us ﬁrst deﬁne what unconditionally gradient stable and unconditionally stable schemes are. Definition 4.1.2. [Ey98] A one-step numerical integration scheme is unconditionally gradient stable if there exists a function J(.) : RN → R such that, for all ∆t > 0 and for all initial data: (i) J(U ) ≥ 0 for all U ∈ RN (ii) J(U ) → ∞ as kU k → ∞ (iii) J(Uk+1 ) ≤ J(Uk ) for all Uk ∈ RN (iv) If J(Uk ) = J(U0 ) for all k ≥ 0 then U0 is a zero of ∇J for (4.1) and (4.2). Now, Cahn-Hilliard inpainting (2.1) and TV-H−1 inpainting (2.27), are not given by gradient ﬂows. Hence, in the context of these inpainting models the meaning of unconditional stability has to be redeﬁned. Namely, in the case of an evolution equation which does not follow a gradient ﬂow, a corresponding discrete time stepping scheme is said to be unconditionally stable if solutions of the diﬀerence equation are bounded within a ﬁnite time interval, independently from the step size ∆t. Definition 4.1.3. Let u be an element of a suitable function space defined on Ω×[0, T ], with Ω ⊂ R2 open and bounded, and T > 0. Let further F be a real valued function and ut = F (u, Dα u) be a partial differential equation with space derivatives Dα u, α = 1, . . . , 4. A corresponding discrete time stepping method Uk+1 = Uk + ∆tFk , (4.8) where Fk is a suitable approximation of F in Uk and Uk+1 , is unconditionally stable, if all solutions of (4.8) are bounded for all ∆t > 0 and all k such that k∆t ≤ T . We start with a theorem proved by Eyre in [Ey98]. Theorem 4.1.4. [Ey98, Theorem 1] Let J satisfy (4.2), and Jc and Je satisfy (4.5)(4.6). If Je (u) additionally satisfies hJ(∇Je )(u)u, ui ≥ −λ 108 (4.9) 4.1 Unconditionally Stable Solvers when λ < 0 in (4.2)(iii), then for any initial condition, the numerical scheme (4.7) is consistent with (4.1), gradient stable for all ∆t > 0, and possesses a unique solution for each time step. The local truncation error for each step is τk = (∆t)2 (J(∇Jc (û)) + J(∇Je (û))) ∇J(u(ξ)), 2 for some ξ ∈ (k∆t, (k +1)∆t) and for some û in the parallelopiped with opposite vertices at Uk and Uk+1 . Remark 4.1.5. Condition (4.9) in Theorem 4.1.4 is equivalent to the requirement that all the eigenvalues of J(∇Je ) dominate the largest eigenvalue −λ of −J(∇J), i.e., (4.9) (4.2) hJ(∇Je )(u)u, ui ≥ −λ ≥ h−J(∇J)(u)u, ui for all u ∈ RN , i.e., λ̂ ≥ |λ| , for all eigenvalues λ̂ > 0 of Je . (4.10) In the following we apply the idea of convexity splitting to our three inpainting models (2.1), (2.27), and (2.42). For this we change from the discrete setting to the continuous setting, i.e., considering functions u in a suitable Hilbert space instead of vectors u in RN . Although the ﬁrst two of these inpainting approaches, i.e., CahnHilliard inpainting and TV-H−1 inpainting, are not given by gradient ﬂows, we shall see that the resulting numerical schemes are still unconditionally stable (in the sense of Deﬁnition 4.1.3) and therefore suitable to solve them accurately and reasonably fast. For inpainting with LCIS (2.42) the results of Eyre can be directly applied, even in the continuous setting, cf. Remark 4.1.1. Nevertheless, also for this case, we additionally present a rigorous analysis, similar to the one done for Cahn-Hilliard- and TV-H−1 inpainting. 4.1.2 Cahn-Hilliard Inpainting In this section we show that the application of convexity splitting to Cahn-Hilliard inpainting (2.1) yields a consistent, unconditionally stable and convergent numerical scheme. Recall that the inpainted version u(x) of f (x) is constructed by following the evolution equation 1 ′ ut = ∆ −ǫ∆u + F (u) + λ(f − u). ǫ 109 4.1 Unconditionally Stable Solvers The application of convexity splitting to Cahn-Hilliard inpainting (2.1) was originally introduced in [BEG07a]. The numerical results presented there already suggested the usefulness of this scheme. Although the authors did not analyze the scheme rigorously, they concluded that based on their numerical results, the scheme is unconditionally stable. In the following we will present this numerical scheme and derive some additional properties based on its rigorous analysis. The original Cahn-Hilliard equation is a gradient ﬂow in H −1 for the energy Z ǫ 1 2 J1 (u) = |∇u| + F (u) dx, ǫ Ω 2 while the ﬁtting term in (2.1) can be derived from a gradient ﬂow in L2 for the energy Z 1 λ(f − u)2 dx. J2 (u) = 2 Ω Note that equation (2.1) as a whole does not result in a gradient system anymore. Hence, we apply the convexity splitting discussed in Section 4.1.1 to both functionals J1 and J2 separately. Namely we split J1 in J1 = J1c − J1e with Z Z ǫ 1 C1 2 C1 2 2 J1c (u) = − F (u) + |∇u| + |u| dx, J1e (u) = |u| dx. 2 ǫ 2 Ω 2 Ω A possible splitting for J2 is J2 = J2c − J2e with Z Z C2 2 C2 2 1 1 −λ(f − u)2 + |u| dx, J2e (u) = |u| dx. J2c (u) = 2 Ω 2 2 Ω 2 To make sure that J1c , J1e and J2c , J2e are strictly convex the constants C1 and C2 have to be chosen such that C1 > 1ǫ , C2 > λ0 , compare [BEG07b]. Then the resulting discrete time-stepping scheme for an initial condition U0 = u0 is given by Uk+1 − Uk = −∇H −1 (J1c (Uk+1 ) − J1e (Uk )) − ∇L2 (J2c (Uk+1 ) − J2e (Uk )), ∆t where ∇H −1 and ∇L2 represent gradient descent with respect to the H −1 and the L2 inner product respectively. This translates to a numerical scheme of the form Uk+1 − Uk 1 + ǫ∆∆Uk+1 − C1 ∆Uk+1 + C2 Uk+1 = ∆F ′ (Uk ) − C1 ∆Uk ∆t ǫ + λ(f − Uk ) + C2 Uk , in Ω. (4.11) 110 4.1 Unconditionally Stable Solvers We imply Neumann boundary conditions on ∂Ω, i.e., ∂∆Uk+1 ∂Uk+1 = = 0, ∂ν ∂ν on ∂Ω, and intend to compute Uk+1 in (4.11) in the spectral domain using the discrete cosine transform (DCT). Note that in order to derive Neumann boundary conditions for (4.11) the standard space H −1 as deﬁned in Appendix A.4 has to be replaced by H∂−1 as deﬁned in Section 2.1.3. Spectral methods, like the DCT, are a classical tool for the discretization of Laplacian operators. In the case of a Laplacian operator, their main advantage is that the resulting matrix is of diagonal type, which allows fast computations. In addition, fast numerical methods for the Fourier transform exist. Let Û be the DCT of U and λi the eigenvalues of the discrete Laplacian with Neumann boundary conditions. Then equation (4.11) in Û reads Ûk+1 (i, j) = 1 (1 − C1 ∆t( ∆x 2 λi + 1 λ ) ∆y 2 j + C2 ∆t)Ûk (i, j) ci,j + ∆t \ ′ ǫ ∆F (Uk )(i, j) + ∆tλ(f\ − Uk ) ci,j , where ci,j is deﬁned as ci,j = 1 + C2 ∆t + ǫ∆t 1 1 λj 2 λi + ∆x ∆y 2 2 − C1 ∆t 1 1 λi + λj . ∆x2 ∆y 2 Rigorous Estimates for the Scheme From Theorem 4.1.4 we know that (at least in the spatially discrete framework) the convexity splitting scheme (4.5)-(4.7) is unconditionally stable, i.e., separate numerical schemes for the gradient ﬂows of the energies J1 (u) and J2 (u) are not increasing for all ∆t > 0. But this does not guarantee that our numerical scheme (4.11) is unconditionally stable, since it combines the ﬂows of two energies. In this section we shall analyze the scheme in more detail and derive some rigorous estimates for its solutions. In particular we will show that the scheme (4.11) is unconditionally stable in the sense of Deﬁnition 4.1.3. Our results are summarized in the following theorem. 111 4.1 Unconditionally Stable Solvers Theorem 4.1.6. Let u be the exact solution of (2.1) and uk = u(k∆t) the exact solution at time k∆t, for a time step ∆t > 0 and k ∈ N. Let further Uk be the kth iterate of (4.11) with constants C1 > 1/ǫ, C2 > λ0 . Then the following statements are true: (i) Under the assumption that kutt k−1 , k∇∆ut k2 and kut k2 are bounded, the numerical scheme (4.11) is consistent with the continuous equation (2.1) and of order 1 in time. Under the additional assumption that F ′′ (Uk−1 ) ≤ K (4.12) for a nonnegative constant K, we further have (ii) The solution sequence Uk is bounded on a finite time interval [0, T ], for all ∆t > 0. (iii) The discretization error ek , given by ek = uk − Uk , converges to zero as ∆t → 0. Remark 4.1.7. Note that our assumptions for the consistency of the numerical scheme, only hold if the time derivative of the solution of the continuous equation (2.1) is uniformly bounded. This is true, for smooth and bounded solutions of the equation. Further, since we are interested in bounded solutions Uk of the discrete equation (4.11), it is natural to assume (4.12), i.e., that the nonlinearity F ′′ in the previous time step ∆(k − 1) is bounded. The proof of Theorem 4.1.6 is organized in the following three Propositions 4.1.84.1.10. Proposition 4.1.8. (Consistency (i)) Under the same assumptions as in Theorem 4.1.6 and in particular under the assumption that kutt k−1 , k∇∆ut k2 and kut k2 are bounded, the numerical scheme (4.11) is consistent with the continuous equation (2.1) with local truncation error kτk k−1 = O(∆t). Proof. Let Fk (U ) = 0 represent the diﬀerence equation approximating the PDE at time k∆t. If the discrete solution U is replaced by the exact solution u of (2.1), the value τk = Fk (u) is the local truncation error deﬁned over a time step. Then τk = τk1 + τk2 , 112 4.1 Unconditionally Stable Solvers with uk+1 − uk − ut (k∆t) ∆t τk2 = ǫ∆∆(uk+1 − uk ) − C1 ∆(uk+1 − uk ) + C2 (uk+1 − uk ) uk+1 − uk uk+1 − uk uk+1 − uk = ǫ∆t∆2 − C1 ∆t∆2 + C2 ∆t , ∆t ∆t ∆t τk1 = i.e., 1 uk+1 − uk + ǫ∆2 uk+1 − ∆F ′ (uk ) − λ(f − uk ) − C1 ∆(uk+1 − uk ) + C2 (uk+1 − uk ). ∆t ǫ (4.13) Using standard Taylor series arguments and assuming that kutt k−1 , k∇∆ut k2 and kut k2 are bounded we deduce that kτk k−1 = O(∆t). (4.14) τk = Proposition 4.1.9. (Unconditional stability (ii)) Under the same assumptions as in Theorem 4.1.6 and in particular assuming that (4.12) holds, the solution sequence Uk with k∆t ≤ T for T > 0 fixed, fulfills for every ∆t > 0 k∇Uk k22 + ∆tK1 k∆Uk k22 ≤ eK2 T k∇U0 k22 + ∆tC1 k∆U0 k22 + ∆t T C(Ω, D, λ0 , f ) , (4.15) for suitable constants K1 and K2 , and constant C depending on Ω, D, λ0 , f only. This gives boundedness of the solution sequence on [0, T ]. Proof. We consider our discrete model Uk+1 − Uk 1 +ǫ∆∆Uk+1 −C1 ∆Uk+1 +C2 Uk+1 = ∆F ′ (Uk )−C1 ∆Uk +λ(f −Uk )+C2 Uk , ∆t ǫ multiply the equation with −∆Uk+1 and integrate over Ω. We obtain 1 k∇Uk+1 k22 − h∇Uk , ∇Uk+1 i2 + ǫ k∇∆Uk+1 k22 + C1 k∆Uk+1 k22 + C2 k∇Uk+1 k22 ∆t 1 ′′ F (Uk )∇Uk , ∇∆Uk+1 2 + C1 h∆Uk , ∆Uk+1 i2 = ǫ + h∇λ(f − Uk ), ∇Uk+1 i2 + C2 h∇Uk , ∇Uk+1 i2 . Using Cauchy’s inequality we obtain 1 k∇Uk+1 k22 − k∇Uk k22 + ǫ k∇∆Uk+1 k22 + C1 k∆Uk+1 k22 + C2 k∇Uk+1 k22 2∆t 1 F ′′ (Uk )∇Uk 2 + δ k∇∆Uk+1 k2 + C1 k∆Uk k2 + C1 k∆Uk+1 k2 ≤ 2 2 2 2 2ǫδ 2ǫ 2 2 C2 1 1 C2 k∇Uk k22 + k∇Uk+1 k22 + k∇λ(f − Uk )k22 + k∇Uk+1 k22 . + 2 2 2 2 113 4.1 Unconditionally Stable Solvers Using the estimate k∇λ(f − Uk )k22 ≤ 2λ20 k∇Uk k22 + C(Ω, D, λ0 , f ) and reordering the terms we obtain 1 C2 1 δ C1 2 2 + − k∆Uk+1 k2 + ǫ − k∇Uk+1 k2 + k∇∆Uk+1 k22 2∆t 2 2 2 2ǫ C C2 1 1 F ′′ (Uk )∇Uk 2 + 1 k∆Uk k2 + C(Ω, D, λ0 , f ). + + λ20 k∇Uk k22 + ≤ 2 2 2∆t 2 2ǫδ 2 By choosing δ = 2ǫ2 , the third term on the left side of the inequality is zero. Because of assumption (4.12) we obtain the following bound on the right side of the inequality ′′ F (Uk )∇Uk 2 ≤ K 2 k∇Uk k2 2 2 and we have 1 C2 1 C1 + − k∆Uk+1 k22 k∇Uk+1 k22 + 2∆t 2 2 2 1 C2 K2 C1 2 ≤ + + λ0 + 3 k∇Uk k22 + k∆Uk k22 + C(Ω, D, λ0 , f ). 2∆t 2 4ǫ 2 Now we multiply the above inequality by 2∆t and deﬁne C̃ = 1 + ∆t(C2 − 1), K2 C̃˜ = 1 + ∆t(C2 + 2λ20 + 3 ). 2ǫ Since C2 is chosen greater than λ0 > 1, the ﬁrst coeﬃcient C̃ is positive and we can divide the inequality by it. We obtain k∇Uk+1 k22 C1 C̃˜ C1 2 + ∆t k∆Uk+1 k2 ≤ k∇Uk k22 + ∆t k∆Uk k22 + ∆tC(Ω, D, λ0 , f ), C̃ C̃ C̃ where we updated the constant C(Ω, D, λ0 , f ) by C(Ω, D, λ0 , f )/C̃. ˜ ≥ 1, we can multiply the second term on the right side of the inequality by Since C̃ C̃ this quotient to obtain k∇Uk+1 k22 C1 C̃˜ C1 2 2 2 + ∆t k∆Uk+1 k2 ≤ k∆Uk k2 + ∆tC(Ω, D, λ0 , f ). k∇Uk k2 + ∆t C̃ C̃ C̃ 114 4.1 Unconditionally Stable Solvers We deduce by induction that C1 k∇Uk k22 + ∆t k∆Uk k22 ≤ C̃ C̃˜ C̃ !k k∇U0 k22 +∆t C̃˜ C̃ k−1 X i=0 = (1 + K2 ∆t)k (1 + K1 ∆t)k +∆t !i C1 k∆U0 k22 + ∆t C̃ C(Ω, D, λ0 , f ) C1 2 2 k∇U0 k2 + ∆t k∆U0 k2 C̃ k−1 X (1 + K2 ∆t)i i=0 (1 + K1 ∆t)i C(Ω, D, λ0 , f ). For k∆t ≤ T we have k∇Uk k22 C1 C1 2 2 2 (K2 −K1 )T k∆Uk k2 ≤ e k∇U0 k2 + ∆t k∆U0 k2 + ∆t C̃ C̃ +∆tT e(K2 −K1 )T C(Ω, D, λ0 , f ) C1 (K2 −K1 )T = e k∇U0 k22 + ∆t k∆U0 k22 C̃ +∆t T C(Ω, D, λ0 , f )) , which gives boundedness of the solution sequence on [0, T ] for any T > 0 assuming that (4.12) holds. The convergence of the discrete solution to the continuous one as the time step ∆t → 0 is veriﬁed in the following proposition. Proposition 4.1.10. (Convergence (iii)) Under the same assumptions as in Theorem 4.1.6 and in particular under assumption (4.12) the discretization error ek fulfills, for suitable constants C, C1 , K1 , K2 , k∇ek k22 + ∆t C1 k∆ek k22 ≤ T ∆te4(K1 +K2 )T · C, C̃ (4.16) for k∆t ≤ T and a fixed T > 0. Proof. Let us follow the lines of the consistency proof in (4.13). Then the discretization 115 4.1 Unconditionally Stable Solvers error ek satisﬁes ek+1 − ek + ǫ∆2 ek+1 − C1 ∆ek+1 + C2 ek+1 ∆t 1 1 = (uk+1 − uk ) − (Uk+1 − Uk ) + ǫ∆2 uk+1 − ǫ∆2 Uk+1 ∆t ∆t −C1 ∆uk+1 + C1 ∆Uk+1 + C2 uk+1 − C2 Uk+1 1 ′ = − ∆F (Uk ) − C1 ∆Uk + λ(f − Uk ) + C2 Uk ǫ 1 ′ + ∆F (uk ) + λ(f − uk ) − C1 ∆uk + C2 uk + τk ǫ 1 ′ ′ = − ∆(F (Uk ) − F (uk )) − C1 ∆(Uk − uk ) + C2 (Uk − uk ) − λ(Uk − uk ) + τk . ǫ Multiplication with −∆ek+1 leads to 1 h∇(ek+1 − ek ), ∇ek+1 i2 + ǫ k∇∆ek+1 k22 + C1 k∆ek+1 k22 + C2 k∇ek+1 k22 ∆t 1 = ∆(F ′ (Uk ) − F ′ (uk )), ∆ek+1 2 − C1 h∆(Uk − uk ), ∆ek+1 i2 ǫ + h∇λ(Uk − uk ), ∇ek+1 i2 − C2 h∇(Uk − uk ), ∇ek+1 i2 + ∇∆−1 τk , ∇∆ek+1 2 . Further, because we obtain 1 1 (k∇ek+1 k22 − k∇ek k22 ), k∇ek+1 k22 − h∇ek , ∇ek+1 i2 ≥ ∆t 2∆t 1 (k∇ek+1 k22 − k∇ek k22 ) + ǫ k∇∆ek+1 k22 + C1 k∆ek+1 k22 + C2 k∇ek+1 k22 2∆t 1 ∆(F ′ (Uk ) − F ′ (uk )), ∆ek+1 2 + C1 h∆ek , ∆ek+1 i2 − h∇λek , ∇ek+1 i2 ≤ ǫ + C2 h∇ek , ∇ek+1 i2 + ∇∆−1 τk , ∇∆ek+1 2 . Applying Cauchy’s inequality leads to 1 (k∇ek+1 k22 − k∇ek k22 ) + ǫ k∇∆ek+1 k22 + C1 k∆ek+1 k22 + C2 k∇ek+1 k22 2∆t C1 1 k∆ek k22 + C1 δ1 k∆ek+1 k22 ≤ − (F ′′ (Uk )∇Uk − F ′′ (uk )∇uk ), ∇∆ek+1 2 + ǫ δ1 λ2 C2 + 0 k∇ek k22 + δ3 k∇ek+1 k22 + k∇ek k22 + C2 δ2 k∇ek+1 k22 δ3 δ2 1 + kτk k2−1 + δ4 k∇∆ek+1 k22 . δ4 116 4.1 Unconditionally Stable Solvers Since (4.12) holds, we obtain 1 ′′ (F (Uk )∇Uk − F ′′ (uk )∇uk ), ∇∆ek+1 2 ǫ 1 F ′′ (Uk )∇Uk − F ′′ (uk )∇uk )2 + δ5 k∇∆ek+1 k2 2 2 2ǫδ5 2ǫ K δ5 K k∇Uk k22 + k∇uk k22 + k∇∆ek+1 k22 , ǫδ5 ǫδ5 2ǫ − ≤ ≤ and therefore 1 + C2 (1 − δ2 ) − δ3 k∇ek+1 k22 + C1 (1 − δ1 ) k∆ek+1 k22 2∆t δ5 k∇∆ek+1 k22 + ǫ − δ4 − 2ǫ 1 λ20 C2 1 K C1 ≤ k∇Uk k22 + k∇uk k22 . + + k∆ek k22 + kτk k2−1 + k∇ek k22 + 2∆t δ3 δ2 δ1 δ4 ǫδ5 Next we choose δ1 = 1/2 and multiply the inequality with 2∆t (1 + 2∆t(C2 (1 − δ2 ) − δ3 )) k∇ek+1 k22 + ∆tC1 k∆ek+1 k22 δ5 + 2∆t ǫ − δ4 − k∇∆ek+1 k22 ǫ 2∆t λ20 C2 ) k∇ek k22 + 4∆tC1 k∆ek k22 + kτk k2−1 ≤ 1 + 2∆t( + δ3 δ2 δ4 2K 2 2 k∇Uk k2 + k∇uk k2 . + ∆t ǫ Now choosing all δs such that the coeﬃcients of all terms in the inequality are nonnegative and estimating the last term on the left side from below by zero we get k∇ek+1 k22 + ∆t where C̃˜ C1 2∆t C1 k∆ek+1 k22 ≤ k∇ek k22 + 4∆t k∆ek k22 + kτk k2−1 C̃ C̃ C̃ δ4 C̃ 2K + ∆t k∇Uk k22 + k∇uk k22 , ǫC̃ C̃ = 1 + 2∆t(C2 (1 − δ2 ) − δ3 ), λ2 C2 ). C̃˜ = 1 + 2∆t( 0 + δ3 δ2 Further, k∇ek+1 k22 C1 C̃˜ C1 2 2 2 + ∆t k∇ek k2 + ∆t k∆ek+1 k2 ≤ 4 k∆ek k2 C̃ C̃ C̃ 2 2K 2 2 2 + ∆t k∇Uk k2 + k∇uk k2 kτk k−1 + . δ4 C̃ ǫC̃ 117 4.1 Unconditionally Stable Solvers In addition we apply the consistency result (4.14) and the uniform bound (4.15) to the above inequality and obtain k∇ek+1 k22 C̃˜ C1 C1 2 2 2 + ∆t k∆ek+1 k2 ≤ 4 k∆ek k2 k∇ek k2 + ∆t C̃ C̃ C̃ 2 2K 2 2 + ∆t C2 + k∇uk k2 (∆t) + . δ4 C̃ ǫC̃ To proceed we assume for now that the exact solution is uniformly bounded in H 1 (Ω), i.e., ∃C3 > 0 such that k∇uk k2 ≤ C3 for all k∆t < T. (4.17) This assumption will be proven in Lemma 4.1.11 just after the end of this proof. Now, implementing (4.17) in our computation we have k∇ek+1 k22 C1 C̃˜ C1 2 2 2 + ∆t k∆ek+1 k2 ≤ 4 k∆ek k2 k∇ek k2 + ∆t C̃ C̃ C̃ 2K 2 (∆t)2 + (C2 + C3 ) . + ∆t δ4 C̃ ǫC̃ By induction we deduce that C1 k∇ek k22 + ∆t k∆ek k22 ≤ C̃ !k C1 2 k∆e0 k2 + ∆t C̃ !i k−1 X C̃˜ 2 2K 2 4 (∆t) + (C2 + C3 ) . (4.18) + ∆t C̃ δ4 C̃ ǫC̃ i=0 C̃˜ 4 C̃ k∇e0 k22 Since e0 = U0 − u0 = 0, the ﬁrst term on the right-hand side of the above inequality vanishes. For k∆t ≤ T we ﬁnally have k∇ek k22 + ∆t C1 k∆ek k22 ≤ T ∆te4(K1 +K2 )T · C. C̃ From [BEG07a, BEG07b] we know that the solution uk to the continuous equation globally exists and is uniformly bounded in L2 (Ω). Next we show that assumption (4.17) holds. Lemma 4.1.11. Let uk be the exact solution of (2.1) at time t = k∆t and let T > 0. Then there exists a constant C > 0 such that k∇uk k2 ≤ C for all k∆t < T . 118 4.1 Unconditionally Stable Solvers Proof. Let K(u) = −ǫ∆u + 1ǫ F ′ (u). We multiply the continuous evolution equation (2.1) with K(u) and obtain hut , K(u)i2 = h∆K(u), K(u)i2 + hλ(f − u), K(u)i2 . Let us further deﬁne ǫ J(u) := 2 Z 1 |∇u| dx + ǫ Ω 2 Z F (u) dx. Ω Then we have 1 ′ hut , K(u)i2 = ut , −ǫ∆u + F (u) ǫ 2 1 ′ = h∇ut , ǫ∇ui2 + ut , F (u) ǫ 2 d = J(u), dt since u satisﬁes Neumann boundary conditions. Therefore we get Z d 1 ′ 2 J(u) = − |∇K(u)| dx + hλ(f − u), −ǫ∆ui2 + λ(f − u), F (u) . dt ǫ Ω 2 (4.19) Since F (u) is bounded from below, we only have to show that J(u) is uniformly bounded on [0, T ], and we automatically have that |∇u| is uniformly bounded on [0, T ]. We start with the last term, and recall the following bounds on F ′ (u) (cf. [Te88]): There exist positive constants C1 , C2 such that F ′ (s)s ≥ C1 s2 − C2 , ∀s ∈ R and, for every δ > 0, there exists a constant C3 such that |F ′ (s)| ≤ δC1 s2 + C3 (δ), ∀s ∈ R. Use the last two estimates to obtain the following Z Z 1 ′ λ0 λ0 ′ λ(f − u), F (u) = F (u)f dx − F ′ (u)u dx ǫ ǫ ǫ Ω\D Ω\D 2 Z Z λ0 λ0 C1 λ0 C2 |Ω \ D| ≤ |F ′ (u)| dx · kf kL∞ (Ω) − u2 dx + ǫ Ω\D ǫ ǫ Ω\D ! Z Z C3 (δ) |Ω \ D| λ0 C1 C1 u2 dx + u2 dx − ≤ λ0 C(f, Ω) δ ǫ Ω\D ǫ ǫ Ω\D λ0 C2 |Ω \ D| ǫ Z λ0 C1 u2 dx + C(λ0 , ǫ, δ, Ω, D, f ), (1 − δC(f, Ω)) ≤− ǫ Ω\D + 119 4.1 Unconditionally Stable Solvers where we choose δ < 1/C(f, Ω). Therefore integrating (4.19) over the time interval [0, T ] results in Z T 0 d J(u(t)) dt ≤ dt Z T Z Z 2 T hλ(f − u), −ǫ∆ui2 dt |∇K(u)| dx dt + 0 Ω 0 Z TZ λ0 C1 − u2 dx dt + T · C(λ0 , ǫ, δ, Ω, D, f ). (1 − δC(f, Ω)) ǫ 0 Ω\D − Next we consider the second term on the right side of the last inequality. From Theorem 4.1 in [BEG07a] we know that a solution u of (2.1) is an element in L2 (0, T ; H 2 (Ω)) for all T > 0. Hence ∆u ∈ L2 (0, T ; L2 (Ω)) and the second term is bounded by a constant depending on T . Consequently for each 0 ≤ t ≤ T we get, J(u(t)) ≤ J(u(0)) + C(T ) + T · C(λ0 , ǫ, δ, Ω, D, f ) # Z Z T "Z λ0 C1 2 2 (1 − δC(f, Ω)) u dx dt, − |∇K(u)| dx + ǫ Ω\D 0 Ω and with this, for a ﬁxed T > 0, that |∇u| is uniformly bounded in [0, T ]. 4.1.3 TV-H−1 Inpainting In this section we discuss convexity splitting for TV-H−1 inpainting (2.27). To avoid numerical diﬃculties we approximate an element p in the subdiﬀerential of the total ∇u ), the square root reguvariation functional T V (u) by a smoothed version of ∇ · ( |∇u| larization for instance. With the latter regularization the smoothed version of (2.27) reads ut = −∆∇ · ∇u p |∇u|2 + δ 2 ! + λ(f − u), (4.20) with 0 < δ ≪ 1. In contrast to its second-order analogue, the well-posedness of (2.27) ∇u strongly depends on the smoothing used for ∇ · ( |∇u| ). In fact there are smoothing functions for which (2.27) produces singularities in ﬁnite time. This is caused by the lack of maximum principles which in the second-order case guarantee the well-posedness for all smooth monotone regularizations. In [BGOV05] the authors considered (2.27) with λ = λ0 in all of Ω, i.e., the fourth-order analogue to TV-L2 denoising, which was originally introduced in [OSV03]. They proved well-posedness in one space dimension for a set of smooth monotone regularizations which include the square root smoothing ! ux p (4.21) |ux |2 + δ 2 x 120 4.1 Unconditionally Stable Solvers and, the arctan regularization 2 arctan(ux /δ) , π x (4.22) for 0 < δ ≪ 1. The behavior of the fourth-order PDE in one dimension is also rel- evant for two-dimensional images since a lot of structure involves edges which are one-dimensional objects. In two dimensions similar results are much more diﬃcult to obtain, since energy estimates and the Sobolev lemma involved in its proof might not hold in higher dimensions anymore. In the following we will present the convexity splitting method applied to (2.27) for both the square root and the arctan regularization. Similarly to the convexity splitting for Cahn-Hilliard inpainting, we propose the following splitting for the TVH−1 inpainting equation. The regularizing term in (2.27) can be modeled by a gradient ﬂow in H −1 of the energy J1 (u) = Z Ω |∇u| dx, where |∇u| is replaced by its regularized version, e.g., in J1c − J1e with J1c (u) = Z Ω C1 |∇u|2 dx, and J1e (u) = 2 Z Ω q |∇u|2 + δ 2 , δ > 0. We split J1 −|∇u| + C1 |∇u|2 dx. 2 The ﬁtting term is split into J2 = J2c − J2e analogous to the Cahn-Hilliard inpainting. The resulting time-stepping scheme is given by ∇Uk Uk+1 − Uk + C1 ∆∆Uk+1 + C2 Uk+1 = C1 ∆∆Uk − ∆ ∇ · ( ) + C2 Uk + λ(f − Uk ). ∆t |∇Uk | (4.23) We assume that Uk+1 satisﬁes zero Neumann boundary conditions and use the DCT to solve (4.23). Note again that in the case of Neumann boundary conditions the standard space H −1 as deﬁned in Appendix A.4 has to be replaced by H∂−1 as deﬁned in Section 2.1.3. The constants C1 and C2 have to be chosen such that J1c , J1e , J2c , J2e are all strictly convex. In the following we will demonstrate how to compute the appropriate constants. Let us consider C1 ﬁrst. The functional J1c is strictly convex for all C1 > 0. The choice of C1 for the convexity of J1e depends on the regularization of the total variation we are using. We use the square regularization (4.21), i.e., instead of |∇u| we have Z p G(|∇u|) dx, with G(s) = s2 + δ 2 . 121 4.1 Unconditionally Stable Solvers Setting y = |∇u| we have to choose C1 such that condition for the second derivative gives us that δ2 (δ 2 +y 2 )3/2 is convex. The convexity δ2 1 ⇐⇒ C1 > , 3/2 2 2 δ (δ + y ) C1 > G′′ (y) ⇐⇒ C1 > is suﬃcient as C1 2 2 y − G(y) has its maximum value at y = 0. Next we would like to compare this with the arctan regularization (4.22), i.e., replacing ∇u |∇u| by π2 arctan( ∇u δ ), as proposed in [BGOV05]. Here the convexity condition for the second derivative reads s d 2 arctan( ) > 0. C1 ± ds π δ The ± sign results from the absent absolute value in the regularization deﬁnition. We obtain 2 1 > 0. π δ(1 + s2 /δ 2 ) C1 ± The inequality with a plus sign instead of ± is true for all constants C1 > 0. In the other case we obtain C1 > which is fulﬁlled for all s ∈ R if C1 > δ 2 , π δ 2 + s2 2 δπ . Note that this condition is almost the same as in the case of the square regularization. Now we consider J2 = J2c − J2e . The functional J2c is strictly convex if C2 > 0. For the convexity of J2e we rewrite Z 1 J2e (u) = −λ(f − u)2 + C2 |u|2 dx 2 Ω Z Z C2 2 λ0 C2 2 = |u| dx + − (f − u)2 + |u| dx 2 2 2 D Ω\D Z Z C2 2 C2 λ0 λ0 2 |u| dx + ( − ) |u|2 + λ0 f u − |f | . = 2 2 Ω\D 2 D 2 This is convex for C2 > λ0 , e.g., with C2 = λ0 + 1 we can write J2e (u) = Z D C2 2 |u| dx + 2 Z Ω\D 1 u + λ0 f 2 122 2 λ0 2 − λ0 + |f |2 dx. 2 4.1 Unconditionally Stable Solvers Rigorous Estimates for the Scheme As in Section 4.1.2 for Cahn-Hilliard inpainting, we proceed with a more detailed analysis of (4.23). Throughout this section we consider the square-root regularization of the total variation both in our numerical scheme and in the continuous evolution equation (2.27). Note that similar results are true for other monotone regularizers such as the arctan smoothing. Our results are summarized in the following theorem. Theorem 4.1.12. Let u be the exact solution of (4.20) and uk = u(k∆t) be the exact solution at time k∆t for a time step ∆t > 0 and k ∈ N. Let further Uk be the kth iterate of (4.23) with constants C1 > 1/δ, C2 > λ0 . Then the following statements are true: (i) Under the assumption that kutt k−1 , k∇∆ut k2 and kut k2 are bounded, the numerical scheme (4.23) is consistent with the continuous equation (2.27) and of order 1 in time. (ii) The solution sequence Uk is bounded on a finite time interval [0, T ], for all ∆t > 0. (iii) Let further ek = uk − Uk . If k∇uk k22 + k∇∆uk k22 ≤ K for a constant K > 0, and for all k∆t < T, (4.24) then the error ek converges to zero as ∆t → 0. Remark 4.1.13. Note that assumption (4.24) in Theorem 4.1.12 (iii) does not hold in general. Given the results in [BGOV05] for the one dimensional equation with λ(x) = λ0 on all of Ω, which guarantee the existence of a unique smooth solution for both the square root and the arctan regularization, this assumption nevertheless seem to be reasonable in an heuristic sense. Heuristically speaking, this assumption holds for one dimensional structures, like edges, in the two dimensional image. Rigorously, the wellposedness and regularity of solutions in the two dimensional case with non constant λ is still a matter for further research. The proof of Theorem 4.1.12 is split into three separate Propositions 4.1.14-4.1.16. Proposition 4.1.14. (Consistency (i)) Under the same assumptions as in Theorem 4.1.12 and in particular under the assumption that kutt k−1 , k∇∆ut k2 and kut k2 are bounded, the numerical scheme (4.23) is consistent with the continuous equation (4.20) with local truncation error kτk k−1 = O(∆t). 123 4.1 Unconditionally Stable Solvers Proof. The local truncation error is deﬁned over a time step as satisfying τk = τk1 + τk2 , where τk1 = uk+1 − uk − ut (k∆t), ∆t τk2 = C1 ∆2 (uk+1 − uk ) + C2 (uk+1 − uk ), i.e., uk+1 − uk +∆ ∇· τk = ∆t ∇uk p |∇uk |2 + δ 2 !! − λ(f − uk ) + C1 ∆2 (uk+1 − uk ) + C2 (uk+1 − uk ). (4.25) Using standard Taylor series arguments and assuming that kutt k−1 , k∇∆ut k2 and kut k2 are bounded we deduce that kτk k−1 = O(∆t). (4.26) Proposition 4.1.15. (Unconditional stability (ii)) Under the same assumptions as in Theorem 4.1.12 the solution sequence Uk with k∆t ≤ T for T > 0 fixed, fulfills for every ∆t > 0 k∇Uk k22 + ∆tK1 k∇∆Uk k22 ≤ eK2 T k∇U0 k22 + ∆tK1 k∇∆U0 k22 + ∆tT C(Ω, D, λ0 , f ) , (4.27) for suitable constants K1 , K2 , and a constant C, which depends on Ω, D, λ0 , f only. This gives boundedness of the solution sequence on [0, T ]. Proof. If we multiply (4.23) with −∆Uk+1 and integrate over Ω we obtain 1 k∇Uk+1 k22 − h∇Uk , ∇Uk+1 i2 + C2 k∇Uk+1 k22 + C1 k∇∆Uk+1 k22 ∆t * + ∇U k , ∆Uk+1 = ∆∇ · q + C1 h∇∆Uk , ∇∆Uk+1 i2 2 2 |∇Uk | + δ 2 + h∇ (λ(f − Uk )) , ∇Uk+1 i2 + C2 h∇Uk , ∇Uk+1 i2 . Applying Cauchy’s inequality to the inner products on the right and estimating k∇λ(f − Uk )k22 ≤ 2λ20 k∇Uk k22 + C(Ω, D, λ0 , f ) 124 4.1 Unconditionally Stable Solvers results in 1 k∇Uk+1 k22 − k∇Uk k22 + C2 k∇Uk+1 k22 + C1 k∇∆Uk+1 k22 2∆t * + ∇U C1 k , ∆Uk+1 k∇∆Uk k22 + C1 δ1 k∇∆Uk+1 k22 ≤ ∆∇ · q + δ 2 1 |∇Uk | + δ 2 2 2λ2 C2 + 0 k∇Uk k22 + δ2 k∇Uk+1 k22 + k∇Uk k22 + C2 δ3 k∇Uk+1 k22 + C(Ω, D, λ0 , f ). δ2 δ3 Now, the ﬁrst term on the right side of the inequality can be estimated as follows * + * + ∇U ∇U k k , ∆Uk+1 , ∇∆Uk+1 ∆∇ · q = − ∇∇ · q 2 2 2 2 |∇Uk | + δ |∇Uk | + δ 2 2 2 1 ∇∇ · q ∇Uk + δ4 k∇∆Uk+1 k2 . ≤ 2 δ4 |∇Uk |2 + δ 2 2 Applying Poincaré’s and Cauchy’s inequality to the ﬁrst term leads to 2 ∇∇ · q ∇Uk ≤ O(1/δ)(k∇Uk k2 + k∆Uk k2 + k∇∆Uk k2 ). 2 2 2 |∇Uk |2 + δ 2 2 Interpolating the L2 norm of ∆u by the L2 norms of ∇u and ∇∆u, we obtain 1 + C2 (1 − δ3 ) − δ2 k∇Uk+1 k22 + (C1 (1 − δ1 ) − δ4 ) k∇∆Uk+1 k22 2∆t 1 2λ2 C2 C(1/δ, Ω) C1 C(1/δ, Ω) ≤ + 0+ + + k∇Uk k22 + k∇∆Uk k22 2∆t δ2 δ3 δ4 δ1 δ4 + C(Ω, D, λ0 , f ). For δi = 1/2, i = 1, . . . , 4 we obtain C2 − 1 C1 − 1 1 + k∇∆Uk+1 k22 k∇Uk+1 k22 + 2∆t 2 2 1 ≤ + 4λ20 + 2(C2 + C) k∇Uk k22 + 2 (C1 + C) k∇∆Uk k22 + C(Ω, D, λ0 , f ). 2∆t Since C1 and C2 are chosen such that C1 > 1/δ > 1 and C2 > λ0 > 1, the coeﬃcients in the inequality above are positive. The rest of the proof is similar to the proof of Proposition 4.1.9. We multiply the inequality by 2∆t and set Ca = 1+∆t(C2 −1), Cb = C1 −1, Cc = 1+2∆t(4λ20 +2(C2 +C)), 125 Cd = 4(C1 +C). 4.1 Unconditionally Stable Solvers We obtain Ca k∇Uk+1 k22 + ∆tCb k∇∆Uk+1 k22 ≤ Cc k∇Uk k22 + ∆tCd k∇∆Uk k22 + 2∆tC(Ω, D, λ0 , f ). Dividing by Ca and multiplying the ﬁrst and the second term on the right side of the inequality by Cd and Cc Cb respectively we have k∇Uk+1 k22 Cb Cc Cd + ∆t k∇∆Uk+1 k22 ≤ Ca Ca Cb 2 2 k∇∆Uk k2 k∇Uk k2 + ∆t Ca 2 ∆tC(Ω, D, λ0 , f ). + Ca By induction it follows that k∇Uk+1 k22 + ∆t Cb k∇∆Uk+1 k22 ≤ Ca Cc Cd Ca Cb k∇∆U0 k22 Ca k−1 X Cc Cd i 2 C(Ω, D, λ0 , f ). + ∆t Ca Ca k k∇U0 k22 + ∆t i=0 Therefore we obtain for k∆t ≤ T k∇Uk k22 Cb + ∆t k∇∆Uk k22 ≤ eKT Ca Cb k∇U0 k22 + ∆t k∇∆U0 k22 Ca 2 +∆tT C(Ω, D, λ0 , f ) . Ca Finally we show that the discrete solution converges to the continuous one as ∆t tends to zero. Proposition 4.1.16. (Convergence (iii)) Under the same assumptions as in Theorem 4.1.12 and in particular under assumption (4.24) the error ek fulfills, for suitable nonnegative constants M1 , M2 and M3 , k∇ek k22 + ∆tM1 k∇∆ek k22 ≤ T ∆teM2 T · M3 , (4.28) for k∆t ≤ T and a fixed T > 0. Proof. By our discrete approximation (4.23) and the consistency computation (4.25), 126 4.1 Unconditionally Stable Solvers we have for ek = uk − Uk = ek+1 − ek + C1 ∆2 ek+1 + C2 ek+1 ∆t 1 1 (uk+1 − uk ) − (Uk+1 − Uk ) + C1 ∆2 uk+1 − C1 ∆2 Uk+1 + C2 uk+1 − C2 Uk+1 ∆t ∆t ∇Uk + λ(f − Uk ) + C2 Uk = − C1 ∆2 Uk − ∆ ∇ · q 2 2 |∇Uk | + δ ∇uk − λ(f − uk ) − C1 ∆2 uk − C2 uk + τk − ∆ ∇ · q 2 2 |∇uk | + δ ∇u ∇U k k − ∇ · q = −[−∆ ∇ · q 2 2 |∇Uk | + δ 2 |∇uk | + δ 2 +C1 ∆2 (Uk − uk ) + C2 (Uk − uk ) − λ(Uk − uk )] + τk . Taking the inner product with −∆ek+1 , we have 1 h∇(ek+1 − ek ), ∇ek+1 i2 + C1 k∇∆ek+1 k22 + C2 k∇ek+1 k22 ∆t * + ∇U ∇u k k − ∇ · q , ∆ek+1 = −∆ ∇ · q 2 2 2 2 |∇Uk | + δ |∇uk | + δ 2 2 + C1 ∆ (Uk − uk ), ∆ek+1 2 + h∇λ(Uk − uk ), ∇ek+1 i2 − C2 h∇(Uk − uk ), ∇ek+1 i2 + ∇∆−1 τk , ∇∆ek+1 2 Using the same arguments as in the proof of Proposition 4.1.10 we obtain 1 (k∇ek+1 k22 − k∇ek k22 ) + C1 k∇∆ek+1 k22 + C2 k∇ek+1 k22 2∆t * + ∇U ∇u k k − ∇ · q , ∆ek+1 ≤ −∆ ∇ · q |∇Uk |2 + δ 2 |∇uk |2 + δ 2 2 C1 λ2 + k∇∆ek k22 + C1 δ1 k∇∆ek+1 k22 + 0 k∇ek k22 + δ3 k∇ek+1 k22 δ1 δ3 1 C2 k∇ek k22 + C2 δ2 k∇ek+1 k22 + kτk k2−1 + δ4 k∇∆ek+1 k22 . + δ2 δ4 127 4.1 Unconditionally Stable Solvers We consider the ﬁrst term on the right side of the above inequality in detail, * + ∇U ∇u k k − ∇ · q , ∆ek+1 −∆ ∇ · q 2 2 2 2 |∇Uk | + δ |∇uk | + δ 2 + * ∇Uk , ∇∆ek+1 − ∇ · q ∇uk = ∇ ∇ · q 2 2 |∇Uk | + δ 2 |∇uk | + δ 2 2 2 1 ∇ ∇ · q ∇Uk − ∇ · q ∇uk + 1 k∇∆ek+1 k2 ≤ 2 2 2 |∇Uk |2 + δ 2 |∇uk |2 + δ 2 2 2 2 ∇uk ∇Uk + ∇ ∇ · q ≤ ∇ ∇ · q 2 2 2 2 |∇Uk | + δ |∇uk | + δ 2 2 1 + k∇∆ek+1 k22 2 ≤ C(1/δ, Ω) k∇Uk k22 + k∇∆Uk k22 + k∇uk k22 + k∇∆uk k22 1 + k∇∆ek+1 k22 , 2 for a constant C > 0 (cf. the estimate for the regularizer in the proof of Proposition 4.1.15). Therefore we obtain 1 1 2 + C2 (1 − δ2 ) − δ3 k∇ek+1 k2 + C1 (1 − δ1 ) − δ4 − k∇∆ek+1 k22 2∆t 2 1 C2 λ20 1 C1 ≤ + + k∇∆ek k22 + kτk k2−1 k∇ek k22 + 2∆t δ2 δ3 δ1 δ4 2 2 + C(1/δ, Ω) k∇Uk k2 + k∇∆Uk k2 + k∇uk k22 + k∇∆uk k22 . Proposition 4.1.15 guarantees that Uk is bounded by a constant, and assumption (4.24) that the exact solution uk is bounded. Therefore by following the lines of the proof of Proposition 4.1.10 we ﬁnally have for k∆t ≤ T k∇ek k22 + ∆tM1 k∇∆ek k22 ≤ T ∆teM2 T · M3 , for suitable positive constants M1 , M2 and M3 . 4.1.4 LCIS Inpainting Our last example for the applicability of the convexity splitting method to higher-order inpainting approaches is inpainting with LCIS (2.42). With f ∈ L2 (Ω) our inpainted 128 4.1 Unconditionally Stable Solvers image u evolves in time as ut = −∆(arctan(∆u)) + λ(f − u). In contrast to the other two inpainting methods that we discussed, this inpainting equation is a gradient ﬂow in L2 for the energy Z Z 1 G(∆u) dx + J(u) = λ(f − u)2 , 2 Ω Ω with G′ (y) = arctan(y). Therefore Eyre’s result in Theorem 4.1.4 can be applied directly. The functional J(u) is split into J1 − J2 with Z Z C1 C2 2 1 2 J1 (u) = (∆u) dx + |u| dx, 2 2 Ω 2 Z ZΩ C2 2 1 C1 2 −λ(f − u)2 + (∆u) dx + |u| dx. −G(∆u) + J2 (u) = 2 2 Ω 2 Ω The resulting time-stepping scheme is Uk+1 − Uk + C1 ∆2 Uk+1 + C2 Uk+1 = −∆(arctan(∆Uk )) + C1 ∆2 Uk + λ(f − Uk ) + C2 Uk . ∆t (4.29) Again we impose homogeneous Neumann boundary conditions, use DCT to solve (4.29) and we choose the constants C1 and C2 such that J1 and J2 are all strictly convex and condition (4.9) is satisﬁed. The functional J1 is convex for all C1 , C2 > 0. The ﬁrst term in J2 is convex if C1 > 1. This follows from its second variation, namely Z d 2 (C1 ∆(u + sw) − arctan(∆(u + sw)))∆v dx ∇ J1e (u)(v, w) = ds s=0 Z 1 = C1 − ∆v∆w dx. 1 + (∆u)2 For J1e being convex ∇2 J1e (u)(v, w) has to be > 0 for all v, w ∈ C ∞ and therefore C1 − 1 > 0. 1 + (∆u)2 C1 > 1 ∀s ∈ R. 1 + s2 Substituting s = ∆u we obtain This inequality is fulﬁlled for all s ∈ R if C1 > 1. We obtain the same condition on C1 for G′ (s) = arctan( δs ). For the convexity of the second term of J2 , the second constant has to fulﬁll C2 > λ0 , cf. the computation for the ﬁtting term in Section 4.1.3. With these choices of C1 and C2 also condition (4.9) of Theorem 4.1.4 is automatically satisﬁed. 129 4.1 Unconditionally Stable Solvers Rigorous Estimates for the Scheme Finally we present rigorous results for (4.29). In contrast to the inpainting equations (2.1) and (2.27), inpainting with LCIS follows a variational principle. Hence, by choosing the constants C1 and C2 appropriately, i.e., C1 > 1, C2 > λ0 (cf. the computations above), Theorem 4.1.4 ensures that the iterative scheme (4.29) is unconditionally gradient stable. Additionally to this property, we present similar results as before for Cahn-Hilliard- and TV-H−1 inpainting. Theorem 4.1.17. Let u be the exact solution of (2.42) and uk = u(k∆t) the exact solution at time k∆t, for a time step ∆t > 0 and k ∈ N. Let further Uk be the kth iterate of (4.29) with constants C1 > 1, C2 > λ0 . Then the following statements are true: (i) Under the assumption that kutt k−1 , k∇∆ut k2 and kut k2 are bounded, the numerical scheme (4.29) is consistent with the continuous equation (2.42) and of order 1 in time. (ii) The solution sequence Uk is bounded on a finite time interval [0, T ], for all ∆t > 0. (iii) Let further ek = uk − Uk . If k∇∆uk k22 ≤ K, for a constant K > 0, and for all k∆t < T, (4.30) then the error ek converges to zero as ∆t → 0. Remark 4.1.18. Note that the consistency result of Theorem 4.1.17 is weaker than than the one following from Theorem 4.1.4. In fact Eyre’s theorem shows that the scheme is of order 2, which is a stronger result than Theorem 4.1.17 (i). Note that assumption (4.30) does not hold in general. Given the results of [BG04] for the denoising case λ(x) = λ0 in all of Ω and for smooth initial data and smooth f , this assumption nevertheless seems to be reasonable in an heuristic sense. Rigorously, the well-posedness and regularity of solutions in the two-dimensional case with non constant λ is a matter for future research. The proof of Theorem 4.1.17 is organized in the following three Propositions 4.1.194.1.21. Since the proof of consistency follows the lines of Proposition 4.1.8 and Proposition 4.1.14, we just state the result. 130 4.1 Unconditionally Stable Solvers Proposition 4.1.19. (Consistency (i)) Under the same assumptions as in Theorem 4.1.17 and in particular assuming that kutt k−1 , k∇∆ut k2 and kut k2 are bounded, we have kτk k−1 = O(∆t). Next we would like to show the boundedness of a solution of (4.29) in the following proposition. Proposition 4.1.20. (Unconditional stability (ii)) Under the same assumptions as in Theorem 4.1.17 the solution sequence Uk fulfills, for k∆t < T k∇Uk k22 + ∆tK1 k∇∆Uk k22 ≤ eK2 T k∇U0 k22 + ∆tK1 k∇∆U0 k22 +∆tT C(Ω, D, λ0 , f )) for suitable constants K1 , K2 , and constant C depending on Ω, D, λ0 , f only. Proof. If we multiply (4.29) with −∆Uk+1 and integrate over Ω we obtain 1 k∇Uk+1 k22 − h∇Uk , ∇Uk+1 i2 + C2 k∇Uk+1 k22 + C1 k∇∆Uk+1 k22 ∆t = h∆arctan(∆Uk ), ∆Uk+1 i2 + C1 h∇∆Uk , ∇∆Uk+1 i2 + h∇ (λ(f − Uk )) , ∇Uk+1 i2 + C2 h∇Uk , ∇Uk+1 i2 . Using the same arguments as in the proofs of Proposition 4.1.9 and 4.1.15 we obtain 1 k∇Uk+1 k22 − k∇Uk k22 + C2 k∇Uk+1 k22 + C1 k∇∆Uk+1 k22 2∆t λ2 C1 k∇∆Uk k22 + C1 δ1 k∇∆Uk+1 k22 + 0 k∇Uk k22 ≤ h∆arctan(∆Uk ), ∆Uk+1 i2 + δ1 2δ2 C 2 k∇Uk k22 + C2 δ3 k∇Uk+1 k22 + C(Ω, D, λ0 , f ). + δ2 k∇Uk+1 k22 + δ3 Now, the ﬁrst term on the right side of the inequality can be estimated as follows h∆arctan(∆Uk ), ∆Uk+1 i2 = − h∇arctan(∆Uk ), ∇∆Uk+1 i2 1 =− ∇∆Uk , ∇∆Uk+1 1 + (∆Uk )2 2 2 1 1 2 ≤ ∇∆Uk + δ4 k∇∆Uk+1 k2 2 δ4 1 + (∆Uk ) 2 1 k∇∆Uk k22 + δ4 k∇∆Uk+1 k22 . ≤ δ4 131 (4.31) 4.1 Unconditionally Stable Solvers From this we get 1 + C2 (1 − δ3 ) − δ2 k∇Uk+1 k22 + (C1 (1 − δ1 ) − δ4 ) k∇∆Uk+1 k22 2∆t 1 λ20 C2 1 C1 2 ≤ + + + k∇Uk k2 + k∇∆Uk k22 + C(Ω, D, λ0 , f ). 2∆t 2δ2 δ3 δ1 δ4 Analogously to Section 4.1.3, with Ca = 1 + ∆t(C2 − 1), Cb = C1 − 1, Cc = 1 + 2∆t(λ20 + 2C2 ), Cd = 4(C1 + 1), we obtain k∇Uk k22 + ∆t Cb k∇∆Uk k22 ≤ eKT Ca Cb k∇U0 k22 + ∆t k∇∆U0 k22 Ca 2 +∆tT C(Ω, D, λ0 , f ) , Ca which gives boundedness of the solution sequence on [0, T ] for any T > 0 and any ∆t > 0. The convergence of the discrete solution to the continuous one as ∆t → 0 is veriﬁed in the following proposition. Proposition 4.1.21. (Convergence (iii)) Under the same assumptions as in Theorem 4.1.17 and in particular under assumption (4.30), the error ek fulfills, for suitable nonnegative constants M1 , M2 and M3 , k∇ek k22 + ∆tM1 k∇∆ek k22 ≤ T ∆teM2 T · M3 , (4.32) for k∆t ≤ T and a fixed T > 0. Proof. Since all the computations in the convergence proof for (4.29) are the same as in Section 4.1.3 for (4.23) except of the estimate for the regularizer ∆ (arctan(∆u)), we only give the details for the latter and leave the rest to the reader. Thus for the inner product involving the regularizer of (4.29) within the convergence proof we apply the arguments from (4.31) and obtain h−∆ (arctan(∆Uk ) − arctan(∆uk )) , ∆ek+1 i2 = h∇ (arctan(∆Uk ) − arctan(∆uk )) , ∇∆ek+1 i2 1 ≤ k∇∆Uk k22 + k∇∆uk k22 + k∇∆ek+1 k22 . 2 132 4.1 Unconditionally Stable Solvers By assumption (4.30) and by following the same steps as in the proof of Proposition 4.1.16 we ﬁnally have for k∆t ≤ T k∇ek k22 + ∆tM1 k∇∆ek k22 ≤ T ∆teM2 T · M3 , for suitable positive constants M1 , M2 and M3 . 4.1.5 Numerical Discussion Numerical results for all three inpainting schemes have already been presented in Sections 2.1.2, 2.2.5, and 2.3.1. In Figures 4.1, 4.2, and 4.3 one additional numerical result for each approach is presented. In all three inpainting schemes, for the discretization in space we used ﬁnite differences and spectral methods, i.e., the discrete cosine transform or the fast Fourier transform, to simplify the inversion of the Laplacian ∆ for the computation of Uk+1 . Note that in the case of Cahn-Hilliard inpainting (2.1) special attention has to be paid to the correct choice of the mesh size ∆x, which has to be of the order ǫ, also cf. the numerical examples in Section 3.1.3 and [Gl03, VR03]. Further, recall that Cahn-Hilliard inpainting is applied in a two-step procedure with changing ǫ value, cf. Section 3.1.3 for details. For Cahn-Hilliard- and TV-H−1 inpainting the optimal time step size ∆t turned out to be ∆t = 1 or 10 (depending also on the size of ǫ and λ0 ). For the numerical computation of inpainting with LCIS the time step size ∆t was chosen to be equal to 0.01. Figure 4.1: Destroyed binary image and the solution of Cahn-Hilliard inpainting with λ0 = 109 and switching ǫ value: u(200) with ǫ = 0.8, u(500) with ǫ = 0.01 Although the proposed discrete schemes in this paper are unconditionally stable, their numerical performance is still far of being in real-time. The reason is the damping 133 4.1 Unconditionally Stable Solvers Figure 4.2: TV-H−1 inpainting: u(1000) with a regularization parameter ǫ = 0.01 and λ0 = 102 Figure 4.3: LCIS inpainting result u(1000) with δ = 0.1 and λ0 = 103 . introduced by the conditions on the constants C1 , and C2 in all three schemes (4.11), (4.23), and (4.29). In Figure 4.2 for instance, we need around 1000 iterations to receive the restored image. If now additionally the data dimension is large, e.g., when we have to process 2D images of high resolution, of sizes 3000 × 3000 pixels for instance, or even 3D image data, each iteration step itself is computationally expensive and we are far from real-time computations. As already discussed in Section 1.3.3 in more detail, the ﬁeld of research on fast numerical solvers for higher-order inpainting models is still in its infancy. The construction of such solvers is very favorable though, in particular for the use of these inpainting models in practice. TV approaches for example already turned out to be an eﬀective tool for the reconstruction of medical images as the 134 4.2 A Dual Solver for TV-H−1 Minimization ones gained from PET (Positron Emission Tomography) measurements (cf. [JHC98], for instance). These imaging approaches need to deal with 3D or even 4D image date (including time dependence) in a fast and robust way. Motivated by this we are thinking about alternative ways to reduce the dimensionality of the data and hence speed up the reconstruction process. This will be the interest of Section 4.3 where we present a domain decomposition approach to be applied, among others, to TV inpainting models. 4.2 A Dual Solver for TV-H−1 Minimization In this section we are concerned with the numerical minimization of total variation functionals with an H −1 constraint. We present an algorithm for its solution, which is based on the dual formulation of total variation and show its application in several areas of image processing, among them also TV-H−1 inpainting. In fact a major motivation for this algorithm is the creation of a fast numerical method to solve TV-H−1 inpainting. Namely, we shall see in Section 4.3 that this dual solver enables us to apply a domain decomposition algorithm to TV-H−1 inpainting. 4.2.1 Introduction and Motivation Let Ω ⊂ R2 be a bounded and open domain with Lipschitz boundary. For a given function f ∈ L2 (Ω) we are interested in the numerical realization of the following minimization problem min u∈BV (Ω) J(u), where J(u) = kT u − f k2−1 + 2α |Du| (Ω), (4.33) where T ∈ L(L2 (Ω)) is a bounded linear operator and α > 0 is a tuning parameter. The function |Du| (Ω) is the total variation of u and k.k−1 is the norm in H −1 (Ω), the dual of H01 (Ω). Please compare the Appendix for the deﬁnition of these terms. Problem (4.33) is a model for minimizing the total variation of a function u which obeys an H −1 constraint, i.e., kT u − f k−1 is small, for a given function f ∈ L2 (Ω). In the terminology of inverse problems this means that from an observed datum f one wants to determine the original function u, from which a priori we know that T u = f and u has some regularity properties modeled by the total variation and the H −1 norm. 135 4.2 A Dual Solver for TV-H−1 Minimization Minimization problems like this have important applications in a wide range of image processing tasks. We give their overview in the following subsections. The main interest of this section is the numerical solution of (4.33). In [Ch04] Chambolle proposes an algorithm to solve total variation minimization with an L2 constraint and T = Id. This approach was extended to more general operators T in a subsequent work [BBAC04]. In the following we shall introduce a generalization of Chambolle’s algorithm for the case of an H −1 norm in the problem. Moreover, we present strategies to extend the use of this algorithm from problems with T = Id to problems (4.33) with an arbitrary linear operator T . Additional to the theory of this algorithm we present its applications in image processing, in particular for image denoising, image decomposition and inpainting. Finally we show how to speed up numerical computations by considering a domain decomposition approach for our problem. Note that the existence and uniqueness of minimizers for (4.33) is guaranteed. In fact the existence of a unique solution of (4.33) follows as a special case from the following theorem. Theorem 4.2.1. [LV08, Theorem 3.1] Given Ω ⊂ R2 , open, bounded and connected, with Lipschitz boundary, f ∈ H −s (R2 ) (s > 0), f = 0 outside of Ω̄, α > 0, and T ∈ L(L2 (Ω)) an injective continuous linear operator such that T 1Ω 6= 0, then the minimization problem min u∈BV (Ω) kT u − f k2−s + 2α |Du| (Ω), s > 0, where k·k−s denotes the norm in H −s (R2 ), has a unique solution in BV (Ω). Proof. The proof is a standard application of methods from variational calculus and can be found, e.g., in [LV08]. The main ingredients in the proof, in order to guarantee compactness, are the Poincaré-Wirtinger inequality (cf. [AV94] for instance) which bounds the L2 - and the L1 - norm by the total variation, and the fact that L2 (Ω) can be embedded in H −1 (Ω). In the subsequent two subsections we shall present two main applications of TVH−1 minimization (4.33) for image denoising / decomposition, and image inpainting. 136 4.2 A Dual Solver for TV-H−1 Minimization TV-H−1 Minimization for Image Denoising and Image Decomposition In taking T = Id in (4.33), we encounter two interesting areas in image processing, namely image denoising and image decomposition. In many image denoising models a given noisy image f is decomposed into its piecewise smooth part u and its oscillatory, noisy part v, i.e., f = u + v. Similarly, in image decomposition the piecewise smooth part u represents the structure/cartoon part of the image, and the oscillatory part v the texture part of the image. We call the latter task also cartoon-texture decomposition. The most famous model within this range is the TV-L2 denoising model proposed by Rudin, Osher and Fatemi [ROF92] J(u) = ku − f k22 + 2α |Du| (Ω) → min u∈BV (Ω) . (4.34) This model produces very good results for removing noise and preserving edges in structured images, meaning images without texture-like components, i.e., high oscillatory edges. Unfortunately it fails in the presence of the latter. Namely it cannot separate pure noise from high oscillatory edges but removes both equally. To overcome this situation, Y. Meyer [Me01] suggested to replace the L2 −ﬁdelity term by a weaker norm. Namely he proposes the following model: J(u) = ku − f k∗ + 2α |Du| (Ω) → min , u∈BV (Ω) where the k·k∗ is deﬁned as follows. Definition 4.2.2. Let G denote the Banach space consisting of all generalized functions f (x, y) which can be written as f (x, y) = ∇ · (~g (x, y)), ~g = (g1 , g2 ), g1 , g2 ∈ L∞ (Ω), ~g · ~n = 0 on ∂Ω, where ~n is the unit normal on ∂Ω. Then kf k∗ is the induced norm of G defined as p kf k∗ = inf g1 (x, y)2 + g2 (x, y)2 ∞ . f =∇·~g L (Ω) In fact, the space G is the dual space of W01,1 (Ω). In [Me01] Meyer further introduces two other spaces with similar properties to G but we are not going into detail here. We only mention that these spaces are intrinsically appropriate for modeling textured or oscillatory patterns and in fact they provide for them norms which are smaller than the L2 norm. 137 4.2 A Dual Solver for TV-H−1 Minimization The drawback of Y. Meyer’s model is that it can not be solved directly with respect to the minimizer u and therefore has to be approximated, cf. [VO04]. Thereby the ∗−norm is replaced by q 1 2 2 2 g1 + g2 , k(u + ∇ · ~g ) − f k2 + 2µ Lp (Ω) with µ > 0 and p ≥ 1. In the case p = 2 the second term in the above expression is equivalent to the H −1 norm. In particular v = ∇ · ~g corresponds to v ∈ H −1 (Ω). Indeed, for v ∈ H −1 (Ω), there is a unique v ∗ ∈ H01 (Ω) such that kvk2−1 = k∇v ∗ k22 q 2 −1 2 2 2 = ∇∆ v 2 = g1 + g2 . 2 Limiting to the case p = 2 and the limit µ → 0 the TV-H−1 denoising model was created, cf. [OSV03, MG01, LV08], i.e., ku − f k2−1 + 2α |Du| (Ω) → min u∈BV (Ω) . (4.35) Numerical experiments showed that (4.35) gives much better results than (4.34) under the presence of oscillatory data, cf. [OSV03, LV08]. In Section 4.2.3 we will present some numerical results that support this claim. TV-H−1 Inpainting The second application we are interested in is TV-H−1 inpainting (2.27). This inpainting approach was already discussed in Section 2.2 in great detail. Let us recall the basic setup of this inpainting approach: the inpainted image u of f ∈ L2 (Ω), shall evolve via (2.27), i.e., ut = ∆p + λ(f − u), p ∈ ∂ |Du| (Ω), where ∂ |Du| (Ω) denotes the subdiﬀerential of the total variation, cf. Appendix A.5, and λ is the indicator function of the domain outside of the missing domain D multiplied by a constant λ0 ≫ 1. As Cahn-Hilliard inpainting in Section 2.1, also TV-H−1 inpainting does not follow a variational principle. In fact the most natural formulation of (4.33) in terms of inpainting would be in the setting T = 1Ω\D and α = 1/λ0 . Similarly to the Cahn-Hilliard case, cf. (2.4), this only results in an optimality condition which describes a second-order anisotropic diﬀusion inside of the inpainting domain D, i.e., p = 0 in 138 4.2 A Dual Solver for TV-H−1 Minimization D. Hence it is not immediately clear how a numerical method for the minimization problem (4.33) is applicable to solve the inpainting equation (2.27). How this is possible though, will be explained in the following sections, cf. especially (4.47) and (4.48). Numerical Solution for TV-H−1 Minimization The numerical solution of TV-H−1 minimization depends on the speciﬁc problem at hand. In [LV08] Lieu and Vese proposed a numerical method to solve TV-H−1 denoising / decomposition (4.35) by using the Fourier representation of the H −1 norm on the whole Rd , d ≥ 1. Therein the space H −1 (Rd ) is deﬁned to be the Hilbert space equipped with the inner product hf, gi−1 = and associated norm kf k−1 = transform of g in L2 (Rd ), i.e., ĝ(ξ) := q Z 1 + |ξ|2 −1 fˆĝ¯ dξ hf, f i−1 , cf. also [DL88]. Here ĝ denotes the Fourier 1 (2π)2d Z e−ixξ g(x) dx, Rd ξ ∈ Rd , (4.36) and ḡ the complex conjugate of g. Equivalently to this deﬁnition H −1 (Rd ) is the dual space of H 1 (Rd ). Note that we consider functions g deﬁned in Rd rather than on a bounded domain which can be done by considering zero extensions of the image function. With this deﬁnition of the H −1 norm the corresponding optimality condition for total variation denoising/decomposition (4.35) reads ^ ¯ ¯ fˆ − û αp + 2 Re −1 = 0 in Ω 2 1 + |ξ| ∇u · ~n = 0 |∇u| u=0 (4.37) on ∂Ω outside Ω̄, g} denotes the inverse Fourier transform of f , deﬁned in analogy to (4.36), Re where {f denotes the real part of a complex number, and ~n is the outward-pointing unit normal vector on ∂Ω. For the numerical computation of a solution of (4.37), we approximate an element p of the subdiﬀerential of |Du| (Ω), by its relaxed version p ≈ ∇ · (∇u/|∇u|ǫ ), 139 (4.38) 4.2 A Dual Solver for TV-H−1 Minimization where |∇u|ǫ = p |∇u|2 + ǫ. Equation (4.37) leads to solve a second-order PDE rather than a fourth-order PDE, resulting in a better CFL condition for the numerical scheme, cf. [LV08]. In the case of TV-H−1 inpainting (2.27) the situation is completely diﬀerent since (2.27) does not fulﬁll a variational principle, cf. Section 4.2.1. In Section 4.1.3 we proposed the semi-implicit scheme (4.23) for the solution of (2.27). Let Uk denote the approximate solution to the exact solution u(kτ ) (where τ denotes the time step). Then this scheme reads Uk+1 − Uk ∇Uk )) + C1 ∆2 Uk+1 + C2 Uk+1 = C1 ∆2 Uk − ∆(∇ · ( τ |∇Uk |ǫ + C2 Uk + λ(f − Uk ), where ∆2 = ∆∆ and with constants C1 > 1 ǫ and C2 > λ0 . Since usually in inpainting tasks λ0 is chosen comparatively large, e.g., λ0 = 103 , the condition on C2 makes the numerical scheme, although unconditionally stable, quite slow. To our knowledge this is the only existing numerical method proposed for TV-H−1 inpainting, except for the one presented in this section. In the following we are going to present a method introduced by Chambolle [Ch04] for TV-L2 minimization (4.34) and its generalization for the TV-H−1 case (4.33). This algorithm will give us the opportunity to address TV-H−1 minimization in a general way. 4.2.2 The Algorithm Preliminaries Throughout this section k·k denotes the norm in X = L2 (Ω) in the continuous setting and the Euclidean norm in X = RN ×M in the discrete setting. In the discrete setting the continuous image domain Ω = [a, b] × [c, d] ⊂ R2 is approximated by a ﬁnite grid {a = x1 < . . . < xN = b} × {c = y1 < . . . < yM = d} with equidistant step-size h = xi+1 − xi = b−a N = d−c M = yj+1 − yj equal to 1 (one pixel). The digital image u is an element in X. We denote u(xi , yj ) = ui,j for i = 1, . . . , N and j = 1, . . . , M . Further we deﬁne Y = X × X with Euclidean norm k·kY and inner product h·, ·iY . Moreover the operators gradient ∇, divergence ∇· and Laplacian ∆ in the discrete setting are deﬁned as follows: 140 4.2 A Dual Solver for TV-H−1 Minimization The gradient ∇u is a vector in Y given by forward diﬀerences (∇u)i,j = ((∇x u)i,j , (∇y u)i,j ), with (∇x u)i,j ( ui+1,j − ui,j = 0 if i < N if i = N, (∇y u)i,j = ( ui,j+1 − ui,j 0 if j < M if j = M, for i = 1, . . . , N , j = 1, . . . , M . We further introduce a discrete divergence ∇· : Y → X deﬁned, by analogy with the continuous setting, by ∇· = −∇∗ (∇∗ is the adjoint of the gradient ∇). That is, the discrete divergence operator x x pi,j − pi−1,j (∇ · p)ij = pxi,j x −pi−1,j for every p = (px , py ) ∈ Y . is given by backward diﬀerences y y if 1 < i < N pi,j − pi,j−1 + pyi,j if i = 1 y −pi,j−1 if i = N like if 1 < j < M if j = 1 if j = M, Finally we deﬁne the discrete Laplacian as ∆ = ∇ · ∇, i.e., ui+1,j − 2ui,j + ui−1,j if 1 < i < N (∆u)i,j = ui+1,j − ui,j if i = 1 ui−1,j − ui,j if i = N ui,j+1 − 2ui,j + ui,j−1 if 1 < j < M + ui,j+1 − ui,j if j = 1 ui,j−1 − ui,j if j = M, and its inverse operator ∆−1 , as in the continuous setting (cf. Appendix A.4), i.e., u = ∆−1 f is the unique solution of ( − (∆u)i,j = fi,j ui,j = 0 1 < i < N, 1 < j < M i = 1, N ; j = 1, M. Moreover, without always indicating it, when in the discrete setting, instead of minimizing J(u) = kT u − f k2−1 + 2α |Du| (Ω), we consider the discretized functional X 2 ∇∆−1 ((T u)i,j − fi,j ) + 2α (∇u)i,j , Jδ (u) := 1≤i≤N 1≤j≤M with |y| = p y12 + y22 for every y = (y1 , y2 ) ∈ R2 . To give a meaning to (T u)i,j we assume that T is applied to the piecewise interpolant û of the matrix (ui,j ). 141 4.2 A Dual Solver for TV-H−1 Minimization Chambolle’s Algorithm for Total Variation Minimization In [Ch04] Chambolle proposes an algorithm to compute numerically a minimizer of J(u) = ku − f k22 + 2α |Du| (Ω). His algorithm is based on considerations of the convex conjugate of the total variation and on exploiting the corresponding optimality condition. It amounts to computing the minimizer u of J as u = f − PαK (f ), where PK denotes the orthogonal projection over L2 (Ω) on the convex set K which is the closure of the set ∇ · ξ : ξ ∈ Cc1 (Ω; R2 ), |ξ(x)|∞ ≤ 1 ∀x ∈ R2 . To compute numerically the projection PαK (f ) Chambolle uses a ﬁxed point algorithm. More precisely, in two dimensions the following semi-implicit gradient descent algorithm is given to approximate PαK (f ): Choose τ > 0, let p(0) = 0 and, for any n ≥ 0, iterate (n+1) (n+1) (n) , pi,j = pi,j + τ (∇(∇ · p(n) − f /α))i,j − (∇(∇ · p(n) − f /α))i,j pi,j so that (n) (n+1) pi,j pi,j + τ (∇(∇ · p(n) − f /α))i,j . = 1 + τ (∇(∇ · p(n) − f /α))i,j (4.39) For τ ≤ 1/8 the iteration α∇ · p(n) converges to PαK (f ) as n → ∞ (compare [Ch04, Theorem 3.1]). All this will be explained in more detail in the context of TVH−1 minimization in the following subsection. A Generalization of Chambolle’s Algorithm for TV-H−1 Minimization The main contribution of this section is to generalize Chambolle’s algorithm to the case of an H −1 constrained minimization of the total variation and to the case where T is an arbitrary linear and bounded operator. In short we shall see how to solve (4.33) using a similar strategy as in [Ch04]. We start with solving the simpliﬁed problem when T = Id and as a second step present a method how to use this solution in order 142 4.2 A Dual Solver for TV-H−1 Minimization to solve the general case (4.33). Hence for the time being we consider the minimization problem min{J(u) = ku − f k2−1 + 2α |Du| (Ω)}. (4.40) u We proceed by exploiting the optimality condition of (4.40), i.e., 1 0 ∈ ∂ |Du| (Ω) + ∆−1 (u − f ) . α This can be rewritten as ∆−1 (f − u) ∈ ∂ |Du| (Ω). α (4.41) Since s ∈ ∂f (x) ⇐⇒ x ∈ ∂f ∗ (s), where f ∗ is the convex conjugate (or Fenchel transform) of f , it follows that −1 ∗ ∆ (f − u) . u ∈ ∂ |D·| (Ω) α Here ( 0 |D·| (Ω) (v) = χK (v) = +∞ ∗ if v ∈ K otherwise, where, as before, K is the closure of the set ∇ · ξ : ξ ∈ Cc1 (Ω; R2 ), |ξ(x)|∞ ≤ 1 ∀x ∈ R2 . Rewriting the above inclusion again we have f f −u 1 ∈ + ∂ |D·| (Ω)∗ α α α ∆−1 (f − u) α , i.e., with w = ∆−1 (f − u)/α it reads 0 ∈ (−∆w − f /α) + 1 ∂ |D·| (Ω)∗ (w) in Ω α w=0 on ∂Ω. In other words, w is a minimizer of w − ∆−1 f /α2 1 H (Ω) 0 2 143 + 1 |D·| (Ω)∗ (w), α (4.42) 4.2 A Dual Solver for TV-H−1 Minimization where H01 (Ω) = v ∈ H 1 (Ω) : v = 0 on ∂Ω and kvkH 1 (Ω) = k∇vk. Because of (4.42), 0 for w to be a minimizer of the above functional it is necessary that |D·| (Ω)∗ (w) = 0, i.e., w ∈ K. Hence a minimizer w fulﬁlls w = P1K (∆−1 f /α), where P1K is the orthogonal projection on K over H01 (Ω), i.e., P1K (u) = argminv∈K ku − vkH 1 (Ω) . 0 Hence the solution u of problem (4.40) is given by u = f + ∆ P1αK (∆−1 f ) , where −∆ denotes the Laplacian with zero Dirichlet boundary conditions as before. Computing the nonlinear projection P1αK (∆−1 f ) in the discrete setting amounts to solving the following problem: 2 −1 min ∇ α∇ · p − ∆ f i,j : p ∈ Y, |pi,j | ≤ 1 ∀i = 1, . . . , N ; j = 1, . . . , M . (4.43) Analogously to [Ch04] we use the Karush-Kuhn-Tucker conditions for the above constrained minimization. Then there exist βi,j ≥ 0 such that the corresponding Euler- Lagrange equation reads ∇ ∆ α∇ · p − ∆−1 f i,j + βi,j pi,j = 0, ∀i = 1, . . . , N ; j = 1, . . . , M. Since βi,j (|pi,j |2 − 1) = 0, either βi,j > 0 and |pi,j | = 1 or |pi,j | < 1 and βi,j = 0. Now, following the arguments in [Ch04], in both cases this yields βi,j = ∇∆ ∇ · pn − ∆−1 f /α i,j , ∀i = 1, . . . , N ; j = 1, . . . , M. Then the gradient descent algorithm for solving (4.43) reads: for an initial p0 = 0, iterate for n ≥ 0 pn+1 i,j pni,j − τ ∇∆ ∇ · pn − ∆−1 f /α i,j . = 1 + τ (∇∆ (∇ · pn − ∆−1 f /α))i,j (4.44) Redoing the convergence proof in [Ch04] we end up with a similar result: Theorem 4.2.3. Let τ ≤ 1/64. Then, α∇ · pn converges to P1αK (∆−1 f ) as n → ∞. 144 4.2 A Dual Solver for TV-H−1 Minimization Proof. The proof proceeds similarly to the proof in [Ch04]. For the sake of completeness and clarity we still present the detailed proof here, keeping close to the notation in [Ch04]. By induction we easily see that for every n ≥ 0, pni,j ≤ 1 for all i, j. Indeed, starting with pn , with |pni,j | ≤ 1 for all i = 1, . . . , N ; j = 1, . . . , M , we have pni,j + τ ∇(−∆) ∇ · pn − ∆−1 f /α i,j n+1 ≤ 1. pi,j ≤ 1 + τ (∇(−∆) (∇ · pn − ∆−1 f /α))i,j Now, let us ﬁx an n ≥ 0 and consider ∇ ∇ · pn+1 − ∆−1 (f /α) Y . We want to show that this norm is decreasing with n. In what follows we will abbreviate k·kY by k·k. We have ∇ ∇ · pn+1 − ∆−1 (f /α) 2 = ∇∇ · (pn+1 − pn ) + ∇ ∇ · pn − ∆−1 (f /α) 2 2 = ∇ ∇ · pn − ∆−1 (f /α) +2 ∇∇ · (pn+1 − pn ), ∇ ∇ · pn − ∆−1 (f /α) 2 2 + ∇∇ · (pn+1 − pn ) . Inserting η = (pn+1 − pn )/τ in the above equation and integrating by parts in the second term we get ∇ ∇ · pn+1 − ∆−1 (f /α) 2 = ∇ ∇ · pn − ∆−1 (f /α) 2 +2τ η, ∇∆ ∇ · pn − ∆−1 (f /α) 2 + τ 2 k∇∇ · ηk2 . By further estimating k∇∇ · ηk ≤ κ kηk, where κ = k|∇∇ · k| = supkpk≤1 k∇∇ · pk the norm of the operator ∇∇· : Y → Y , we deduce ∇ ∇ · pn+1 − ∆−1 (f /α) 2 ≤ ∇ ∇ · pn − ∆−1 (f /α) 2 h i +τ 2 η, ∇∆ ∇ · pn − ∆−1 (f /α) 2 + κ2 τ kηk2 . The operator norm κ will be bounded by the end of the proof. For now we are going to show that the term multiplied by τ is always negative as long as pn+1 6= pn and 2 τ ≤ 1/κ2 , and hence that ∇ ∇ · pn − ∆−1 (f /α) is decreasing. To do so we consider 2 η, ∇∆ ∇ · pn − ∆−1 (f /α) 2 + κ2 τ kηk2 X 2ηi,j ∇∆ ∇ · pn − ∆−1 (f /α) i,j + κ2 τ |ηi,j |2 . (4.45) = 1≤i≤N 1≤j≤M Now, from the ﬁxed point equation we have h i ηi,j = − ∇∆ ∇ · pn − ∆−1 (f /α) i,j + ∇∆ ∇ · pn − ∆−1 (f /α) i,j · pn+1 . i,j 145 4.2 A Dual Solver for TV-H−1 Minimization Setting ρi,j = ∇∆ ∇ · pn − ∆−1 (f /α) i,j · pn+1 i,j and inserting the above expression for ηi,j into (4.45) we have for every i, j 2ηi,j ∇∆ ∇ · pn − ∆−1 (f /α) i,j + κ2 τ |ηi,j |2 2 = (κ2 τ − 1) |ηi,j |2 − ∇∆ ∇ · pn − ∆−1 (f /α) i,j + |ρi,j |2 . n −1 Since pn+1 i,j ≤ 1 it follows that |ρi,j | ≤ ∇∆ ∇ · p − ∆ (f /α) i,j , and hence 2ηi,j ∇∆ ∇ · pn − ∆−1 (f /α) i,j + κ2 τ |ηi,j |2 ≤ (κ2 τ − 1) |ηi,j |2 . The last term is negative or zero if and only if κ2 τ − 1 ≤ 0. Hence, if τ ≤ 1/κ2 , we see that ∇ ∇ · pn − ∆−1 (f /α) is nonincreasing with n. For τ < 1/κ2 it is immediately clear that the norm is even decreasing, unless η = 0, that is, pn+1 = pn . 2 The same holds for κ2 τ = 1. Indeed, in this case, if ∇ ∇ · pn+1 − ∆−1 (f /α) = ∇ ∇ · pn − ∆−1 (f /α) 2 it follows that 0 = 2 η, ∇∆ ∇ · pn − ∆−1 (f /α) 2 + τ k∇∇ · ηk2 ≤ 2 η, ∇∆ ∇ · pn − ∆−1 (f /α) 2 + κ2 τ kηk2 X 2 n −1 − ∇∆ ∇ · p − ∆ (f /α) i,j + |ρi,j |2 , = 1≤i≤N 1≤j≤M and therefore ∇∆ ∇ · pn − ∆−1 (f /α) i,j ≤ |ρi,j |. Since in turn |ρi,j | ≤ ∇∆ ∇ · pn − ∆−1 (f /α) i,j n −1 we deduce |ρi,j | = ∇∆ ∇ · p − ∆ (f /α) i,j for each i, j. But this can only be if either ∇∆ ∇ · pn − ∆−1 (f /α) i,j = 0 or pn+1 i,j = 1. In both cases, the ﬁxed point n iteration (4.44) yields pn+1 i,j = pi,j for all i, j. Now, since ∇ ∇ · pn − ∆−1 (f /α) is decreasing with n, the norm is uniformly bounded and hence there exists an m ≥ 0 such that m = lim ∇ ∇ · pn − ∆−1 (f /α) . n→∞ Moreover the sequence pn has converging subsequences. Let p̄ be the limit of a subsequence (pnk ) and p̄′ be the limit of (pnk +1 ). Inserting pnk +1 and pnk into the ﬁxed 146 4.2 A Dual Solver for TV-H−1 Minimization point equation (4.44) and passing to the limit we have p̄′i,j p̄i,j − τ ∇∆ ∇ · p̄ − ∆−1 f /α i,j . = 1 + τ (∇∆ (∇ · p̄ − ∆−1 f /α))i,j Repeating the previous calculations we see that since m = ∇ ∇ · p̄ − ∆−1 (f /α) = ∇ ∇ · p̄′ − ∆−1 (f /α) , it is true that η̄i,j = (p̄′i,j − p̄i,j )/τ = 0 for every i, j, that is, p̄ = p̄′ . Hence p̄ is a ﬁxed point of (4.44), i.e., ( i = 1, . . . , N −1 −1 ∇∆ ∇ · p̄ − ∆ f /α i,j + ∇∆ ∇ · p̄ − ∆ f /α i,j p̄i,j = 0, ∀ j = 1, . . . , M which is the Euler equation for a solution of (4.43). One can deduce that p̄ solves (4.43) and that α∇ · p̄ is the projection P1K (∆−1 f ). Since this projection is unique, we deduce that all the sequence α∇ · pn converges to P1K (∆−1 f ). The theorem is proved if we can show that κ2 ≤ 64. By deﬁnition κ = k|∇∇ · k| = sup k∇∇ · pk . kpk≤1 Then for every i, j, we have k∇∇ · pk2 = 2 X (∇∇ · p)i,j . 1≤i≤N 1≤j≤M For more clarity let us set u := ∇ · p ∈ X for now. With the convention that p0,j = pN,j = pi,0 = pi,M = 0 we get k∇∇ · pk2 = = X 1≤i<N 1≤j≤M = X 1≤i<N 1≤j≤M = 1≤i≤N 1≤j≤M (ui+1,j − ui,j )2 + X 1≤i≤N 1≤j<M 1≤i≤N 1≤j≤M (ui,j+1 − ui,j )2 ((∇ · p)i+1,j − (∇ · p)i,j )2 + X h 1≤i<N 1≤j≤M + 2 2 2 X X (∇ u) (∇u) (∇ u) + = x i,j y i,j i,j 1≤i≤N 1≤j<M ((∇ · p)i,j+1 − (∇ · p)i,j )2 i2 pxi+1,j − pxi,j + pyi+1,j − pyi+1,j−1 − pxi,j − pxi−1,j + pyi,j − pyi,j−1 X h 1≤i≤N 1≤j<M X i2 , pxi,j+1 − pxi−1,j+1 + pyi,j+1 − pyi,j − pxi,j − pxi−1,j + pyi,j − pyi,j−1 147 4.2 A Dual Solver for TV-H−1 Minimization and further k∇∇ · pk2 ≤ 8 · X pxi+1,j 2 + pxi,j 2 + py 2 2 y + p i+1,j i+1,j−1 1≤i<N 1≤j≤M 2 2 2 2 + pxi,j + pxi−1,j + pyi,j + pyi,j−1 X 2 x 2 y 2 y 2 x pi,j+1 + pi−1,j+1 + pi,j+1 + pi,j +8· 1≤i≤N 1≤j<M x 2 x 2 y 2 y 2 + pi,j + pi−1,j + pi,j + pi,j−1 ≤ 64 · kpk2 ≤ 64. Remark 4.2.4. In our numerical computations we stop the fixed point iteration (4.44) as soon as the distance between the iterates is small enough, i.e., n+1 p − pn ≤ e · pn+1 , (4.46) where e is a chosen error bound. Then, in summary, to minimize (4.40) we apply the following algorithm Algorithm (P) • For an initial p0 = 0, iterate (4.44) until (4.46); • Set P1αK (∆−1 f ) = α∇ · pn̂ , where n̂ is the ﬁrst iterate of (4.44) which fulﬁlls (4.46); • Compute a minimizer u of (4.40) by u = f + ∆ P1αK (∆−1 f ) = f + ∆ α∇ · pn̂ . The second step is to use the presented algorithm for (4.40) in order to solve (4.33), i.e., min{J(u) = kT u − f k2−1 + 2α |Du| (Ω)}, u for an arbitrary bounded linear operator T . To do so we ﬁrst approximate a minimizer of (4.33) iteratively by a sequence of minimizers of “surrogate” functionals Js . This approach is inspired by similar methods used, e.g., in [BBAC04, DTV07]. Let τ > 0 be a ﬁxed stepsize. Starting with an initial condition u0 = f , we solve for k ≥ 0 uk+1 = argminu Js (u, uk ), 148 (4.47) 4.2 A Dual Solver for TV-H−1 Minimization where Js (u, uk ) = 2 2 1 1 u − u k + u − f + (Id − T )uk + |Du| (Ω). 2τ 2α −1 −1 Note that a ﬁxed point of Js is a potential minimizer for J. In the case of TVH−1 inpainting (4.47) was actually used as a ﬁxed point approach in the proof of a stationary solution of (2.27), cf. Section 2.2. A rigorous proof of convergence properties is still missing and is a matter for future investigation. Note however that in the case of image inpainting, i.e., T = 1Ω\D and f is replaced by 1Ω\D f , the optimality condition of (4.47) indeed describes a fourth-order diﬀusion inside of the inpainting domain D. Hence, in this case, minimizing (4.47) rather describes the behavior of solutions of the inpainting approach (2.27) than directly minimizing (4.33), cf. also Subsection 4.2.1. Despite the missing theory, the numerical results obtained by using this scheme for inpainting issues indicate its correct asymptotic behavior, see Section 4.2.3. Now, the corresponding optimality condition to (4.47) reads 1 1 0 ∈ ∂ |Du| (Ω) + ∆−1 (u − uk ) + ∆−1 u − f + (Id − T )uk , τ α which can be rewritten as ∆ −1 f1 − u f2 − u + τ α ∈ ∂ |Du| (Ω), where f1 = uk , f2 = f + (Id − T )uk . Setting f1 α + f2 τ α+τ ατ µ= , α+τ f= we end up with the same inclusion as (4.41), i.e., ∆−1 (f − u) ∈ ∂ |Du| (Ω), µ and Algorithm (P) for solving (4.40) can be directly applied. 149 4.2 A Dual Solver for TV-H−1 Minimization Algorithm TV-H−1 : • In the case T = Id directly apply Algorithm (P) to compute a minimizer of (4.33). • In the case T 6= Id iterate (4.47) by solving Algorithm (P) in every iteration step until the two subsequent iterates uk and uk+1 are suﬃciently close. 4.2.3 Applications In this section we present applications of our new algorithm for solving (4.33) for image denoising, decomposition and inpainting, and present numerical results. For comparison, we also present results for the TV-L2 model in [ROF92] on the same images. Now, in order to compute the minimizer u of (4.33), we have the following algorithm. Note that in our numerical examples e in (4.46) is chosen to be 10−4 . Image Denoising and Decomposition In the case of image denoising and image decomposition the operator T = Id and thus Algorithm (P) can be directly applied. For image denoising the signal to noise ratio (SNR) is computed as SN R = 20 log hf i2 σ , with hf i2 the average value of the pixels fi,j and σ the standard deviation of the noise. For our numerical results the parameter α in (4.33) was chosen so that the best residual-mean-squared-error (RMSE) is obtained. We deﬁne the RMSE as v X 1 u u RM SE = (ui,j − ûi,j )2 , N M t 1≤i≤N 1≤j≤M where û is the original image without noise, cf. [LV08]. Numerical examples for image denoising with TV-H−1 minimization and their comparison with the results obtained by the TV-L2 approach are presented in Figures 4.4-4.7. In both examples the superiority of the TV-H−1 minimization approach with respect to the separation of noise and edges is clearly visible. We also apply (4.33) for texture removal in images, i.e., image decomposition, and compare the numerical results with those of the TV-L2 approach, cf. Figure 4.8. 150 4.2 A Dual Solver for TV-H−1 Minimization (a) original (b) noisy, SN R = 25.2 Figure 4.4: Image of a horse and its noisy version with additive white noise The cartoon-texture decomposition in this example works better in the case of TVH−1 minimization, since this approach diﬀerentiates between small oscillations and strong edges, better than the TV-L2 approach. Image Inpainting In order to apply our algorithm to TV-H−1 inpainting we follow the method of surrogate functionals from Section 4.2.2. In fact it turns out that a ﬁxed point of the corresponding optimality condition of (4.47) with T = 1Ω\D and f replaced by 1Ω\D f is indeed a stationary solution of (2.27). This approach is also motivated by the ﬁxed point approach used in [BHS08] in order to prove existence of a stationary solution of (2.27). Hence a stationary solution to (2.27) can be computed iteratively by the following algorithm: Take u0 = f , with any trivial (zero) expansion to the inpainting domain, and solve for k ≥ 0 1 1 k 2 k 2 ||u − 1Ω\D f − (1 − 1Ω\D )u ||−1 → uk+1 , min |Du| (Ω) + ||u − u ||−1 + 2τ 2α u∈BV (Ω) (4.48) for positive iterationsteps τ > 0, and α = 1/λ0 . Now, as before, let f1 = uk , f2 = 1Ω\D f + (1 − 1Ω\D )uk and f1 α + f2 τ α+τ ατ , µ= α+τ f= 151 4.2 A Dual Solver for TV-H−1 Minimization (a) f = u + v (b) TV-L2 : u (c) TV-L2 : v (d) TV-H−1 : u (e) TV-H−1 : v Figure 4.5: Denoising results for the image of a horse in Figure 4.4. Results from the TV-L2 denoising model compared with TV-H−1 denoising with α = 0.05 for both. 152 4.3 Domain Decomposition for TV Minimization (a) original (b) noisy, SN R = 29.4 Figure 4.6: Image of the roof of a house in Scotland and its noisy version with additive white noise then we end up with the same inclusion as (4.41), i.e., ∆−1 (f − u) ∈ ∂ |Du| (Ω), µ and Algorithm (P) can be directly applied. Compare Figure 4.9 for a numerical example. In the following section we shall see how the dual algorithm of the present section can be used in order to formulate a domain decomposition approach for TV-H−1 inpainting. 4.3 Domain Decomposition for TV Minimization Domain decomposition methods are a special instance of subspace splitting methods and were introduced as techniques for solving partial diﬀerential equations based on a decomposition of the spatial domain of the problem into several subdomains [Li88, BPWX91, Xu92, CM94, QV99, XZ02, LXZ03, BDHP03, NS05]. The initial equation restricted to the subdomains deﬁnes a sequence of new local problems. The main goal is to solve the initial equation via the solution of the local problems. This procedure induces a dimension reduction which is the major aspect responsible for the success of such a method. Indeed, one of the principal motivations is the formulation of solvers which can be easily parallelized. The current section is mainly based on a joint work with Massimo Fornasier developed in [FS07]. Therein we are concerned with the numerical minimization by means of 153 4.3 Domain Decomposition for TV Minimization (a) f = u + v (b) TV-L2 : u (c) TV-L2 : v (d) TV-H−1 : u (e) TV-H−1 : v Figure 4.7: Denoising results for the image of the roof in Figure 4.6. Results from the TV-L2 denoising model with α = 0.05 compared with TV-H−1 denoising with α = 0.01 subspace splittings of energy functionals in Hilbert spaces involving convex constraints coinciding with a semi-norm for a subspace. In more detail: Let H be a real separable Hilbert space. We are interested in the numerical minimization in H of the general form of functionals J(u) := kT u − f k2H + 2αψ(u), 154 (4.49) 4.3 Domain Decomposition for TV Minimization (a) f = u + v (b) TV-L2 : u (c) TV-L2 : v (d) TV-H−1 : u (e) TV-H−1 : v Figure 4.8: Decomposition into cartoon and texture of a synthetic image. Results from the TV-L2 model with α = 1 and TV-H−1 minimization with α = 0.1. 155 4.3 Domain Decomposition for TV Minimization Figure 4.9: TV-H−1 inpainting result for the image of the two statues with α = 1/λ0 = 0.005. where T ∈ L(H) is a bounded linear operator, f ∈ H is a datum, and α > 0 is a ﬁxed constant. The function ψ : H → R+ ∪ {+∞} is a semi-norm for a suitable subspace Hψ of H. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. In particular, we investigate splittings into arbitrary orthogonal subspaces H = V1 ⊕ V2 for which we may have ψ(πV1 (u) + πV2 (v)) 6= ψ(πV1 (u)) + ψ(πV2 (v)), u, v ∈ H, (4.50) where πVi is the orthogonal projection onto Vi . With this splitting we want to minimize J by suitable instances of the following alternating algorithm: Pick an initial V1 ⊕ V2 ∋ (0) (0) u1 + u2 := u(0) ∈ HΨ , for example u(0) = 0, and iterate (n+1) (n) u ≈ argminv1 ∈V1 J(v1 + u2 ) 1 (n+1) (n+1) u2 ≈ argminv2 ∈V2 J(u1 u(n+1) := u(n+1) + u(n+1) . 1 2 + v2 ) (4.51) Diﬀerent to situations already encountered in the literature we shall propose a domain decomposition algorithm for the minimization of functionals for which we neither can guarantee smoothness nor additivity with respect to its local problems. In [TT98, TX02], for instance, the authors consider the minimization of a convex function F which is assumed to be Gateaux diﬀerentiable in a real reﬂexive Banach space and prove convergence for an asynchronous space decomposition method applied to this 156 4.3 Domain Decomposition for TV Minimization general problem. In contrast to this work our functional J consists of a non-smooth part which is not Gateaux-diﬀerentiable and hence the analysis carried out in [TT98, TX02] is not directly applicable to our problem. Further domain decomposition methods have been proposed for the case when ψ is not a smooth function, but it is additive, i.e., ψ(πV1 (u) + πV2 (v)) = ψ(πV1 (u)) + ψ(πV2 (v)), u, v ∈ H, see [Ca97] for instance. The additivity of ψ is crucial for the proof of convergence of the algorithm and cannot be directly generalized for functions ψ without this property. To our knowledge no results and methods are presented in the literature related to the situation described by our fundamental assumptions, where ψ is not smooth (in particular is not diﬀerentiable) and is not additive (only subadditive), cf. (4.50). In [FS07] we present a subspace minimization algorithm (4.51) for the general problem in (4.49) with (4.50). On each subspace an iterative proximity-map algorithm is implemented via oblique thresholding (cf. Section 4.3.3), which is the main new tool introduced in [FS07]. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. In particular we apply the theory and the algorithms developed in [FS07] to adapt domain decompositions to the minimization of functionals with total variation constraints. Therefore interesting solutions may be discontinuous, e.g., along curves in 2D. These discontinuities may cross the interfaces of the domain decomposition patches. Hence, the crucial diﬃculty is the correct treatment of interfaces, with the preservation of crossing discontinuities and the correct matching where the solution is continuous instead. We consider the minimization of the functional J in the following setting: Let Ω ⊂ Rd , for d = 1, 2, be a bounded open set with Lipschitz boundary. We are interested in the case when H = L2 (Ω), Hψ = BV (Ω) and ψ(u) = |Du|(Ω), the variation of u. Then a domain decomposition Ω1 ∪ Ω2 ⊂ Ω ⊂ Ω̄1 ∪ Ω̄2 induces the space splitting into Vi := {u ∈ L2 (Ω) : supp(u) ⊂ Ωi }, i = 1, 2. This means we shall solve (2.26), i.e., we will minimize the functional J(u) := kT u − f k2L2 (Ω) + 2α|Du|(Ω) 157 4.3 Domain Decomposition for TV Minimization by means of the alternating subspace minimizations (4.51). Moreover, we shall see that also the TV-H−1 minimization problem (4.33) from Section 4.2, i.e., J(u) := kT u − f k2H −1 (Ω) + 2α|Du|(Ω), can be treated within this framework, cf. Section 4.3.7. In contrast to [Ca97], in the case of total variation minimization (2.26) and (4.33) in general we have |D(uΩ1 + uΩ2 )|(Ω) 6= |DuΩ1 |(Ω1 ) + |DuΩ2 |(Ω2 ), compare (4.55) for a more detailed presentation. We limit ourself to mentioning that, to our knowledge, the work contained in this section is the ﬁrst in presenting a successful domain decomposition approach to total variation minimization. The motivation is that several approaches are directed to the solution of the Euler-Lagrange equations associated to the functional J, which determine a singular elliptic PDE involving the 1Laplace operator. Due to the fact that |Du|(Ω) is not diﬀerentiable, one has to discretize its subdiﬀerential, and its characterization is indeed hard to implement numerically in a correct way. The lack of a simple characterization of the subdiﬀerential of the total variation especially raises signiﬁcant diﬃculties in dealing with discontinuous interfaces between patches of a domain decomposition. There is another and partially independent attempt of addressing domain decomposition methods for total variation minimization within the group of Martin Burger at the University of Münster (Germany)1 , in particular the diploma thesis of Jahn Müller [Mu08]2 , in cooperation with Sergej Gorlatch. Their approach diﬀers from ours in the sense that their domain decomposition is based on computing a minimizer of the functional J by a primal-dual Newton method for the function u and its dual variable p. There, the function u and its dual p are deﬁned in alternating grid points, i.e., the degrees of freedom of the dual variable are placed in the center between the pixels of u. Their numerical results are very satisfactory, but their theory is still not conclusive. Our current approach overcomes the diﬃculties, which may arise when attempting the direct solution of the Euler-Lagrange equations, by minimizing the functional via an iterative proximity map algorithm, as proposed, e.g., in [Ch04, CW05, DDD04, 1 http://wwwmath.uni-muenster.de/u/burger/index/ en.html available online under http://wwwmath.uni-muenster.de/num/Arbeitsgruppen/ag burger/ organization/burger/pictures/DA%20Mueller.pdf 2 158 4.3 Domain Decomposition for TV Minimization DFL08, DTV07, Fo07]. Using this algorithm for the solution of the subproblems has additionally the advantage that its numerical solution is already quite established in the literature, e.g., in [Ch04], and that is is very simple to implement and very ﬂexible with respect to acceleration for instance. Further this algorithm is perfectly designed to provide us with desirable convergence properties, as the monotonicity of the energy J, which decreases at each iteration, cf. Sections 4.3.4 and 4.3.5 for details. Note that another speciﬁc example for the minimization problem (4.49) is that of ℓ1 minimization, which found growing interest in recent years in the context of sparse recovery and compressed sensing (cf. e.g., the review papers [Ba07, Ca06]). The application of the following domain decomposition method, i.e., subspace correction algorithm, to ℓ1 minimization is presented in [FS07] as a second example for the use of such a method. We shall not present this part of our work here, since it is clearly beyond the scope of this thesis. It is also worth mentioning that, due to the generality of our setting, our approach can be extended to more general subspace decompositions, not only those arising from a domain splitting. This can open room to more sophisticated multiscale algorithms where Vi are multilevel spaces, e.g., originating in a wavelet decomposition. In what follows we trace the results developed in [FS07, Sc09]. 4.3.1 Preliminary Assumptions We begin this section with a short description of the generic notation used in this section. In the following H is a real separable Hilbert space endowed with the norm k·kH, and Ω ⊂ Rd denotes an open bounded set with Lipschitz boundary. For some countable index set Λ we denote by ℓp = ℓp (Λ), 1 ≤ p ≤ ∞, the space of real sequences u = (uλ )λ∈Λ with norm kukp = kukℓp := X λ∈Λ |uλ |p !1/p , 1≤p<∞ and kuk∞ := supλ∈Λ |uλ | as usual. If (vλ ) is a sequence of positive weights then we deﬁne the weighted spaces ℓp,v = ℓp,v (Λ) = {u, (uλ vλ ) ∈ ℓp (Λ)} with norm !1/p X p p kukp,v = kukℓp,v = k(uλ vλ )kp = vλ |uλ | λ∈Λ 159 4.3 Domain Decomposition for TV Minimization (with the standard modiﬁcation for p = ∞). Depending on the context, the symbol ≃ may deﬁne an equivalence of norms or an isomorphism of spaces or sets. More speciﬁc notation will be deﬁned, where they turn out to be useful. Remaining notation follows the preﬁx of the thesis. The convex constraint function ψ We are given a function ψ : H → R+ ∪ {+∞} with the following properties: (Ψ1) ψ is sublinear, i.e., ψ(u + v) ≤ ψ(u) + ψ(v) for all u, v ∈ H; (Ψ2) ψ is 1-homogeneous, i.e., ψ(λu) = |λ|ψ(u) for all λ ∈ R. (Ψ3) ψ is lower-semicontinuous in H, i.e., for any converging sequence un → u in H ψ(u) ≤ lim inf ψ(un ). n∈N Associated with ψ we assume that there exists a dense subspace Hψ ⊂ H for which ψ|Hψ is a seminorm and Hψ endowed with the norm kukHψ := kukH + ψ(u), is a Banach space. We do not assume instead that Hψ is reﬂexive in general; note that due to the dense embedding Hψ ⊂ H we have Hψ ⊂ H ≃ H∗ ⊂ (Hψ )∗ , and the duality h·, ·i(Hψ )∗ ×Hψ extends the scalar product on H. In particular, H is weakly-∗-dense in (Hψ )∗ . In the following we require (H1) bounded subsets in Hψ are sequentially bounded in some other topology τ ψ of Hψ ; (H2) ψ is lower-semicontinuous with respect to the topology τ ψ , i.e., for any sequence un in Hψ which converges to u ∈ Hψ with the τ ψ −topology satisﬁes ψ(u) ≤ lim inf n∈N ψ(un ); In practice, we will always require also that 160 4.3 Domain Decomposition for TV Minimization (H3) Hψ = {u ∈ H : ψ(u) < ∞}. We list in the following the speciﬁc examples we consider in this section. Examples 4.3.1. 1. Let Ω ⊂ Rd , for d = 1, 2 be a bounded open set with Lipschitz boundary, and H = L2 (Ω). Let V (u, Ω) be defined as in (2.25) and for u ∈ BV (Ω) let |D(u)|(Ω) = V (u, Ω) total variation of the finite Radon measure Du, cf. Appendix A.5. We define ψ(u) = V (u, Ω) and it is immediate to see that Hψ must coincide with BV (Ω). Due to the embedding L2 (Ω) ⊂ L1 (Ω) and the Sobolev embedding [AFP00, Theorem 3.47] we have kukHψ = kuk2 + V (u, Ω) ≃ kuk1 + |Du|(Ω) = kukBV . Hence (Hψ , k · kHψ ) is indeed a Banach space. It is known that V (·, Ω) is lowersemicontinuous with respect to L2 (Ω) [AFP00, Proposition 3.6]. We say that a sequence (un )n in BV (Ω) converges to u ∈ BV (Ω) with the weak-∗-topology if (un )n converges to u in L1 (Ω) and Dun converges to Du with the weak-∗-topology in the sense of finite Radon measures. Bounded sets in BV (Ω) are sequentially weakly-∗compact [AFP00, Proposition 3.13], and V (·, Ω) is lower-semicontinuous with respect to the weak-∗-topology. 2. Let H = RN endowed with the Euclidean norm, and Q : RN → Rn , for n ≤ N , is a fixed linear operator. We define ψ(u) = kQukℓn1 . Clearly Hψ = RN and all the requested properties are trivially fulfilled. One particular example of this finite dimensional situation is associated with the choice of Q : RN → RN −1 given by Q(u)i := N (ui+1 − ui ), i = 0, . . . , N − 2. In this case ψ(u) = kQukℓ1 is the discrete variation of the vector u and the model can be seen as an approximation to the situation encountered in the first example, by discrete sampling and finite differences, i.e., setting ui := u( Ni ) and u ∈ BV (0, 1). Bounded subspace decompositions In the following we will consider orthogonal decompositions of H into closed subspaces. We will also require that such a splitting is bounded in Hψ . Assume that V1 , V2 are two mutually orthogonal, and complementary subspaces of H, i.e., H = V1 ⊕ V2 , and πVi are the corresponding orthogonal projections, for i = 1, 2. Moreover we require the mapping property πVi |Hψ : Hψ → Viψ := Hψ ∩ Vi , 161 i = 1, 2, 4.3 Domain Decomposition for TV Minimization continuously in the norm of Hψ , and that Range(πVi |Hψ ) = Viψ is closed. This implies that Hψ splits into the direct sum Hψ = V1ψ ⊕ V2ψ . Example 4.3.2. Let Ω1 ⊂ Ω ⊂ Rd , for d = 1, 2, be two bounded open sets with Lipschitz boundaries, and Ω2 = Ω \ Ω1 . We define Vi := {u ∈ L2 (Ω) : supp(u) ⊂ Ωi }, i = 1, 2. Then πVi (u) = u1Ωi . For ψ(u) = V (u, Ω), by [AFP00, Corollary 3.89], Viψ = BV (Ω) ∩ Vi is a closed subspace of BV (Ω) and πVi (u) = u1Ωi ∈ Viψ , i = 1, 2, for all u ∈ BV (Ω). 4.3.2 A Convex Variational Problem and Subspace Splitting We are interested in the minimization in H (actually in Hψ ) of the functional J(u) := kT u − f k2H + 2αψ(u), where T ∈ L(H) is a bounded linear operator, f ∈ H is a datum, and α > 0 is a ﬁxed constant. In order to guarantee the existence of its minimizers we assume that: (C) J is coercive in H, i.e., {J ≤ C} := {u ∈ H : J(u) ≤ C} is bounded in H for all constants C > 0. Example 4.3.3. Assume Ω ⊂ Rd , for d = 1, 2 be a bounded open set with Lipschitz boundary, H = L2 (Ω) and ψ(u) = V (u, Ω) (compare Examples 4.3.1.1). In this case we deal with total variation minimization. It is well-known that if T 1Ω 6= 0 then condition (C) is indeed satisfied, see [Ve01, Proposition 3.1] and [CL97]. Lemma 4.3.4. Under the assumptions above, J has minimizers in Hψ . Proof. The proof is a standard application of the direct method of calculus of variations and can be found for example in [CW05]. Let (un )n ⊂ H, a minimizing sequence. By assumption (C) we have kun kH + ψ(un ) ≤ C for all n ∈ N. Therefore by (H1) we can extract a subsequence in Hψ converging in the topology τ ψ . Possibly passing to a further subsequence we can assume that it also converges weakly in H. By lower-semicontinuity of kT u − f k2H with respect to the weak topology of H and the lower-semicontinuity of ψ with respect to the topology τ ψ , ensured by assumption (H2), we have the desired existence of minimizers. 162 4.3 Domain Decomposition for TV Minimization The minimization of J is a classical problem [ET76] which was recently reconsidered by several authors, [Ch04, CW05, DDD04, DTV07, SCD02, Ti96], with emphasis on the computability of minimizers in particular cases. They studied essentially the same algorithm for the minimization. For ψ with properties (Ψ1 − Ψ3), there exists a closed convex set Kψ ⊂ H such that ψ ∗ (u) = sup {hv, ui − ψ(v)} v∈H 0 if u ∈ Kψ = χKψ (u) = +∞ otherwise. See also Examples 4.3.6.2 below. In the following we assume furthermore that Kψ = −Kψ . For any closed convex set K ⊂ H we denote PK (u) = argminv∈K ku − vkH the orthogonal projection onto K. For Sψ α := I − PαKψ , called the generalized thresholding map in the signal processing literature, the iteration (n) u(n+1) = Sψ + T ∗ (f − T u(n) )) α (u (4.52) converges weakly to a minimizer u ∈ Hψ of J, for any initial choice u(0) ∈ Hψ , provided that T and f are suitably rescaled so that kT k < 1. For particular situa- tions, e.g., H = ℓ2 (Λ) and ψ(u) = kukℓ1,w , one can prove the convergence in norm [DDD04, DTV07]. As it is pointed out, for example in [DFL08, Fo07], this algorithm converges at a poor rate, unless T is non-singular or has additional special spectral properties. For this reason acceleration by means of projected steepest descent iterations [DFL08] and domain decomposition methods [Fo07] were proposed. The particular situation considered in [Fo07] is H = ℓ2 (Λ) and ψ(u) = kukℓ1 (Λ) . In this case one takes advantage of the fact that for a disjoint partition of the index set Λ = Λ1 ∪ Λ2 we have the splitting ψ(uΛ1 + uΛ2 ) = ψ(uΛ1 ) + ψ(uΛ2 ) for any vector uΛi supported on Λi , i = 1, 2. Thus, a decomposition into column subspaces (i.e., componentwise) of the operator T (if identiﬁed with a suitable matrix) is realized, and alternating minimizations on these subspaces are performed by means of iterations of the type (4.52). This leads, e.g., to the following sequential algorithm: Pick an initial 163 4.3 Domain Decomposition for TV Minimization (0,L) u Λ1 (0,M ) + u Λ2 := u(0) ∈ ℓ1 (Λ), for example u(0) = 0, and iterate ( (n+1,0) (n,L) u Λ1 = u Λ1 (n+1,ℓ) (n+1,ℓ+1) ∗ ((f − T u(n,M ) ) − T u(n+1,ℓ) ) u + T ℓ = 0, . . . , L − 1 u = S α Λ2 Λ2 Λ1 Λ1 Λ1 Λ1 ( Λ1 (n+1,0) (n,M ) u Λ2 = u Λ2 (n+1,ℓ) (n+1,ℓ+1) ∗ ((f − T u(n+1,L) ) − T u(n+1,ℓ) ) ℓ = 0, . . . , M − 1 u + T u = S α Λ1 Λ1 Λ2 Λ2 Λ2 Λ2 Λ2 (n+1) (n+1,L) (n+1,M ) u := uΛ1 + u Λ2 . (4.53) Here the operator Sα is the soft-thresholding operator which acts componentwise Sα v = (Sα vλ )λ∈Λ and deﬁned by Sα (x) = x − sgn(x)α, |x| > α 0, otherwise. (4.54) The expected beneﬁt from this approach is twofold: 1. Instead of solving one large problem with many iteration steps, we can solve approximatively several smaller subproblems, which might lead to an acceleration of convergence and a reduction of the overall computational eﬀort, due to possible conditioning improvements; 2. The subproblems do not need more sophisticated algorithms, simply reproduce at smaller dimension the original problem, and they can be solved in parallel. The nice splitting of ψ as a sum of evaluations on subspaces does not occur, for instance, when H = L2 (Ω), ψ(u) = V (u, Ω) = |Du|(Ω), and Ω1 ∪ Ω2 ⊂ Ω ⊂ Ω̄1 ∪ Ω̄2 is a disjoint decomposition of Ω. Indeed, cf. [AFP00, Theorem 3.84], we have Z − |D(uΩ1 + uΩ2 )|(Ω) = |DuΩ1 |(Ω1 ) + |DuΩ2 |(Ω2 ) + |u+ Ω1 (x) − uΩ2 (x)|dH1 (x) . | ∂Ω1 ∩∂Ω2 {z } additional interface term (4.55) Here one should not confuse Hd with any Hψ since the former indicates the Hausdorﬀ measure of dimension d. The symbols v + and v − denote the left and right approximated limits at jump points [AFP00, Proposition 3.69]. The presence of the additional R − boundary interface term ∂Ω1 ∩∂Ω2 |u+ Ω1 (x) − uΩ2 (x)|dH1 (x) does not allow to use in a straightforward way iterations as in (4.52) to minimize the local problems on Ωi . 164 4.3 Domain Decomposition for TV Minimization Moreover, also in the sequence space setting mentioned above, the hope for a better conditioning by column subspace splitting as in [Fo07] might be ill-posed, no such splitting needs to be well conditioned in general (good cases are provided in [Tr08] instead). Therefore, one may want to consider arbitrary subspace decompositions and, in order to deal with these more general situations, we investigate splittings into arbitrary orthogonal subspaces H = V1 ⊕ V2 for which we may have ψ(πV1 (u) + πV2 (v)) 6= ψ(πV1 (u)) + ψ(πV2 (v)). In principle, in this work we limit ourself to consider the detailed analysis for two subspaces V1 , V2 . Nevertheless, the arguments can be easily generalized to multiple subspaces V1 , . . . , VN, see, e.g., [Fo07], and in the numerical experiments we will also test this more general situation. With this splitting we want to minimize J by suitable instances of the following (0) (0) alternating algorithm: Pick an initial V1 ⊕ V2 ∋ u1 + u2 := u(0) ∈ HΨ , for example u(0) = 0, and iterate (n+1) (n) ≈ argminv1 ∈V1 J(v1 + u2 ) u 1 (n+1) (n+1) u2 ≈ argminv2 ∈V2 J(u1 + v2 ) u(n+1) := u(n+1) + u(n+1) . 1 2 (4.56) We use “≈” (the approximation symbol) because in practice we never perform the exact minimization, as in (4.53). In the following section we discuss how to realize the approximation to the individual subspace minimizations. As pointed out above, this cannot just reduce to a simple iteration of the type (4.52). 4.3.3 Local Minimization by Lagrange Multipliers Let us consider, for example, argminv1 ∈V1 J(v1 + u2 ) = argminv1 ∈V1 kT v1 − (f − T u2 )k2H + 2αψ(v1 + u2 ). (4.57) First of all, observe that {u ∈ H : πV2 u = u2 , J(u) ≤ C} ⊂ {J ≤ C}, hence the former set is also bounded by assumption (C). By the same argument as in Lemma 4.3.4, the minimization (4.57) has a solution. It is useful to introduce an auxiliary functional 165 4.3 Domain Decomposition for TV Minimization Js1 , called the surrogate functional of J (cf. also (4.47) in Section 4.2.2 for a similar approach): Assume a, u1 ∈ V1 and u2 ∈ V2 and deﬁne Js1 (u1 + u2 , a) := J(u1 + u2 ) + ku1 − ak2H − kT (u1 − a)k2H. (4.58) A straightforward computation shows that Js1 (u1 + u2 , a) = ku1 − (a + πV1 T ∗ (f − T u2 − T a))k2H + 2αψ(u1 + u2 ) + ϕ(a, f, u2 ), where ϕ is a function of a, f, u2 only. We want to realize an approximate solution to (0) (4.57) by using the following algorithm: For u1 ∈ V1ψ , (ℓ+1) u1 (ℓ) = argminu1 ∈V1 Js1 (u1 + u2 , u1 ), ℓ ≥ 0. (4.59) Before proving the convergence of this algorithm, we need to investigate ﬁrst how to (n+1) compute practically u1 (n) for given u1 . To this end we need to introduce further concepts and to recall some useful results. Generalized Lagrange multipliers for nonsmooth objective functions Let us begin this subsection with recalling the concept of a subdiﬀerential, cf. also Appendix A.2. Definition 4.3.5. For a locally convex space V and for a convex function F : V → R∪{−∞, +∞}, we define the subdiﬀerential of F at x ∈ V , as ∂F (x) = ∅ if F (x) = ∞, otherwise ∂F (x) := ∂FV (x) := {x∗ ∈ V ∗ : hx∗ , y − xi + F (x) ≤ F (y) ∀y ∈ V }, where V ∗ denotes the dual space of V . It is obvious from this definition that 0 ∈ ∂F (x) if and only if x is a minimizer of F . Since we deal with several spaces, namely, H, Hψ , Vi , Viψ , it will turn out to be useful to distinguish sometimes in which space (and associated topology) the subdifferential is defined by imposing a subscript ∂V F for the subdifferential considered on the space V . Examples 4.3.6. 1. Let V = ℓ1 (Λ) and let F (x) := kxk1 be the ℓ1 −norm. We have ∂k · k1 (x) = {ξ ∈ ℓ∞ (Λ) : ξλ ∈ ∂| · |(xλ ), λ ∈ Λ} where ∂| · |(z) = {sgn(z)} if z 6= 0 and ∂| · |(0) = [−1, 1]. 166 (4.60) 4.3 Domain Decomposition for TV Minimization 2. Assume, V = H and ϕ ≥ 0 is a proper lower-semicontinuous convex function. For F (u; z) = ku − zk2H + 2ϕ(u), we define the function proxϕ (z) := argminu∈V F (u; z), which is called the proximity map in the convex analysis literature, e.g., [ET76, CW05], and generalized thresholding in the signal processing literature, e.g., [DDD04, DFL08, DTV07, Fo07]. Observe that by ϕ ≥ 0 the function F is coercive in H and by lowersemicontinuity and strict convexity of the term ku − zk2H this definition is well-posed. In particular, proxϕ (z) is the unique solution of the following differential inclusion 0 ∈ (u − z) + ∂ϕ(u). It is well known [ET76, RW98] that the proximity map is nonexpansive, i.e., kproxϕ (z1 ) − proxϕ (z2 )kH ≤ kz1 − z2 kH. In particular, if ϕ is a 1-homogeneous function then proxϕ (z) = (I − PKϕ )(z), where Kϕ is a suitable closed convex set associated to ϕ, see for instance [CW05]. Under the notations of Deﬁnition 4.3.5, we consider the following constrained minimization problem argminx∈V {F (x) : G(x) = 0}, (4.61) where G : V → R is a bounded linear operator on V . We have the following useful result. Theorem 4.3.7. (Generalized Lagrange multipliers for nonsmooth objective functions, Theorem 1.8, [BP96]) If F is continuous at a point of ker G and G∗ has closed range in V then a point x0 ∈ ker G is an optimal solution of (4.61) if and only if ∂F (x0 ) ∩ RangeG∗ 6= ∅. Oblique thresholding We want to exploit Theorem 4.3.7 in order to produce an algorithmic solution to each iteration step (4.59). 167 4.3 Domain Decomposition for TV Minimization Theorem 4.3.8 (Oblique thresholding). For u2 ∈ V2ψ and for z ∈ V1 the following statements are equivalent: (i) u∗1 = argminu∈V1 ku − zk2H + 2αψ(u + u2 ); (ii) there exists η ∈ Range(πV2 |Hψ )∗ ≃ (V2ψ )∗ such that 0 ∈ u∗1 − (z − η) + α∂Hψ ψ(u∗1 + u2 ). Moreover, the following statements are equivalent and imply (i) and (ii). (iii) there exists η ∈ V2 such that u∗1 = (I −PαKψ )(z+u2 −η)−u2 = Sψ α (z+u2 −η)−u2 ∈ V1 ; (iv) there exists η ∈ V2 such that η = πV2 PαKψ (η − (z + u2 )). Proof. Let us show the equivalence between (i) and (ii). The problem in (i) can be reformulated as u∗1 = argminu∈Hψ {F (u) := ku − zk2H + 2αψ(u + u2 ), πV2 (u) = 0}. The latter is a special instance of (4.61). Moreover, F is continuous in V1ψ ⊂ V1 = ker πV2 in the norm-topology of Hψ (while in general it is not in V1 with the norm topology of H). Recall now that πV2 |Hψ is assumed to be a bounded and surjective map with closed range in the norm topology of Hψ (see Section 4.3.1). This means that (πV2 |Hψ )∗ is injective and that Range(πV2 |Hψ )∗ ≃ (V2ψ )∗ is closed. Therefore, by an application of Theorem 4.3.7, the optimality of u∗1 is equivalent to the existence of η ∈ Range(πV2 |Hψ )∗ ≃ (V2ψ )∗ such that −η ∈ ∂Hψ F (u∗1 ). Due to the continuity of ku − zk2H in Hψ , we have, by [ET76, Proposition 5.6], that ∂Hψ F (u∗1 ) = 2(u∗1 − z) + 2α∂Hψ ψ(u∗1 + u2 ). Thus, the optimality of u∗1 is equivalent to 0 ∈ u∗1 − (z − η) + α∂Hψ ψ(u∗1 + u2 ). This concludes the equivalence of (i) and (ii). Let us show now that (iii) implies (ii). The condition in (iii) can be rewritten as ξ = (I − PαKψ )(z + u2 − η), 168 ξ = u∗1 + u2 . 4.3 Domain Decomposition for TV Minimization Since ψ ≥ 0 is 1-homogeneous and lower-semicontinuous, by Examples 4.3.6.2, the latter is equivalent to 0 ∈ ξ − (z + u2 − η) + α∂Hψ(ξ) or, by (H3), ξ = argminu∈Hku − (z + u2 − η)k2H + 2αψ(u) = argminu∈Hψ ku − (z + u2 − η)k2H + 2αψ(u). The latter optimal problem is equivalent to 0 ∈ ξ − (z + u2 − η) + α∂Hψ ψ(ξ) or 0 ∈ u∗1 − (z − η) + α∂Hψ ψ(u∗1 + u2 ). Since V2 ⊂ (V2ψ )∗ ≃ Range(πV2 |Hψ )∗ we deduce that (iii) implies (ii). We prove now the equivalence between (iii) and (iv). We have u∗1 = (I − PαKψ )(z + u2 − η) − u2 ∈ V1 = z − η − PαKψ (z + u2 − η). By applying πV2 to both sides of the latter equality we get 0 = −η − πV2 PαKψ (z + u2 − η). By recalling that Kψ = −Kψ , we obtain the ﬁxed point equation η = πV2 PαKψ (η − (z + u2 )). (4.62) Conversely, assume η = πV2 PαKψ (η − (z + u2 )) for some η ∈ V2 . Then (I − PαKψ )(z + u2 − η) − u2 = z − η − PαKψ (z + u2 − η) = z − πV2 PαKψ (η − (z + u2 )) − PαKψ (z + u2 − η) = z − (I − πV2 )PαKψ (z + u2 − η) = z − πV1 PαKψ (z + u2 − η) = u∗1 ∈ V1 . Remark 4.3.9. 1. Unfortunately in general we have V2 ( (V2ψ )∗ which excludes the complete equivalence of the previous conditions (i)-(iv). For example, in the case H = i ℓ2 (Λ) and ψ(u) = kukℓ1 , Λ = Λ1 ∪ Λ2 , Vi = ℓΛ 2 (Λ) := {u ∈ ℓ2 (Λ) : supp(u) ⊂ Λi }, ψ Λ2 i = 1, 2, we have V2 = ℓ1 (Λ) = {u ∈ ℓ1 (Λ) : supp(u) ⊂ Λ2 }, hence, V2 ( (V2ψ )∗ ≃ Λ2 Λ2 2 ℓΛ ∞ (Λ) = {u ∈ ℓ∞ (Λ) : supp(u) ⊂ Λi }. It might well be that η ∈ ℓ∞ (Λ) \ ℓ2 (Λ). However, since ψ(uΛ1 + uΛ2 ) = ψ(uΛ1 ) + ψ(uΛ2 ) in this case, we have 0 ∈ u∗1 − z + α∂k · k1 (u∗1 ) and therefore may choose any η in ∂k · k1 (u2 ). Following [Fo07], u2 is assumed 169 4.3 Domain Decomposition for TV Minimization to be the result of soft-thresholding iterations, hence u2 is a finitely supported vector. Therefore, by Examples 4.3.6.1, we can choose η to be also a finitely supported vector, 2 hence η ∈ ℓΛ 2 (Λ) = V2 . This means that the existence of η ∈ V2 as in (iii) or (iv) of the previous theorem may occur also in those cases for which V2 ( (V2ψ )∗ . In general, we can only observe that V2 is weakly-∗-dense in (V2ψ )∗ . 2. For H of finite dimension – which is the relevant case in numerical applications – all the spaces are independent of the particular attached norm and coincide with their duals, hence all the statements (i)-(iv) of the previous theorem are equivalent in this case. A simple constructive test for the existence of η ∈ V2 as in (iii) or (iv) of the previous theorem is provided by the following iterative algorithm: η (0) ∈ V2 , η (m+1) = πV2 PαKψ (η (m) − (z + u2 )), m ≥ 0. (4.63) Proposition 4.3.10. The following statements are equivalent: (i) there exists η ∈ V2 such that η = πV2 PαKψ (η − (z + u2 )) (which is in turn the condition (iv) of Theorem 4.3.8) (ii) the iteration (4.63) converges weakly to any η ∈ V2 that satisfies (4.62). In particular, there are no fixed points of (4.62) if and only if kη (m) kH → ∞, for m → ∞. For the proof of this Proposition we need to recall some classical concepts and results. Definition 4.3.11. A nonexpansive map T : H → H is strongly nonexpansive if for (un − vn )n bounded and kT (un ) − T (vn )kH − kun − vn kH → 0 we have un − vn − (T (un ) − T (vn )) → 0, n → ∞. Proposition 4.3.12. [BR77, Corollaries 1.3, 1.4, and 1.5] Let T : H → H be a strongly nonexpansive map. Then ﬁxT = {u ∈ H : T (u) = u} = 6 ∅ if and only if (T n u)n converges weakly to a fixed point u0 ∈ ﬁxT for any choice of u ∈ H. Proof of Proposition 4.3.10. Orthogonal projections onto convex sets are strongly nonexpansive [BBL97, Corollary 4.2.3]. Moreover, composition of strongly nonexpansive maps are strongly nonexpansive [BR77, Lemma 2.1]. By an application of Proposition 4.3.12 we immediately have the result, since any map of the type T (ξ) = Q(ξ) + ξ0 170 4.3 Domain Decomposition for TV Minimization is strongly nonexpansive whenever Q is, (this is a simple observation from the definition of strongly nonexpansive map). Indeed, we are looking for ﬁxed points of η = πV2 PαKψ (η − (z + u2 )) or, equivalently, of ξ = πV2 PαKψ (ξ) − (z + u2 ). | {z } | {z } :=ξ0 :=Q In Examples 4.3.6, we have already observed that u∗1 = proxαψ(·+u2 ) (z). For consistency with the terminology of generalized thresholding in signal processing, we call the map proxαψ(·+u2 ) an oblique thresholding and denote it by Sαψ,V1 ,V2 (z; u2 ) := proxαψ(·+u2 ) (z). The attribute “oblique” emphasizes the presence of an additional subspace which contributes to the computation of the solution. By using results in [CW05, Subsection 2.3] (see also [ET76, II.2-3]) we can already infer that kSαψ,V1 ,V2 (z1 ; u2 ) − Sαψ,V1 ,V2 (z2 ; u2 )kH ≤ kz1 − z2 kH, for all z1 , z2 ∈ V1 . For a ﬁnite dimensional H (and, more generally, for any situation where for any choice of u2 ∈ V2ψ and z ∈ V1 there exists η ∈ V2 which solves (4.62)) we can show the nonexpansiveness of Sαψ,V1 ,V2 (·; u2 ) by a simple direct argument which exploits the or- thogonality of V1 and V2 and the relationships revealed by Theorem 4.3.8. For the sake of completeness, we would like to report this short argument. Lemma 4.3.13. Assume that dim(H) < ∞. The oblique thresholding Sαψ,V1 ,V2 (·; u2 ) is a nonexpansive map for any choice of u2 ∈ V2 . Proof. Let us denote ξ = z + u2 − η, where η = η(z, u2 ) = πV2 PαKψ (η − (z + u2 )). We have ξ = πV2 PαKψ (ξ) + (z + u2 ), and, by using the latter equivalence, Sαψ,V1 ,V2 (z; u2 ) = ξ − PαKψ (ξ) − u2 = πV2 PαKψ (ξ) − PαKψ (ξ) + z = z − πV1 PαKψ (ξ). Then Sαψ,V1 ,V2 (z; u2 ) + u2 − η = ξ − πV1 PαKψ (ξ) = (I − πV1 PαKψ )(ξ). Let us denote ξi = zi +u2 −ηi , where ηi = η(zi , u2 ) = πV2 PαKψ (ηi −(zi +u2 )). We have the equivalence S ψ,V1 ,V2 (z1 ; u2 ) − Sαψ,V1 ,V2 (z2 ; u2 ) + η2 − η1 = (I − πV1 PαKψ )(ξ1 ) − (I − πV1 PαKψ )(ξ2 ) {z } | {z } |α ∈V1 ∈V2 171 4.3 Domain Decomposition for TV Minimization which implies, by orthogonality of V1 and V2 , kSαψ,V1 ,V2 (z1 ; u2 ) − Sαψ,V1 ,V2 (z2 ; u2 )k2H + kη2 − η1 k2H ≤ kz1 − z2 k2H + kη2 − η1 k2H. In the latter inequality we also used the nonexpansiveness of I − πV1 PαKψ . This concludes the proof: kSαψ,V1 ,V2 (z1 ; u2 ) − Sαψ,V1 ,V2 (z2 ; u2 )kH ≤ kz1 − z2 kH. Convergence of the subspace minimization In light of the results of the previous subsection, the iterative algorithm (4.59) can be equivalently rewritten as (ℓ+1) u1 (ℓ) (ℓ) = Sαψ,V1 ,V2 (u1 + πV1 T ∗ (f − T u2 − T u1 ); u2 ). (4.64) In certain cases, e.g., in ﬁnite dimensions, the iteration can be explicitly computed by Oblique Thresholding: (ℓ+1) u1 (ℓ) (ℓ) = Sαψ (u1 + πV1 T ∗ (f − T u2 − T u1 ) + u2 − η (ℓ) ) − u2 , where η (ℓ) ∈ V2 is any solution of the ﬁxed point equation (ℓ) (ℓ) η = πV2 PαKψ (η − (u1 + πV1 T ∗ (f − T u2 − T u1 ) + u2 )). The computation of η (ℓ) can be (approximately) implemented by the algorithm (4.63). Theorem 4.3.14. Assume that u2 ∈ V2ψ and kT k < 1. Then the iteration (4.64) (0) converges weakly to a solution u∗1 ∈ V1ψ of (4.57) for any initial choice of u1 ∈ V1ψ . Proof. For the sake of completeness, we report the proof of this theorem, which follows the same strategy already proposed in [DDD04], compare also similar results in [CW05]. In particular we want to apply Opial’s ﬁxed point theorem: Theorem 4.3.15. [Op67] Let the mapping A from H to H satisfy the following conditions: (i) A is nonexpansive: for all z, z ′ ∈ H, kAz − Az ′ kH ≤ kz − z ′ kH; (ii) A is asymptotically regular: for all z ∈ H, kAn+1 z − An zkH → 0, for n → ∞; 172 4.3 Domain Decomposition for TV Minimization (iii) the set F = ﬁxA of fixed points of A in H is not empty. Then for all z ∈ H, the sequence (An z)n∈N converges weakly to a fixed point in F. A simple proof of this theorem can be found in the appendix of [DDD04]. We need to prove that A(u1 ) := Sαψ,V1 ,V2 (u1 + πV1 T ∗ (f − T u2 − T u1 ); u2 ) fulﬁlls the assumptions of the Opial’s theorem on V1 . Step 1. As stated at the beginning of this section, there exist solutions u∗1 ∈ V1ψ to (4.57). With similar argument to the one used to prove the equivalence of (i) and (ii) in Theorem 4.3.8, the optimality of u∗1 can be readily proved as equivalent to 0 ∈ −πV1 T ∗ (f − T u2 − T u∗1 ) + η + α∂Hψ ψ(u∗1 + u2 ), for some η ∈ (V2ψ )∗ . By adding and subtracting u∗1 we obtain 0 ∈ u∗1 − ((u∗1 + πV1 T ∗ (f − T u2 − T u∗1 )) −η) + α∂Hψ ψ(u∗1 + u2 ), {z } | :=z Applying the equivalence of (i) and (ii) in Theorem 4.3.8, we deduce that u∗1 is a ﬁxed point of the following equation u∗1 = Sαψ,V1 ,V2 (u∗1 + πV1 T ∗ (f − T u2 − T u∗1 ); u2 ), hence ﬁxA 6= ∅. Step 2. The algorithm produces iterations which are asymptotically regular, i.e., (ℓ) (ℓ+1) ku1 − u1 kH → 0. Indeed, by using kT k < 1 and C := 1 − kT k2 > 0, we have the following estimates (ℓ) (ℓ) (ℓ) J(u1 + u2 ) = Js1 (u1 + u2 , u1 ) (ℓ+1) + u 2 , u1 ) (ℓ+1) + u 2 , u1 ≥ Js1 (u1 ≥ Js1 (u1 (ℓ) (ℓ+1) (ℓ+1) ) = J(u1 + u2 ), (ℓ) See also (4.67) and (4.68) below. Since (J(u1 + u2 ))ℓ is monotonically decreasing and bounded from below by 0, it is necessarily a convergent sequence. Moreover, (ℓ) (ℓ+1) J(u1 + u2 ) − J(u1 (ℓ+1) + u2 ) ≥ Cku1 (ℓ+1) and the latter convergence implies ku1 (ℓ) − u1 kH → 0. 173 (ℓ) − u1 k2H, 4.3 Domain Decomposition for TV Minimization Step 3. We are left with showing the nonexpansiveness of A. By nonexpansiveness of Sαψ,V1 ,V2 (·; u2 ) we obtain kSαψ,V1 ,V2 (u11 + πV1 T ∗ (f − T u2 − T u11 ; u2 ) − Sαψ,V1 ,V2 (u21 + πV1 T ∗ (f − T u2 − T u21 ; u2 )kH ≤ ku11 + πV1 T ∗ (f − T u2 − T u11 ) − (u21 + πV1 T ∗ (f − T u2 − T u21 )kH = k(I − πV1 T ∗ T πV1 )(u11 − u21 )kH ≤ ku11 − u21 kH In the latter inequality we used once more the fact that kT k < 1. We do not insist on conditions for strong convergence of the iteration (4.64), which is not a relevant issue, see, e.g., [CW05, DTV07] for a further discussion of this issue. Indeed, the practical realization of (4.56) will never solve completely the subspace minimizations. Let us conclude this section mentioning that all the results presented here hold symmetrically for the minimization on V2 , provided that the notation is suitably adjusted. 4.3.4 Convergence of the Sequential Alternating Subspace Minimization We return to the algorithm (4.56). In the following we denote ui = πVi u for i = 1, 2. Let (0,L) us express explicitly the algorithm as follows: Pick an initial V1 ⊕V2 ∋ u1 u(0) Hψ , (0,M ) +u2 := u(0) ∈ for example = 0, and iterate ( (n+1,0) (n,L) u1 = u1 (n+1,ℓ+1) (n,M ) (n+1,ℓ) = argminu1 ∈V1 Js1 (u1 + u2 , u1 ) ℓ = 0, . . . , L − 1 ( u1 (n+1,0) (n,M ) u2 = u2 (n+1,m) (n+1,L) (n+1,m+1) + u 2 , u2 ) m = 0, . . . , M − 1 u2 = argminu2 ∈V2 Js2 (u1 (n+1) (n+1,M ) (n+1,L) + u2 . u := u1 (4.65) Note that we do prescribe a ﬁnite number, L and M respectively, of inner iterations for each subspace. In this section we want to prove convergence of the algorithm for any choice of L and M . Observe that, for a ∈ Vi and kT k < 1, kui − ak2H − kT ui − T ak2H ≥ Ckui − ak2H, 174 (4.66) 4.3 Domain Decomposition for TV Minimization for C = (1 − kT k2 ) > 0. Hence J(u) = Jsi (u, ui ) ≤ Jsi (u, a), (4.67) Jsi (u, a) − Jsi (u, ui ) ≥ Ckui − ak2H. (4.68) and Theorem 4.3.16 (Convergence properties). The algorithm in (4.65) produces a sequence (u(n) )n∈N in Hψ with the following properties: (i) J(u(n) ) > J(u(n+1) ) for all n ∈ N (unless u(n) = u(n+1) ); (ii) limn→∞ ku(n+1) − u(n) kH = 0; (iii) the sequence (u(n) )n∈N has subsequences which converge weakly in H and in Hψ , both endowed with the topology τ ψ ; (iv) if we additionally assume, for simplicity, that dim H < ∞, (u(nk ) )k∈N is a strongly converging subsequence, and u(∞) is its limit, then u(∞) is a minimizer of J whenever one of the following conditions holds (∞) (∞) (a) ψ(u1 + η2 ) + ψ(u2 i = 1, 2; (∞) (∞) + η1 ) − ψ(u1 + u2 ) ≤ ψ(η1 + η2 ) for all ηi ∈ Vi , (b) ψ is differentiable at u(∞) with respect to Vi for one i ∈ {1, 2}, i.e., there ∂ ψ(u(∞) ) := ζi ∈ (Vi )∗ such that exists ∂V i (∞) hζi , vi i = lim ψ(u1 (∞) + u2 t→0 (∞) + tvi ) − ψ(u1 t (∞) + u2 ) , for all vi ∈ Vi . Proof. Let us ﬁrst observe that (n) (n) (n) (n,L) (n) J(u(n) ) = Js1 (u1 + u2 , u1 ) = Js1 (u1 (n+1,1) By deﬁnition of u1 (n,L) Js1 (u1 (n+1,1) and the minimal properties of u1 (n) (n+1,0) + u 2 , u1 (n+1,1) ) ≥ Js (u1 (n+1,0) + u 2 , u1 (n) ). in (4.65) we have (n+1,0) + u 2 , u1 ). From (4.67) we have (n+1,1) Js1 (u1 (n) (n+1,0) + u 2 , u1 (n+1,1) ) ≥ Js1 (u1 Putting together these inequalities we obtain (n+1,1) J(u(n) ) ≥ J(u1 175 (n) (n+1,1) + u 2 , u1 (n) + u2 ) ). 4.3 Domain Decomposition for TV Minimization In particular, from (4.68) we have (n+1,1) J(u(n) ) − J(u1 (n) (n+1,1) + u2 ) ≥ Cku1 (n+1,0) 2 kH. − u1 After L steps we conclude with the estimate (n+1,L) J(u(n) ) ≥ J(u1 and (n+1,L) J(u(n) ) − J(u1 By deﬁnition of (n+1,1) u2 (n) + u2 ) ≥ C L−1 X (n) + u2 ), (n+1,ℓ+1) ku1 ℓ=0 (n+1,ℓ) 2 kH. − u1 and its minimal properties we have (n+1,L) J(u1 (n) (n+1,L) (n+1,1) + u2 ) ≥ Js2 (u1 + u2 (n+1,0) , u2 ). By similar arguments as above we ﬁnally ﬁnd the decreasing estimate (n+1,L) J(u(n) ) ≥ Js2 (u1 (n+1,M ) + u2 ) = J(u(n+1) ), (4.69) and ≥C L−1 X ℓ=0 J(u(n) ) − J(u(n+1) ) (n+1,ℓ+1) ku1 (n+1,ℓ) 2 kH + − u1 M −1 X m=0 (n+1,m+1) ku2 ! (n+1,m) 2 kH − u2 . (4.70) From (4.69) we have J(u(0) ) ≥ J(u(n) ). By the coerciveness condition (C) (u(n) )n∈N is uniformly bounded in Hψ , hence there exists a H-weakly- and τ ψ -convergent subsequence (u(nj ) )j∈N . Let us denote by u(∞) the weak limit of this subsequence. For simplicity, we rename such a subsequence as (u(n) )n∈N . Moreover, since the sequence (J(u(n) ))n∈N is monotonically decreasing and bounded from below by 0, it is also convergent. From (4.70) and the latter convergence we deduce that ! M −1 L−1 X X (n+1,ℓ+1) (n+1,ℓ) 2 (n+1,m+1) (n+1,m) 2 ku1 − u1 kH + ku2 − u2 kH → 0, n → ∞. ℓ=0 m=0 In particular, by the standard inequality triangle inequality, we have also (a2 + b2 ) ku(n) − u(n+1) kH → 0, ≥ 1 2 (a + b)2 (4.71) for a, b > 0 and the n → ∞. We would like now to show that the following outer lower semicontinuity holds, 0 ∈ lim ∂J(u(n) ) ⊂ ∂J(u(∞) ). n→∞ 176 (4.72) 4.3 Domain Decomposition for TV Minimization For this we need to assume that H-weakly-convergence and τ ψ −convergence do imply strong convergence in H. This is the case, e.g., when dim(H) < ∞. The optimality (n+1,L) condition for u1 is equivalent to (n+1,L) (n+1) 0 ∈ u1 − z1 (n,M ) + α∂V1 ψ(· + u2 (n+1,L) )(u1 ), (4.73) where (n+1) z1 (n+1,L−1) := u1 (n,M ) + πV1 T ∗ (f − T u2 (n+1,L−1) − T u1 ). Analogously we have (n+1,M ) 0 ∈ u2 (n+1) − z2 (n+1,L) + α∂V2 ψ(· + u1 (n+1,M ) )(u2 ), (4.74) where (n+1) z2 (n+1,M −1) := u2 (n+1,L) + πV2 T ∗ (f − T u1 (n+1,M −1) − T u2 ). Due to the strong convergence of the sequence u(n) and by (4.71) we have the following limits for n → ∞ (n+1) := u1 (n+1) := u2 ξ1 ξ2 (n+1,L) − z1 (n+1) → ξ1 := −πV1 T ∗ (f − T u2 (n+1,M ) − z2 (∞) − T u1 (n+1) → ξ2 := −πV2 T ∗ (f − T u2 (∞) (∞) − T u1 ) ∈ V1 , (∞) ) ∈ V2 , and (n+1) ξ1 Moreover, we have (n+1) + ξ2 → ξ := T ∗ (T u(∞) − f ). 1 (n+1) (n,M ) (n+1,L) ∈ ∂V1 ψ(· + u2 )(u1 ), − ξ1 α meaning that 1 (n+1) (n+1,L) (n+1,L) (n,M ) (n,M ) h− ξ1 , η1 − u1 i + ψ(u1 + u2 ) ≤ ψ(η1 + u2 ), α for all η1 ∈ V1 . Analogously we have 1 (n+1) (n+1,M ) (n+1,L) (n+1,M ) (n+1,L) , η2 − u2 i + ψ(u1 + u2 ) ≤ ψ(η2 + u1 ), h− ξ2 α for all η2 ∈ V2 . By taking the limits for n → ∞ and by (4.71) we obtain 1 (∞) (∞) h− ξ1 , η1 − u1 i + ψ(u(∞) ) ≤ ψ(η1 + u2 ), α for all η1 ∈ V1 . (4.75) 1 (∞) (∞) (∞) h− ξ2 , η2 − u2 i + ψ(u1 ) ≤ ψ(η2 + u1 ), α for all η2 ∈ V2 . (4.76) 177 4.3 Domain Decomposition for TV Minimization These latter conditions are rewritten in vector form as ! ξ1 (∞) (∞) (∞) (∞) + α ∂V1 ψ(· + u2 )(u1 ) × ∂V2 ψ(· + u1 )(u2 ) . 0∈ ξ2 (4.77) Observe now that 2ξ + 2α∂Hψ(u(∞) ) = 2T ∗ (T u(∞) − f ) + 2α∂Hψ(u(∞) ) = ∂J(u(∞) ). If 0 ∈ ξ + α∂Hψ(u(∞) ) then we would have the required minimality condition. While the inclusion (∞) ∂Hψ(u(∞) ) ⊂ ∂V1 ψ(· + u2 (∞ (∞) )(u1 ) × ∂V2 ψ(· + u1 (∞) )(u2 ), easily follows from the deﬁnition of a subdiﬀerential, the converse inclusion, which would imply from (4.77) the desired minimality condition, does not hold in general. Thus, we show the converse inclusion under one of the following two conditions: (∞) (a) ψ(u1 (∞) + η2 ) + ψ(u2 (∞) + η1 ) − ψ(u1 (∞) + u2 ) ≤ ψ(η1 + η2 ) for all ηi ∈ Vi , i = 1, 2; (b) ψ is diﬀerentiable at u(∞) with respect to Vi for a single i ∈ {1, 2}, i.e., there ∂ ψ(u(∞) ) := ζi ∈ (Vi )∗ such that exists ∂V i (∞) hζi , vi i = lim t→0 ψ(u1 (∞) + u2 (∞) + tvi ) − ψ(u1 t (∞) + u2 ) , for all vi ∈ Vi . Let us start with condition (a). We want to show that 1 h− ξ, η − u(∞) i + ψ(u(∞) ) ≤ ψ(η), α for all η ∈ H, or, equivalently, that 1 1 (∞) (∞) (∞) (∞) h− ξ1 , η1 − u1 i + h− ξ2 , η2 − u2 i + ψ(u1 + u2 ) ≤ ψ(η1 + η2 ), α α for all ηi ∈ Vi , By the diﬀerential inclusions (4.75) and (4.76) we have 1 1 (∞) (∞) (∞) (∞) (∞) (∞) h− ξ1 , η1 − u1 i + h− ξ2 , η2 − u2 i + 2ψ(u1 + u2 ) ≤ ψ(u1 + η2 ) + ψ(u2 + η1 ), α α for all ηi ∈ Vi , hence 1 1 (∞) (∞) (∞) (∞) h− ξ1 , η1 − u1 i + h− ξ2 , η2 − u2 i + ψ(u1 + u2 ) α α (∞) (∞) (∞) (∞) ≤ ψ(u1 + η2 ) + ψ(u2 + η1 ) − ψ(u1 + u2 ), 178 for all ηi ∈ Vi . 4.3 Domain Decomposition for TV Minimization An application of condition (a) concludes the proof of the required diﬀerential inclusion. Let us now show the inclusion under the assumption of condition (b). Without loss of generality, we assume that ψ is diﬀerentiable at u(∞) with respect to V2 . First of all we deﬁne ψ̃(u1 , u2 ) := ψ(u1 + u2 ). Since ψ is convex, by an application of [RW98, Corollary 10.11] we have ∂V1 ψ(· + u2 )(u1 ) ≃ ∂u1 ψ̃(u1 , u2 ) = {ζ1 ∈ V1∗ : ∃ζ2 ∈ V2∗ : (ζ1 , ζ2 )T ∈ ∂ ψ̃(u1 , u2 ) ≃ ∂Hψ(u1 + u2 )}. Since ψ is diﬀerentiable at u(∞) with respect to V2 , for any (ζ1 , ζ2 )T ∈ ∂ ψ̃(u1 , u2 ) ≃ ∂Hψ(u1 + u2 ) we have necessarily ζ2 = ∂V∂ 2 ψ(u(∞) ) as the unique member of ∂V2 ψ(· + (∞) u1 (∞) )(u2 ). Hence, the following inclusion must also hold ! ξ1 (∞) (∞ (∞) (∞) + α ∂V1 ψ(· + u2 )(u1 ) × ∂V2 ψ(· + u1 )(u2 ) 0∈ ξ2 ! ξ1 ⊂ + α∂V1 ×V2 ψ̃(u1 , u2 ) ξ2 ≃ ξ + α∂Hψ(u(∞) ). Remark 4.3.17. Observe that, by choosing η1 = η2 = 0, condition (a) and (Ψ2) for the case λ = 0 imply that (∞) ψ(u1 (∞) ) + ψ(u2 (∞) + u2 (∞) + u2 ) ≤ ψ(u1 (∞) ) (∞) ) The sublinearity (Ψ1) finally implies the splitting (∞) ψ(u1 (∞) ) + ψ(u2 ) = ψ(u1 Conversely, if ψ(v1 ) + ψ(v2 ) = ψ(v1 + v2 ) for all vi ∈ Vi , i = 1, 2, then condition (a) easily follows. As previously discussed, this latter splitting condition holds only in special cases. Also condition (b) is not in practice always verified, cf. the numerical examples for ℓ1 minimization in [FS07]. Hence, we can affirm that in general we cannot expect convergence of the algorithm to minimizers of J, although it certainly converges to points for which J is smaller than the starting choice J(u(0) ). However, as we will show in the numerical experiments related to total variation minimization (Sections 4.3.6 and 4.3.7), the computed limit can be very close to the expected minimizer. 179 4.3 Domain Decomposition for TV Minimization 4.3.5 A Parallel Alternating Subspace Minimization and its Convergence (n,L) The most immediate modiﬁcation to (4.65) is provided by substituting u1 of instead (n+1,L) u1 in the second iteration, producing the following parallel algorithm: ( (n+1,0) (n,L) u1 = u1 (n,M ) (n+1,ℓ) (n+1,ℓ+1) , u1 ) ℓ = 0, . . . , L − 1 = argminu1 ∈V1 Js1 (u1 + u2 ( u1 (n+1,0) (n,M ) u2 = u2 (n+1,m+1) (n,L) (n+1,m) ) m = 0, . . . , M − 1 u2 = argminu2 ∈V2 Js2 (u1 + u 2 , u2 (n+1) (n+1,L) (n+1,M ) u := u1 + u2 . Unfortunately, this modiﬁcation violates the monotonicity property J(u(n) ) (4.78) ≥ J(u(n+1) ) and the overall algorithm does not converge in general. In order to preserve the mono- tonicity of the iteration with respect to J a simple trick can be applied, i.e., modifying (n+1,L) u(n+1) := u1 (n+1,M ) + u2 by the average of the current iteration and the previous one. This leads to the following parallel algorithm: ( (n+1,0) (n,L) u1 = u1 (n+1,ℓ+1) (n,M ) (n+1,ℓ) = argminu1 ∈V1 Js1 (u1 + u2 , u1 ) ℓ = 0, . . . , L − 1 ( u1 (n+1,0) (n,M ) u2 = u2 (n+1,m+1) (n,L) (n+1,m) u2 = argminu2 ∈V2 Js2 (u1 + u 2 , u2 ) m = 0, . . . , M − 1 (n+1,L) (n+1,M ) (n) +u2 +u u(n+1) := u1 . 2 (4.79) In this section we prove similar convergence properties of this algorithm to that of (4.65). Theorem 4.3.18 (Convergence properties). The algorithm in (4.79) produces a sequence (u(n) )n∈N in Hψ with the following properties: (i) J(u(n) ) > J(u(n+1) ) for all n ∈ N (unless u(n) = u(n+1) ); (ii) limn→∞ ku(n+1) − u(n) kH = 0; (iii) the sequence (u(n) )n∈N has subsequences which converge weakly in H and in Hψ , both endowed with the topology τ ψ ; (iv) if we additionally assume that dim H < ∞, (u(nk ) )k∈N is a strongly converging subsequence, and u(∞) is its limit, then u(∞) is a minimizer of J whenever one of the following conditions holds 180 4.3 Domain Decomposition for TV Minimization (∞) (∞) (a) ψ(u1 + η2 ) + ψ(u2 i = 1, 2; (∞) + η1 ) − ψ(u1 (∞) + u2 ) ≤ ψ(η1 + η2 ) for all ηi ∈ Vi , (b) ψ is differentiable at u(∞) with respect to Vi for one i ∈ {1, 2}, i.e., there ∂ ψ(u(∞) ) := ζi ∈ (Vi )∗ such that exists ∂V i (∞) hζi , vi i = lim ψ(u1 (∞) + u2 t→0 (∞) (∞) + tvi ) − ψ(u1 t + u2 ) , for all vi ∈ Vi . Proof. With the same argument as in the proof of Theorem 4.3.16, we obtain J(u (n) )− (n+1,L) J(u1 + (n) u2 ) (n) (n+1,M ) ku1 M −1 X ku2 ≥C and J(u(n) ) − J(u1 + u2 L−1 X )≥C Hence, by summing and halving ℓ=0 m=0 (n+1,ℓ+1) (n+1,ℓ) 2 kH. − u1 (n+1,m+1) (n+1,m) 2 kH. − u2 1 (n+1,L) (n) (n) (n+1,M ) + u2 ) + J(u1 + u2 )) J(u(n) ) − (J(u1 2 ! M −1 L−1 X C X (n+1,ℓ+1) (n+1,m+1) (n+1,m) 2 (n+1,ℓ) 2 ku2 − u2 kH . ku1 − u1 kH + ≥ 2 m=0 ℓ=0 By convexity we have 2 (n+1) − f = T T u H (n+1,L) (u1 (n+1,M ) + u2 2 ) + u(n) ! 2 − f H 1 1 (n+1,L) (n) (n) (n+1,M ) + u2 ) − f k2H + kT (u1 + u2 ) − f k2H. ≤ kT (u1 2 2 Moreover, by sublinearity (Ψ1) and 1-homogeneity (Ψ2) we have 1 (n+1,L) (n) (n) (n+1,M ) ψ(u1 + u2 ) + ψ(u1 + u2 ) ψ(u(n+1) ) ≤ 2 By the last two inequalities we immediately show that 1 (n+1,L) (n) (n) (n+1,M ) J(u(n+1) ) ≤ J(u1 + u2 ) + J(u1 + u2 ) , 2 hence J(u(n) ) − J(u(n+1) ) C ≥ 2 L−1 X ℓ=0 (n+1,ℓ+1) ku1 (n+1,ℓ) 2 kH + − u1 M −1 X ℓ=0 181 (n+1,ℓ+1) ku2 ! (n+1,ℓ) 2 kH − u2 ≥ 0. (4.80) 4.3 Domain Decomposition for TV Minimization Since the sequence (J(u(n) ))n∈N is monotonically decreasing and bounded from below by 0, it is also convergent. From (4.80) and the latter convergence we deduce ! M −1 L−1 X X (n+1,ℓ+1) (n+1,m+1) (n+1,m) 2 (n+1,ℓ) 2 ku2 − u2 kH → 0, n → ∞. ku1 − u1 kH + m=0 ℓ=0 In particular, by the standard inequality triangle inequality, we have also L−1 X ℓ=0 (n+1,ℓ+1) ku1 − (n+1,ℓ) 2 u1 kH (a2 ≥ C ′′ ≥ = + b2 ) L−1 X ≥ 1 2 (a + (n+1,ℓ+1) ku1 ℓ=0 (n+1,L) ′′ C ku1 (n+1,L) C ′′ ku1 b)2 − (4.81) for a, b > 0 and the !2 (n+1,ℓ) u1 kH (n) − u1 k2H (n) (n) + u1 − 2u1 k2H. Analogously we have M −1 X ℓ=0 (n+1,ℓ+1) ku2 (n+1,ℓ) 2 kH − u2 (n+1,M ) ≥ C ′′ ku2 (n) (n) + u2 − 2u2 k2H. By denoting C ′′ = 21 C ′′′ we obtain C 2 ≥ L−1 X ℓ=0 CC ′′′ (n+1,ℓ+1) ku1 (n+1,L) − (n+1,ℓ) 2 u1 kH + (n+1,M ) ku1 + u2 4 ≥ CC ′′′ ku(n+1) − u(n) k2H. M −1 X ℓ=0 (n+1,ℓ+1) ku2 − ! (n+1,ℓ) 2 u2 kH + u(n) − 2u(n) k2H Therefore, we ﬁnally have ku(n) − u(n+1) kH → 0, n → ∞. (4.82) The rest of the proof follows by analogous arguments to that of Theorem 4.3.16. 4.3.6 Domain Decomposition for TV-L2 Minimization In this section we present the application of the illustrated theory and algorithms to the domain decomposition for total variation minimization. In the following we consider the minimization of the functional J in the setting of Example 4.3.1.1. Speciﬁcally, let Ω ⊂ Rd , for d = 1, 2, be a bounded open set with Lipschitz boundary. We are 182 4.3 Domain Decomposition for TV Minimization interested in the case when H = L2 (Ω), Hψ = BV (Ω) and ψ(u) = V (u, Ω). Then the domain decomposition Ω = Ω1 ∪Ω2 , as described in Examples 4.3.2.1, induces the space splitting into Vi := {u ∈ L2 (Ω) : supp(u) ⊂ Ωi }, and Viψ = BV (Ω) ∩ Vi , i = 1, 2. In particular, we can consider multiple subspaces, since the algorithms and their analysis presented in the previous sections can be easily generalized to these cases, see [Fo07, Section 6]. As before uΩi = πVi (u) = 1Ωi u is the orthogonal projection onto Vi . To exemplify the kind of diﬃculties one may encounter in the numerical treatment of the interfaces ∂Ωi ∩ ∂Ωj , we present ﬁrst an approach based on the direct discretization of the subdiﬀerential of J in this setting. We show that this method can work properly in many cases, but it fails in others, even for simple 1D examples, due to the occurrance of exceptions which cannot be captured by this formulation. Instead of insisting on dealing with these exceptions and strengthening the formulation, we show that the general theory and algorithms presented previously work properly and deal well with interfaces for both d = 1, 2. The “Naive” Direct Approach In light of (4.55), the ﬁrst subiteration in (4.56) is given by (n+1) u1 (n) ≈ argminv1 ∈V1 kT (v1 + u2 ) − f k2L2 (Ω) + 2α (|D(v1 )|(Ω1 ) Z + (n)− + v − u dH 1 d−1 . 2 ∂Ω1 ∩∂Ω2 We would like to describe the conditions that characterize subdiﬀerentials of functionals of the type Z + u − z dHd−1 , |θ {z } interface condition where θ ⊂ ∂Ω, in order to handle the boundary conditions that are imposed at the Γ(u) = |D(u)| (Ω) + interface. Since we are interested in emphasizing the diﬃculties of this approach, we do not insist on the details of a rigorous derivation of these conditions, and we limit ourself to mentioning the main facts. It is well known [Ve01, Proposition 4.1] that, if no interface condition is present, ξ ∈ 183 4.3 Domain Decomposition for TV Minimization ∂|D(·)|(Ω)(u) implies that ∇u in Ω ξ = −∇ · |∇u| ∇u ·ν =0 on ∂Ω. |∇u| The previous conditions fall short of fully characterizing ξ ∈ ∂|D(·)|(Ω)(u), additional conditions would be required [AK06, Ve01], but the latter are, unfortunately, hard to implement numerically. This is the source of the failures of this direct method. The presence of the interface further modiﬁes and aggreviates this situation and for ∂Γ(u) 6= ∅ we need to enforce Z Z ∇u |(u + ws)+ − z| − |u+ − z| · νw dHd−1 + lim dHd−1 ≥ 0, s→0 s ∂Ω |∇u| θ ∀w ∈ C ∞ (Ω̄). The latter condition is implied by the following natural boundary condi- tions: ∇u |∇u| · ν = 0 on ∂Ω \ θ, ∇u − · ν ∈ ∂| · |(u+ − z) on θ. |∇u| (4.83) Note that the conditions above are again not suﬃcient to characterize elements in the subdiﬀerential of Γ. Implementation of the subdifferential approach in TV-L2 interpolation Let D ⊂ Ω ⊂ Rd be open and bounded domains with Lipschitz boundaries. We assume that a function f ∈ L2 (Ω) is given only on Ω \ D, possibly with noise disturbance. The problem is to reconstruct a function u in the damaged domain D ⊂ Ω which nearly coincides with f on Ω \ D. In 1D this is a classical interpolation problem, while in 2D it has assumed the name of “inpainting” due to its applications in image restoration. TV- interpolation/inpainting with L2 ﬁdelity is solved by the minimization of the functional 2 J(u) = 1Ω\D (u − f )L2 (Ω) + 2α|D(u)|(Ω), (4.84) where 1Ω\D denotes the indicator function of Ω \ D. Hence, in this case T is the multiplier operator T u = 1Ω\D u. We consider in the sequel the problem for d = 1, 184 4.3 Domain Decomposition for TV Minimization so that Ω = (a, b) is an interval. We may want to minimize (4.84) iteratively by a subgradient descent method, u(n+1) − u(n) ∇u(n+1) = −∇ · ( ) + 2λ(u(n) − f ) in Ω (n) τ |∇u | ∂u(n+1) 1 =0 |∇u(n) | ∂n where (4.85) on ∂Ω, ( λ0 = λ(x) = 0 1 4α Ω\D D. We can also attempt the minimization by the following domain decomposition algorithm: We split Ω into two intervals Ω = Ω1 ∪ Ω2 and deﬁne two alternating minimiza- tions on Ω1 and Ω2 with the interface θ = ∂Ω1 ∩ ∂Ω2 (n+1) (n+1) (n) − u1 τ u1 = −∇ · ( (n+1) ∂u1 (n) |∇u | ∂n 1 ∇u1 (n) |∇u1 | (n) ) + 2λ1 (u1 − f ) on ∂Ω1 \ θ =0 1 − (n+1) ∂u1 (n) |∇u | ∂n 1 (n+1)+ ∈ ∂ |·| (u1 1 in Ω1 (n)− − u2 ) on θ, (n) in Ω2 and (n+1) (n) − u2 τ u2 1 (n+1) = −∇ · ( (n+1) ∇u2 (n) |∇u2 | ∂u2 (n) |∇u | ∂n =0 ∂u2 (n) |∇u | ∂n ∈ ∂ |·| (u2 ) + 2λ2 (u2 − f ) on ∂Ω2 \ θ 2 1 2 (n+1) (n+1)+ (n+1)− − u1 ) on θ. In this setting ui denotes the restriction of u ∈ BV (Ω) to Ωi . The ﬁtting parameter λ is also split accordingly into λ1 and λ2 on Ω1 and Ω2 respectively. Note that we enforced the interface conditions (4.83), with the hope to match correctly the solution at the internal boundaries. The discretization in space is done by ﬁnite diﬀerences. We only explain the details for the ﬁrst subproblem on Ω1 because the procedure is analogous for the second one. 185 4.3 Domain Decomposition for TV Minimization Let i = 1, . . . , N denote the space nodes supported in Ω1 . We let h = b−a N and u(i) := u(ih). The gradient and the divergence operator are discretized by backward diﬀerences and forward diﬀerences respectively, 1 (u(i) − u(i − 1)) h 1 ∇ · u(i) = (u(i + 1) − u(i)) h r 1 |∇u|(i) = ǫ2 + 2 (u(i) − u(i − 1))2 , h ∇u(i) = for i = 2, . . . , N − 1 and ǫ > 0. Note that we have replaced the total variation |∇u| p here by its regularized form ǫ2 + |∇u|2 . The discretized equation on Ω1 turns out to be (n+1) u1 (i) = (n) u1 (i) + (n) 2τ λ(i)(u1 (i) τ − f (i)) + 2 h (n+1) u1 (n+1) − with cn1 (i) = q (n) (n+1) (i + 1) − u1 cn1 (i + 1) u1 (i) (n+1) (i) − u1 cn1 (i) (i − 1) ! , (n) ǫ2 + (u1 (i) − u1 (i − 1))2 /h2 and i = 2, . . . , N − 1. The Neumann boundary conditions on the external portion of the boundary are enforced by 1 u1 (1) cn1 (1) = 1 u1 (2). cn2 (2) The interface conditions on the internal boundaries are computed by solving the following subdiﬀerential inclusion (n+1) −(u1 (n+1) (N ) − u1 (n) (n+1) (N − 1)) ∈ cn1 (N ) h · ∂| · |(u2 (N ) − u1 (N )). For the solution of this subdiﬀerential inclusion we recall that the soft-thresholding operator u = Sα (x) that has been deﬁned in (4.54) provides the unique solution of the subdiﬀerential inclusion 0 ∈ (u − x) + α∂| · |(u). We reformulate our subdiﬀerential inclusion as (n) h i (n) (n+1) 0 ∈ v − (u2 (N ) − u1 (N − 1)) + cn1 (N )h · ∂| · |(v), (n+1) with v := u2 (N ) − u1 (N ) and get (n) (n+1) v = Scn1 (N )h (u2 (N ) − u1 186 (N − 1)). 4.3 Domain Decomposition for TV Minimization (n+1) Therefore the interface condition on θ reads u1 (n) (N ) = u2 (N ) − v. In the left column of Figure 4.10 three one dimensional signals are considered. The right column shows the result of the application of the domain decomposition method for total variation minimization described above. The support of the signals is split into two intervals. The interface developed by the two intervals is marked by a red dot. In all three examples we ﬁxed λ0 = 1 and τ = 1/2. The ﬁrst example 4.10(a)-4.10(b) shows a step function which has its step directly at the interface of the two intervals. The total variation minimization (4.85) is applied with D = ∅. This example conﬁrms that jumps are preserved at the interface of the two domains. The second and third example 4.10(c)-4.10(f) present the behavior of the algorithm when interpolation across the interface is performed, i.e., D 6= ∅. In the example 4.10(c)-4.10(d) the computation at the interface is correctly performed. But the computation at the interface clearly fails in the last example 4.10(e)-4.10(f), compare the following remark. (n) (n+1) Remark 4.3.19. Evaluating the soft thresholding operator at u2 (N ) − u1 (N − (n+1) 1) implies that we are treating implicitly the computation of u1 at the interface. Namely, the interface condition can be read as (n+1) u1 (n) (n) (n+1) (N ) = u2 (N ) − Θ(n+1) · [u2 (N ) − u1 (n) (N − 1) (n+1) − sgn(u2 (N ) − u1 where Θ(n+1) = ( (n) (n+1) 1, |u2 (N ) − u1 0, otherwise . (n) (N − 1)) · c1 (N )h], (n) (N − 1)| − c1 (N )h > 0 The solution of the implicit problem is not immediate and one may prefer to modify (n) the situation in order to obtain an explicit formulation by computing Scn1 (N )h (u2 (N ) − (n) (n) (n+1) (N − 1)). The problem here is that, u1 (N − 1)) instead of Scn1 (N )h (u2 (N ) − u1 with this discretization, we cannot capture differences in the steepness of u1 and u2 (n) (n) (n) at the interface because u1 (N ) = u2 (N ) for all n. Indeed the condition |u2 (N ) − (n) (n) u1 (N − 1)| − c1 (N )h > 0 is never satisfied and the interface becomes always a (n) Dirichlet boundary condition. Even if we change the computation of c1 (N ) from q (n) (n) ǫ2 + (u1 (N ) − u1 (N − 1))2 /h2 to a forward difference q (n) (n) ǫ2 + (u1 (N + 1) − u1 (N ))2 /h2 187 4.3 Domain Decomposition for TV Minimization Initial Condition 500 iterations 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 Initial Condition Interface Inpainting Region −1.5 10 20 Reconstruction Interface Inpainting Region 30 40 50 60 70 80 90 100 −1.5 10 20 30 40 (a) Initial Condition 70 80 90 100 60 70 80 90 100 60 70 80 90 100 500 iterations 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 Initial Condition Interface Inpainting Region 10 20 Reconstruction Interface Inpainting Region 30 40 50 60 70 80 90 100 −1.5 10 20 30 40 (c) 50 (d) Initial Condition 500 iterations 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 Initial Condition Interface Inpainting Region −1.5 60 (b) 1.5 −1.5 50 10 20 Reconstruction Interface Inpainting Region 30 40 50 60 70 80 90 100 −1.5 (e) 10 20 30 40 50 (f) Figure 4.10: Examples of TV-L2 inpainting in 1D where the domain was split in two. (a)-(f): λ = 1 and τ = 1/2 (as it is indeed done in the numerical examples presented in Figure (4.10)) the method fails when the gradients are equal in absolute value on the left and the right side of the 188 4.3 Domain Decomposition for TV Minimization interface. We do not insist on trying to capture heuristically all the possible exceptions. We can expect that this approach to the problem may become even more deﬁcient and more complicated to handle in 2D. Instead, we want to develop an algorithm in the spirit of (4.53) which allows to deal with the problem in a transparent way. The Novel Approach Based on Subspace Corrections and Oblique Thresholding We want to implement algorithm (4.65) for the minimization of J. To solve its subitera(n,M ) tions we compute the minimizer by means of oblique thresholding. Denote u2 = u2 (n+1,ℓ+1) u1 = u1 (n+1,ℓ) , and z = u1 pute the minimizer (n+1,ℓ) + πV1 T ∗ (f − T u2 − T u1 , ). We would like to com- u1 = argminu∈V1 ku − zk2L2 (Ω) + 2α|D(u + u2 )|(Ω) by (z + u2 − η) − u2 , u1 = (I − PαK|D(·)|(Ω) )(z + u2 − η) − u2 = S|D(·)|(Ω) α for any η ∈ V2 . It is known [Ch04] that K|D(·)|(Ω) is the closure of the set n o d div ξ : ξ ∈ Cc1 (Ω) , |ξ(x)| ≤ 1 ∀x ∈ Ω . The element η ∈ V2 is a limit of the corresponding ﬁxed point iteration (4.63). In order to guarantee concrete computability and the correctness of this procedure, we need to discretize the problem and approximate it in ﬁnite dimensions, compare Examples 4.3.1.2 and Remark 4.3.9.2. In contrast to the approach of the previous subsection, where we used the discretization of the subdiﬀerential to solve the subiterations, in the following we directly work with discrete approximations of the functional J. In dimension d = 1 we consider vectors u ∈ H := RN , u = (u1 , u2 , . . . , uN ) with gradient ux ∈ RN given by ( ui+1 − ui if i < N (ux )i = 0 if i = N, for i = 1, . . . , N . In this setting, instead of minimizing J(u) := kT u − f k2L2 Ω + 2α|D(u)|(Ω), 189 4.3 Domain Decomposition for TV Minimization we consider the discretized functional X Jδ (u) := 1≤i≤N ((T u)i − gi )2 + 2α|(ux )i | . As in Section 4.2.2, we assume that T is applied on the piecewise linear interpolant û of the vector (ui )N i=1 (we make a similar assumption for d = 2). To highlight the relationship between continuous and discrete setting we introduce a step-size h ∼ min{1/N, 1/M } in the discrete deﬁnition of J by deﬁning a new functional Jδh equal to h times the expression Jδ above. One can show that as h → 0, Jδh Γ− converges to the continuous functional J, see [Ch04, Br02]. In particular, piecewise linear interpolants ûh of the minimizers of the discrete functional Jδh converge to minimizers of J. This observation clearly justiﬁes our discretization approach. For the deﬁnition of the set K|D(·)|(Ω) in ﬁnite dimensions we further introduce discrete divergence in one dimension ∇· : H → H (resp. ∇· : H × H → H in two dimensions), deﬁned, by analogy with the continuous setting, by ∇· = −∇∗ (∇∗ is the adjoint of the gradient ∇). That is, the discrete divergence operator is given by backward diﬀerences, in one dimension by pi − pi−1 (∇ · p)i = pi −pi−1 if 1 < i < N if i = 1 if i = N, The discrete setting in dimension d = 2 is analogous to the one speciﬁed at the beginning of Section 4.2.2, with the only diﬀerence being that we have the L2 norm in the deﬁnition of Jδ instead of the H −1 norm. With these deﬁnitions the set Kk(·)x k N in one dimension is given by ℓ1 {∇ · p : p ∈ H, |pi | ≤ 1 ∀i = 1, . . . , N } . and in two dimensions Kk∇(·)k N ×M is given by ℓ1 {∇ · p : p ∈ H × H, |pi,j | ≤ 1 ∀i = 1, . . . , N and j = 1, . . . , M } . For the computation of the projection in oblique thresholding we use the dual algorithm of Chambolle, which has been presented in Section 4.2.2 for dimension d = 2. For d = 1 a similar algorithm for the computation of PαKk(·)x k N (f )is given: ℓ1 190 4.3 Domain Decomposition for TV Minimization We choose τ > 0, let p(0) = 0 and for any n ≥ 0, (n+1) pi (n) = for i = 1, . . . , N . pi + τ ((∇ · p(n) − f /α)x )i , 1 + τ ((∇ · p(n) − f /α)x )i (4.86) In this case the convergence of α∇ · p(n) to the corresponding projection as n → ∞ is guaranteed for τ ≤ 1/4. In this special setting the oblique thresholding algorithm reads Oblique Thresholding for TV-L2 Minimization Start with an initial condition u0 = 0, z given as above, and uj the respective projection of the most recent iterate on L2 (Ωj ). Then iterate = Id − PαK|D(·)|(Ω) ((z + uj ) − η) − uj , uk+1 i i = 1, 2, i 6= j, where η is computed via the ﬁxed point iteration Let η 0 = 0 ∈ V2 and iterate η m+1 = πV2 PαK|D(·)|(Ω) (η m − (uj + z)) , m ≥ 0. Domain decomposition In one dimension the domain Ω = [a, b] is split into two intervals Ω1 = [a, N2 ] and Ω2 = [ N2 + 1, b]. The interface ∂Ω1 ∩ ∂Ω2 is located between i = ⌈N/2⌉ in Ω1 and i = ⌈N/2⌉ + 1 in Ω2 . In two dimensions the domain Ω = [a, b] × [c, d] is split in an analogous way with respect to its rows. In particular we have Ω1 = [a, N2 ] × [c, d] and Ω2 = [ N2 + 1, b] × [c, d], compare Figure 4.11. The splitting in more than two domains is done similarly: Set Ω = Ω1 ∪ . . . ∪ ΩN, the domain Ω decomposed into N disjoint domains Ωi , i = 1, . . . , N. Set s = ⌈N/N⌉. Then Ω1 = [1, s] × [c, d] for i = 2 : N − 1 Ωi = [(i − 1)s + 1, is] × [c, d] end ΩN = [(N − 1)s + 1, N ] × [c, d]. 191 4.3 Domain Decomposition for TV Minimization a = x1 Ω1 x⌈N/2⌉ x⌈N/2⌉+1 ——- ——- ∂Ω1 ∩ ∂Ω2 ——- ——- Ω2 b = xN Figure 4.11: Decomposition of the discrete image in two domains Ω1 and Ω2 with interface ∂Ω1 ∩ ∂Ω2 To compute the ﬁxed point η of (4.62) in an eﬃcient way we make the following considerations, which allow to restrict the computation to a relatively small strip around the interface. For u2 ∈ V2ψ and z ∈ V1 a minimizer u1 is given by u1 = argminu∈V1 ku − zk2L2 (Ω) + 2α|D(u + u2 )|(Ω). We further decompose Ω2 = Ω̂2 ∪(Ω2 \ Ω̂2 ) with ∂ Ω̂2 ∩∂Ω1 = ∂Ω2 ∩∂Ω1 , where Ω̂2 ⊂ Ω2 is a strip around the interface ∂Ω2 ∩ ∂Ω1 , as illustrated in Figure 4.12. By using the splitting of the total variation (4.55) we can restrict the problem to an equivalent minimization where the total variation is only computed in Ω1 ∪ Ω̂2 . In other words, we have u1 = argminu∈V1 ku − zk2L2 (Ω) + 2α|D(u + u2 )|(Ω1 ∪ Ω̂2 ). Hence, for the computation of the ﬁxed point η ∈ V2 , we need to carry out the iteration η (m+1) = πV2 PαK|D(·)|(Ω) (η (m) − z + u2 ) only in Ω1 ∪ Ω̂2 . By further observing that η will be supported only in Ω2 , i.e. η(x) = 0 in Ω1 , we may additionally restrict the ﬁxed point iteration on the relatively small strip Ω̂1 ∪ Ω̂2 , where Ω̂1 ⊂ Ω1 is an neighborhood around the interface from the side of Ω1 . Although the computation of η restricted to Ω̂1 ∪ Ω̂2 is not equivalent to the computation of η on whole Ω1 ∪ Ω̂2 , the produced errors are practically negligible, because of the Neumann boundary 192 4.3 Domain Decomposition for TV Minimization Ω1 \ Ω̂1 ——- ——- ——- Ω̂1 ∂Ω1 ∩ ∂Ω2 Ω̂2 ——- ——- ——- Ω2 \ Ω̂2 Figure 4.12: Computation of η only in the strip Ω̂1 ∪ Ω̂2 . conditions involved in the computation of PαK|∇(·)|(Ω 1 ∪Ω̂2 ) . Analogously, one operates for the minimization on Ω2 . Numerical experiments in one and two dimensions We shall present numerical results in one and two dimensions for the algorithm in (4.65), and discuss them with respect to the choice of parameters. In one dimension we consider the same three signals already discussed for the “naive” approach in Figure 4.10. In the left column of Figure 4.13 we report again the one dimensional signals. The right column shows the result of the application of the domain decomposition method (4.65) for total variation minimization. The support of the signals is split in two intervals. The interface developed by the two intervals is marked by a red dot. In all three examples we ﬁxed α = 1 and τ = 1/4. The ﬁrst example 4.57-4.59 shows a step function, which has its step directly at the interface of the two intervals. The total variation minimization (4.65) is applied with T = I. This example conﬁrms that jumps are preserved at the interface of the two domains. The second and third example 4.13(c)-4.13(f) present the behavior of the algorithm when interpolation across the interface is performed. In this case the operator T is given 193 4.3 Domain Decomposition for TV Minimization (a) (b) (c) (d) (e) (f) Figure 4.13: (a)-(f): Examples of the domain decomposition method for TVL2 denoising/inpainting in 1D where the domain was split in two domains with α = 1 and τ = 1/4 194 4.3 Domain Decomposition for TV Minimization by the multiplier T = 1Ω\D , where D is an interval containing the interface point. In contrast to the performance of the interpolation of the “naive” approach for the third example, Figure 4.10(e)-4.10(f), the new approach solves the interpolation across the interface correctly, see Figure 4.13(e)-4.13(f). Figure 4.14: An example of TV-L2 inpainting in where the domain was split in two with α = 10−2 and τ = 1/4 Figure 4.15: An example of TV-L2 inpainting in where the domain was split in ﬁve with α = 10−2 and τ = 1/4 Inpainting results for the two dimensional case are shown in Figures 4.14-4.15. The interface is here marked by a red line in the given image. In the ﬁrst example in Figure 195 4.3 Domain Decomposition for TV Minimization 4.14 the domain is split in two subdomains, in the second example in Figure 4.15 the domain is split in ﬁve subdomains. The Lagrange multiplier α > 0 is chosen equal to 10−2 . The time-step for the computation of PαK|∇(·)|(Ω) is chosen as τ = 1/4. The examples conﬁrm the correct reconstruction of the image at the interface, preserving both continuities and discontinuities as desired. Despite the fact that Theorem 4.3.16 does not guarantee that the algorithm in (4.65) can converge to a minimizer of J (unless one of the conditions in (iv) holds), it seems that for total variation minimization the result is always rather close to the expected minimizer. Let us now discuss the choice of the diﬀerent parameters. As a crucial issue in order to compute the solution at the interface ∂Ω1 ∩∂Ω2 correctly, one has to pay attention to the accuracy up to which the projection PαK|∇(·)|(Ω) is approximated and to the width of the strip Ω̂1 ∪ Ω̂2 for the computation of η. The alternating iterations (4.65) in practice are carried out with L = M = 5 inner iterations. The outer iterations are carried out until the error J(u(n+1) ) − J(u(n) ) is of order O(10−10 ). The ﬁxed point (0) η is computed by iteration (4.63) in at most 10 iterations with initialization ηn = 0 (0) when n = 1 and ηn+1 = ηn , the η computed in the previous iteration, for n > 1. For the computation of the projection PαK|∇(·)|(Ω) by Chambolle’s algorithm (4.39) we choose τ = 1/4. Indeed Chambolle points out in [Ch04] that, in practice, the optimal constant for the stability and convergence of the algorithm is not 1/8 but 1/4. Moreover, if the derivative along the interface is high, i.e., if there is a step along the interface, one has to be careful concerning the accuracy of the computation for the projection. The stopping criterion for the iteration (4.39) consists in checking that the maximum variation between pni,j and pn+1 is less than 10−3 . With less accuracy, artifacts on i,j the interface can appear. This error tolerance may need to be further decreased for very large α > 0. Furthermore, the size of the strip varies between 6 and 20 pixels, also depending on the size of α > 0, and on whether inpainting is carried out via the interface or not (e.g., the second and third example in Figure 4.13 failed in reproducing the interface correctly with a strip of size 6 but computed it correctly with a strip of size 20). Parallel Implementation The principal reasons to develop domain decomposition algorithms is the formulation of solvers which can be easily parallelized. In Section 4.3.5, equation (4.79), we also propose a parallelized version for the subspace minimization 196 4.3 Domain Decomposition for TV Minimization using oblique thresholding. As for the alternating algorithm, we provide a detailed analysis of the convergence properties of its modiﬁcation for parallel computation, cf. Theorem 4.3.18. Based on this, I developed a parallel implementation of the domain decomposition algorithm for total variation minimization in Matlab (using MatlabMPI1 provided by MIT2 ). Both the parallel code and the sequential code are available online3 . Numerical tests exploring the computational performance of the parallel algorithm are a matter of future research. 4.3.7 Domain Decomposition for TV-H−1 Minimization As already pointed out earlier in this chapter, one of the drawbacks of using TVH−1 minimization in applications is its slow numerical performance. The semi-implicit scheme for TV-H−1 inpainting in Section 4.1.3 suﬀers from a damping of the iterates dependent on the choice of the ﬁdelity parameter; and the new dual algorithm for TVH−1 minimization from Section 4.2.1 , i.e., Algorithm (P), is conditioned to time steps τ ≤ 1/64 in order to guarantee convergence. If now, additionally, the data dimension is large, e.g., when we have to process 2D images of high resolution, of sizes 3000 × 3000 pixels for instance, or even 3D image data, each iteration step itself is computationally expensive and we are far away from real-time computations. Motivated by this, we wish to apply the theory developed in the previous subsections for the solution of the TV-H−1 minimization problem (4.33), i.e., the minimization of J(u) := kT u − f k2H −1 (Ω) + 2α|Du|(Ω). To do so, we shall use the algorithm developed in Section 4.2.2 for the computation of the oblique thresholding operator. This part of the present work is contained in [Sc09]. We follow the notation from the previous sections and from Section 4.2.2. We are in the same setting as for TV-L2 minimization, with the only diﬀerence being that the ﬁdelity term in (4.33) is minimized in the weaker H −1 norm instead of the norm in L2 (Ω). However, as it turns out later, we fulﬁll all the necessary properties, assumed in 1 The author thanks Jahn Müller from the University of Münster for his help with the implementation in MatlabMPI 2 see http://www.ll.mit.edu/mission/isr/matlabmpi/matlabmpi.html 3 http://homepage.univie.ac.at/carola.schoenlieb/webpage_tvdode/tv_dode_numerics.htm 197 4.3 Domain Decomposition for TV Minimization Section 4.3.1, in order to apply the general theory to the case of TV-H−1 minimization. Then we want to minimize J in (4.33) by the alternating algorithm (4.65). As in Section 4.3.3, the subproblem on Ω1 reads n o argminu1 ∈V1 J(u1 + u2 ) = 2α |D(u1 + u2 )| (Ω) + kT u1 − (f − T u2 )k2−1 . (4.87) Diﬀerently to the surrogate functionals suggested in Section 4.3.3, we follow the approach presented in Section 4.2.2, i.e., Algorithm (P) and use the slightly diﬀerent surrogate functionals (4.47). This means that we introduce the following sequence of functionals for this subminimization problem, i.e., let u0 = 0, for k ≥ 0 let 2 2 1 1 Js1 (u1 +u2 , uk1 ) = |D(u1 + u2 )| (Ω)+ u1 − uk1 + u1 − (f − T u2 + (Id − T )uk1 ) . 2τ −1 2α −1 and realize an approximate solution to (4.87) by using the following algorithm: For u01 ∈ BV (Ω1 ), uk+1 = argminu1 ∈V1 Js1 (u1 + u2 , uk1 ), 1 k ≥ 0. (4.88) As in the general theory in Section 4.3.3, we are going to use Theorem 4.3.7 to solve the subminimization problem (4.88) via its reformulation on Ω, i.e., uk+1 = argminu∈BV (Ω) {F (u), πV2 (u) = 0} , 1 with F (u) = J1s (u + u2 , uk1 ). Now, since L2 (Ω) ⊂ L2 (R2 ) ⊂ H −1 (Ω) (by zero extensions of functions on Ω to R2 ), our functional F is continuous on V1ψ ⊂ V1 = ker πV2 in the norm topology of BV (Ω). Further πV2 |BV (Ω) is a bounded and surjective map ∗ with closed range in the norm topology of BV (Ω), i.e., πV2 |BV (Ω) is injective and ∗ ψ Range πV |BV (Ω) ∼ = (V )∗ is closed. By applying Theorem 4.3.7, we know that 2 2 the optimality of uk+1 1 (BV (Ω2 ))∗ such that is equivalent to the existence of an η ∈ Range(πV2 |BV (Ω) )∗ ∼ = −η ∈ ∂BV (Ω) F (uk+1 1 ). Now ∂BV (Ω) F (uk+1 1 )= where 1 −1 k+1 + u ) ∆ (u1 − z) + ∂BV (Ω) D(uk+1 2 (Ω) 1 µ z= z1 α + z2 τ ατ , µ= , α+τ α+τ 198 (4.89) 4.3 Domain Decomposition for TV Minimization with z1 = uk1 , z2 = f − T u2 + (Id − T )uk1 . Then the optimality of uk+1 is equivalent to 1 1 k+1 − z) + η + ∂ D(u + u ) 0 ∈ ∆−1 (uk+1 (Ω). 2 BV (Ω) 1 1 µ The latter is equivalent to 1 uk+1 + u2 ∈ ∂BV (Ω) |D.| (Ω)∗ ( ∆−1 (z − uk+1 1 1 ) − η), µ i.e, u2 + z z − uk+1 1 1 1 ∈ + ∂BV (Ω) |D.| (Ω)∗ ( ∆−1 (z − uk+1 1 ) − η). µ µ µ µ By letting w = ∆−1 (z − uk+1 )/µ − η we have 0 ∈ (−∆(w + η) − (u2 + z)/µ) + 1 ∂ |D.| (Ω)∗ (w), µ BV (Ω) or, in other words, w is a minimizer of w − (∆−1 (u2 + z)/µ − η)2 1 H (Ω) 0 2 + 1 |D·| (Ω)∗ (w). µ Following the same procedure as in Section 4.2.2 we deduce that w = P1K (∆−1 (u2 + z)/µ − η), where P1K denotes the orthogonal projection on K over H01 (Ω) like in Section 4.2.2. Then a minimizer uk+1 of (4.88) can be computed as 1 uk+1 = −∆ Id − P1µK 1 ∆−1 (z + u2 ) − µη − u2 . By applying πV2 to both sides of the latter equality we get 0 = µ∆η + πV2 ∆P1µK ∆−1 (u2 + z) − µη . Assuming necessary zero boundary conditions on ∂Ω, the resulting ﬁxed point equation for η reads η= 1 πV2 P1µK µη − ∆−1 (u2 + z) . µ As before for the general theory, the ﬁxed point can be computed via the iteration η 0 ∈ V2 , η m+1 = 1 πV2 P1µK µη m − ∆−1 (u2 + z) , µ 199 m ≥ 0. 4.3 Domain Decomposition for TV Minimization Oblique Thresholding for TV-H−1 Minimization Start with an initial condition u0 = 0, z as in (4.89) and uj the respective projection of the most recent iterate on L2 (Ωj ). Then iterate uk+1 = −∆ Id − P1µK i ∆−1 (z + uj ) − µη − uj , i = 1, 2, i 6= j, (4.90) where η in (4.90) is computed via the ﬁxed point iteration Let η 0 = 0 ∈ V2 and iterate η m+1 = 1 πV P1 µη m − ∆−1 (uj + z) , µ 2 µK m ≥ 0. (4.91) In sum we solve (4.33) by the alternating subspace minimizations where each subminimization problem is computed by the oblique thresholding algorithm for TVH−1 minimization: As before the projection P1µK is computed by Algorithm (P). Unsurprisingly, we are especially interested in the inpainting setting T = 1Ω\D , compare also with (4.48) in Section 4.2.3. In Figures 4.16 and 4.17 the given images have been divided in four subdomains, marked by red lines, and the image is inpainted by TV-H−1 inpainting computed by the sequential algorithm (4.65) by means of oblique thresholding. Note that for these examples we used a diﬀerent decomposition of the spatial domain Ω, where the four interfaces form a cross and meet in the middle of the domain. Special care has to be taken about this meeting point in the middle, i.e., the solution has to be averaged over the subdomains there. 200 4.3 Domain Decomposition for TV Minimization (a) Given image (b) Intermediate inpainting iterate (c) Inpainting result Figure 4.16: TV-H−1 inpainting computation for a model example on four domains with α = 0.01. Figure 4.17: TV-H−1 inpainting computation on four domains with α = 0.005. 201 Chapter 5 Applications Digital inpainting methods provide an important tool in the restoration of images in a wide range of applications. In my Ph.D. thesis I focus on two special applications. The ﬁrst is the inpainting of ancient frescoes. In particular we discuss the CahnHilliard equation for the inpainting of binary structure and TV-H−1 inpainting for the reconstruction of the grayvalues in the recently discovered Neidhart frescoes in Vienna, cf Section 5.1. The second application has originated in a project of Andrea Bertozzi taking place at UCLA (University of California Los Angeles) and is about the inpainting of roads in satellite images. It is presented in Section 5.2. 5.1 Restoration of Medieval Frescoes In the course of an ongoing interdisciplinary project1 we aim to use digital inpainting algorithms for the restoration of frescoes. Most of the results presented within this section can be found in [BFMS08]. Particular consideration has been extended to the newly found Neidhart frescoes (Tuchlauben 19, 1010 Vienna). These medieval frescoes from the 14th Century are depicting a cycle of songs of the 13th Century minnesinger Neidhart von Reuental. Hidden behind a wall over years, the frescoes have been damaged during exposure. Advanced mathematical tools were developed speciﬁcally for so-called ”mathematical inpainting/retouching” of digital images. To this end, variational methods and third 1 WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts 202 5.1 Restoration of Medieval Frescoes and fourth-order partial diﬀerential equations have been investigated. Eﬃcient numerical methods for the solution of the devised partial diﬀerential equations have been designed. In the following we discuss our mathematical inpainting methods and present numerical results from their application to the Neidhart frescoes. 5.1.1 Neidhart Frescoes Fragments of 14th century wall frescoes found beneath the crumbling plaster of an old apartment in the heart of Vienna depict a popular medieval cycle of songs of the 13th century minnesinger, Neidhart von Reuental. In the very late 14th century, Michel Menschein, a wealthy Viennese council member and cloth merchant, commissioned local artists to paint the stories in Neidhart’s songs on the walls of his Festsaal (banquet hall). The Neidhart frescoes provide a unique peek into medieval humor, and at the same time, a peek into the taste of a medieval man. Figure 5.1: Part of the Neidhart frescoes. Image courtesy Andrea Baczynski. In Figure 5.1 a part of the Neidhart frescoes is shown1 . The white holes in the fresco are due to the wall which covered the fresco until a few years ago. They occurred 1 The author thanks Andrea Baczynski for providing the fresco data. 203 5.1 Restoration of Medieval Frescoes when the wall was removed. In the following we want to apply digital restoration methods to these frescoes. Thereby the main challenge is to capture the structures in the preserved parts of the fresco and transport them into the damaged parts continuously. Due to their great age and almost 600 years of living by owners and tenants in the apartment, saturation, hue and contrast quality of the colours in the frescoes suﬀered. Digital grayvalue, i.e., colour interpolation, in the damaged parts of the fresco therefore demands sophisticated algorithms taking these detrimental factors into account. 5.1.2 Methods In the following we present the mathematical methods we used in order to reconstruct the damaged parts in the fresco. In particular, two inpainting methods based on higherorder partial diﬀerential equations, i.e., Cahn-Hilliard- (2.1) and TV-H−1 inpainting (2.27), are to be presented. We ﬁnalize this section by proposing a possible strategy to adapt these two inpainting approaches to the requirements of the Neidhart frescoes. Let us start with brieﬂy recalling the Cahn-Hilliard inpainting approach (2.1). The inpainted version u(x) of f (x) ∈ L2 (Ω) is constructed by following the evolution equa- tion 1 ut = ∆(−ǫ∆u + F ′ (u)) + λ(f − u) in Ω, ǫ ∂u ∂∆u = =0 on ∂Ω, ∂ν ∂ν with F (u) a so-called double-well potential, e.g., F (u) = u2 (u − 1)2 , while ( λ0 Ω \ D λ(x) = 0 D is the characteristic function of Ω \ D multiplied by a constant λ0 ≫ 1. A generalization of the Cahn-Hilliard inpainting approach to an approach for gray- value images was already presented in Section 2.2 (also cf. [BHS08]). Namely we consider TV-H−1 inpainting (2.27): Let f ∈ L2 (Ω), |f | ≤ 1 be the given grayvalue image. The inpainted version u(x) of f (x) evolves in time like ∇u ut = −∆ ∇ · + λ(f − u), |∇u| where p ∈ ∂T V (u) was replaced by a relaxed version ! ∇u ∇u =∇· p , p≈∇· |∇u| |∇u|2 + ǫ 204 (5.1) 5.1 Restoration of Medieval Frescoes for an 0 < ǫ ≪ 1. As mentioned in Section 5.1.1, the Neidhart frescoes pose a special challenge con- cerning their digital restoration. We summarize the main issues in the following list: 1. Lack of grayvalue contrast 2. Low colour saturation and hue 3. Damaged parts can be rather large, i.e., the diameter of the damaged domain can be larger than the width of lines which are to be continued into the damaged part Hence we need an inpainting approach which takes into account these possible diﬃculties and solves (or circumvents) them. As we have pointed out earlier in this section, the third issue can be solved by using a higher-order inpainting method such as (2.1) and (2.27). Unfortunately diﬃculties two and three prevent the eﬀective application of these methods. As the contrast between grayvalues is low, the edges (which identify the main structure of an image) are not clearly deﬁned. As inpainting lives and dies with uniqueness of edge continuation (cf. Figure 5.2) we may run into trouble if we do not preprocess the digital images of the fresco in an adequate manner. Figure 5.2: (l.) What is the right solution? (r.) Part of the Neidhart fresco: How should the inpainting algorithm decide in this case? Speciﬁcally we follow two strategies: • Strategy 1: Structure inpainting on binary images with the Cahn-Hilliard equa- tion. Based on the recovered binary structure, the fresco is colourized. We discuss the recolourization, i.e., the ﬁlling in of grayvalues based on the given binary structure, in more detail in Section 5.1.2. Also compare [Fo06] for a similar approach. 205 5.1 Restoration of Medieval Frescoes • Strategy 2: Apply TV-H−1 inpainting in two steps. First with a small λ0 , e.g., λ0 = 1, to merge together ﬁne artifacts in the fresco by diﬀusion. In the second step we choose a large λ0 ≫ 1, e.g., λ0 = 103 , to reconstruct the fresco inside the damaged parts. Figure 5.3: Part of the Neidhart frescoes In the following we present ﬁrst numerical results following these two strategies. For both inpainting methods (2.1) and (2.27) we used the convexity splitting algorithms presented in Section 4.1 for their discretization in time. Strategy 1 - Binary based fresco inpainting We begin with the inpainting of the binary structure of the frescoes by means of (2.1), cf. Figure 5.4-5.5. In our numerical examples we applied (2.1) in two steps (cf. [BEG07b]). In the ﬁrst few time steps we solve (2.1) with a rather big ǫ, e.g., ǫ = 3. We stop when we are suﬃciently close to a steady state. Then we switch the ǫ value to a smaller one, e.g., ǫ = 0.01. Using the steady state from the ﬁrst few time steps of (2.1) with a large ǫ as an initial condition, we apply the iteration now for the switched ǫ. Again we stop when we are suﬃciently close to the steady state. The next step is to recolourize the damaged parts by using the recovered binary structure as underlying information. This can be done in the following way. Motivated by previous work, Fornasier [Fo06], and Fornasier and March [FM07], we propose an inpainting model for grayvalue images which uses a given (or previously obtained) binary structure inside the missing domain D. Thereby the binary structure of the image is usually obtained by a preprocessing step with Cahn-Hilliard inpainting [BEG07a, BEG07b]. Let us describe this method in more detail. 206 5.1 Restoration of Medieval Frescoes Figure 5.4: Cahn-Hilliard inpainting with λ0 = 107 f.l.t.r.: Part of the fresco; binary selection in red; binary selection in black and white; initial condition for the inpainting algorithm where the inpainting region is marked with a gray rectangle; inpainting result after 200 time steps with ǫ = 3; inpainting result after additional 800 time steps with ǫ = 0.01. Let f ∈ L2 (Ω) be a given image which grayvalue in Ω \ D and binary in D. We wish to recover the grayvalue information in D based on the binary structure given by f by means of the following minimization problem: Z µ ∗ |u(x) − f (x)|2 dx u = argmin { 2 Ω\D Z λ + |Lbin (u(x)) − f (x)|2 dx 2 D + |Du| (Ω), u ∈ L2 (Ω)(Ω, R+ ) . (5.2) In our case Lbin is a transformation which projects the grayvalue range of u, e.g., [0, 1], on the binary range {0, 1}. The corresponding Euler-Lagrange equation of (5.2) then reads 0 = p + µ · χΩ\D (u − f ) + λ · χD (Lbin (u) − f ) ∂Lbin (u), ∂u p ∈ ∂ |Du| (Ω), (5.3) where χΩ\D and χD are the characteristic functions of Ω \ D and D respectively. In (5.3) we approximate the subgradient of the total variation by a relaxed version p ≈ p −∇ · (∇u/|∇u|ǫ ), where |∇u|ǫ := |∇u|2 + ǫ. The relaxed version of (5.3) then reads ∇u ∂Lbin 0 = −∇ · (u). + µ · χΩ\D (u − f ) + λ · χD (Lbin (u) − f ) |∇u|ǫ ∂u 207 5.1 Restoration of Medieval Frescoes Figure 5.5: Cahn-Hilliard inpainting with λ0 = 106 f.l.t.r.: Part of the fresco; binary selection in red; binary selection in black and white; initial condition for the inpainting algorithm where the inpainting region is marked with a gray rectangle; inpainting result after 200 time steps with ǫ = 4; inpainting result after additional 800 time steps with ǫ = 0.01. For our purpose Lbin is modeled by a relaxed version of the Heaviside function depending on a (presumably) given threshold τ . Recall that the Heaviside function is given by ( 1 H(x) = 0 x ≥ 0, x < 0. In order to have a diﬀerentiable transformation, we approximate H by H δ with x 2 1 δ 1 + arctan , 0 < δ ≪ 1. H (x) = 2 π δ Let us assume now that the binary part of f in D was obtained such that it is 1 where the lost grayvalue information fg ≥ τ and 0 otherwise, i.e., for x ∈ D we assume ( 1 fg (x) ≥ τ, f (x) = 0 fg (x) < τ. Then we want the binarization Lbin of u to be modeled subject to the same assumption and we ﬁnally deﬁne ( 1 u ≥ τ, Lbin (u) := H (u − τ ) ≈ 0 u < τ. δ (5.4) The derivative of Lbin is then a smooth approximation of the Dirac δ-function in u = τ , i.e., dH δ 1 1 (x) = δ δ (x) = , dx πδ 1 + (x/δ)2 208 0 < δ ≪ 1, 5.1 Restoration of Medieval Frescoes and ∂Lbin (u) = δ δ (u − τ ). ∂x (5.5) In Figure 5.1.2 two numerical examples for the binary based grayvalue inpainting are shown. (a) Initial condition from the example in Figure 5.4 (b) Inpainting result for 5.6(a) (c) Initial condition from the example in Figure 5.5 (d) Inpainting result for 5.6(c) Figure 5.6: Two examples for binary based grayvalue inpainting with µ = λ = 102 and 5000 time steps. The grayvalues inside of the inpainting domain are initialized with the results from Cahn-Hilliard inpainting in Figure 5.4-5.5, cf. 5.6(a) and 5.6(c). Solving (5.2) via a steepest descent approach constitutes 5.6(b) and 5.6(d) Strategy 2 - Grayvalue fresco inpainting We consider (5.1) and apply it for the grayvalue inpainting of the Neidhart frescoes. In Figure 5.7 the algorithm (5.1) has been applied to a small part of the Neidhart frescoes. In this particular case we did not even have to preprocess the image because only plain grayvalue information was to be imported into the inpainting domain, whereas in Figure 5.8 we acted on strategy 2. Namely, we primarily denoised the image by (5.1) with λ(x) = λ0 on the whole image domain and applied the inpainting algorithm ((5.1) with λ(x) = 0 inside the inpainting domain D) on the ”cleaned” image in 209 5.1 Restoration of Medieval Frescoes a second step. Clearly our binary based approach in Figure 5.1.2-5.8 produces superior visual results in the presence of edges. Figure 5.7: TV-H−1 inpainting applied to a part of the Neidhart fresco Figure 5.8: TV-H−1 inpainting following strategy 2. f.l.t.r. part of the Neidhart fresco; preprocessed image; initial condition for the inpainting algorithm where the inpainting domain is marked as a gray rectangle; preliminary inpainting result (algorithm carried out until err ≈ 10−4 ). Inpainting diﬃculties due to the reasons indicated in Figure 5.2 are clearly visible. Conclusion We succeeded in developing methods for the restoration of digital images using mathematical concepts of partial diﬀerential equations. We showed the reconstruction of the grayvalue information in the Neidhart frescoes using basically two strategies. In the ﬁrst strategy Cahn-Hilliard inpainting was used to reconstruct the binary structure in the frescoes. Then the grayvalues were ﬁlled into the missing parts based on the binary information available from the Cahn-Hilliard reconstruction. The main idea of this strategy is to exploit the good reconstruction qualities of Cahn-Hilliard inpainting 210 5.2 Road Reconstruction (smooth connection of edges even across large distances) as much as possible. This approach turned out to be the method of choice in the presence of edges in the neighborhood of the missing parts and when the gaps in the frescoes are large. The second strategy uses TV-H−1 inpainting for the reconstruction of the grayvalue information in the frescoes. This approach produced fairly good results for the inpainting of homogeneous areas in the frescoes. As one immediately observes, parts of the surface of the Neidhart frescoes are more texture-like. Since the inpainting methods we used cannot reproduce texture, they can only deliver visually good results to a limited extent. In Figure 5.9 a part of the fully restorated frescoes is shown1 . A direct comparison between the restorated frescoes and our digital results deﬁnes the next step within this project. Figure 5.9: The restored fresco. Image courtesy Wolfgang Baatz. 5.2 Road Reconstruction This project is about the continuation of roads in aerial images and takes place in Andrea Bertozzi’s group at UCLA2 (cf. also [DB08]). The roads are partially occluded ,e.g., by trees, cf. Figure 5.10, and now the challenge is to reconstruct the roads such that one is able to follow them in aerial images. 1 The author thanks Wolfgang Baatz for providing this data. I would like to thank the UCLA Mathematics Department, and Alan Van Nevel and Gary Hewer from the Naval Air Weapons Station in China Lake, CA for providing the data. 2 211 5.2 Road Reconstruction Figure 5.10: Road data from Los Angeles Our ﬁrst approach is to binarize the road data and apply Cahn-Hilliard inpainting (cf. (2.1) in Section 2.1) to the binary roads 1 . In Figure 5.11 two examples for CahnHilliard inpainting of binary roads are shown. Note that the Cahn-Hilliard inpainting approach is applied to the corrupted road images in two steps. First the large gap(s) in the road are ﬁlled by choosing rather large parameter ǫ in (2.1) and letting the inpainting algorithm run until it reaches a steady state. Using the result from this ﬁrst step, Cahn-Hilliard inpainting is applied again with a small ǫ in order to sharpen the edges in the image. Figure 5.11: Cahn-Hilliard inpainting in two steps, namely with ǫ = 0.1 and ǫ = 0.01 in the ﬁrst row, and ǫ = 1.6 and ǫ = 0.01 in the second row of the Figure 5.2.1 Bitwise Cahn-Hilliard Inpainting In a second approach we shall reconstruct the roads with a bitwise Cahn-Hilliard inpainting approach. Speciﬁcally, one possible generalization of Cahn-Hilliard inpainting 1 Thanks to Shao-Ching Huang (UCLA) for the preparation of the data 212 5.2 Road Reconstruction for grayscale images is to split the grayscale image bit-wise into channels u(x) ; K X uk (x)2−(k−1) , k=1 where K > 0. The Cahn-Hilliard inpainting approach is then applied to each binary channel uk separately, compare Figure 5.12. At the end of the inpainting process the channels are assembled again and the result is the inpainted grayvalue image in lower grayvalue resolution, cf. Figure 5.13. In Figure 5.12 and 5.13 the application of bitwise Cahn-Hilliard inpainting for the restoration of satellite images of roads is demonstrated. One can imagine that the black dots in the ﬁrst picture in Figure 5.13 represent trees that cover parts of the road. The idea of bitwise Cahn-Hilliard inpainting was proposed in [DB08] for inpainting with wavelets based on Allen-Cahn energy. Figure 5.12: What is going on in the channels? The given image (ﬁrst row) and the Cahn-Hilliard inpainting result (second row) for the 1st, 3rd, 4th, and 5th channel Figure 5.13: Bitwise Cahn-Hilliard inpainting with K = 8 binary channels applied to road restoration: (l.) given distorted aerial image of a road; (r.) result of the inpainting process with the assembled binary channels of Figure 5.12. 213 Chapter 6 Conclusion My PhD thesis is mainly concerned with higher-order partial diﬀerential equations in image inpainting. The analysis and numerical realization of Cahn-Hilliard inpainting and TV-H−1 inpainting have been of particular interest. I could extend the understanding of these inpainting approaches by, among other things, giving answers about stationary solutions of the underlying equations, by an analysis for the inpainting error, and by giving an interpretation of these inpainting approaches in terms of a mechanism based on transport and diﬀusion. Moreover, I presented reliable and eﬃcient numerical solvers for higher-order inpainting approaches concluding with a domain decomposition approach for TV-based inpainting methods. In addition, the applicability of the discussed inpainting approaches was brought to the test for the restoration of ancient frescoes and the inpainting of satellite images of roads. In the following I would like to address some open problems that I consider to be interesting for the understanding of higher-order inpainting schemes: • The advantage of fourth-order inpainting models over models of second diﬀerential order is in the smooth continuation of image contents even across large gaps in the image. A motivation for the reasonability of this claim was already given in the Introduction of this thesis. Brieﬂy we can imagine that a fourth-order partial diﬀerential equation requires one boundary condition more than a secondorder equation. In [BEG07b] the authors showed that, in the limit λ0 → ∞, a stationary solution u of the Cahn-Hilliard equation fulﬁlls two conditions on the boundary of the inpainting domain, i.e., u = f and ∇u = ∇f on ∂D, cf. also (2.2) in Section 2.1. Hence during the inpainting process, in addition, to the value 214 of the image function, its gradient is continued into the missing domain. This is also true for our other two inpainting approaches, i.e., TV-H−1 inpainting and inpainting with LCIS, and is motivated by the numerical results in Figures 2.8 and 2.4. A rigorous derivation of this phenomenon, as the one for Cahn-Hilliard inpainting in [BEG07b], is a matter for future research. • Besides the fact that rigorous results for fourth-order partial diﬀerential equa- tions are rare in general, an asymptotic analysis of our three inpainting models would be of high (even practical) interest. More precisely the convergence of a solution of the evolution equations (2.1), (2.27), and (2.42), to a stationary state is still open. Since the inpainted image is the stationary solution of those evolution equations, the asymptotic behavior is of course an issue. Also in practice, the numerical schemes are solved to steady state (up to an approximational error). Note that additionally to the fourth diﬀerential order, a diﬃculty in the convergence analysis of (2.1) and (2.27) is that they do not follow a variational principle. Moreover, regularity results for solutions of higher-order inpainting approaches are mostly missing. For the proofs of Theorem 4.1.16 and Theorem 4.1.21, for instance, we had to assume that the exact solution is bounded on a ﬁnite time interval in a certain Sobolev norm. As we already argued in the remarks after the statement of the theorems, these assumptions seem to be heuristically reasonable, considering earlier results in [BGOV05, BG04]. Nevertheless, a rigorous derivation of such bounds is still missing. • Fast numerical solvers for inpainting with higher-order equations is still a mostly open issue, cf. the introductory Section 1.3.3 and Chapter 4. Let me list two promising approaches, whose applicability for inpainting has to be tested still. – As already pointed out in Section 1.3.3, the work in [GBO09], i.e., the socalled Bregman split method, promises very small computation times for ℓ1 regularized optimization problems in the context of surface reconstruction with missing data points. A future project could be to check the possible eﬀectiveness of the algorithm in [GBO09] for the higher-order TVH−1 inpainting approach and inpainting with LCIS. 215 – In [WAB] Weiss et al. propose an eﬃcient numerical algorithm for constrained total variation minimization, which is based on a recent advance in convex optimization proposed by Nesterov [Ne05]. They show that their scheme allows to obtain a solution of precision ǫ in O(1/ǫ) iterations, i.e., √ in the case of a strongly convex constraint even in only O(1/ ǫ) iterations. An idea, is to speed up the computation for TV-H−1 inpainting, combining this new method from [WAB] with my approach in Section 4.2, where TV-H−1 inpainting is computed iteratively via the minimization of surrogate functionals (4.47). 216 Appendix A Mathematical Preliminaries A.1 Distributional Derivatives Take Ω ⊂ Rd , d ≥ 1, to be an open set. We deﬁne D(Ω) := Cc∞ (Ω), where Cc∞ (Ω) is the set of C ∞ functions with compact support in Ω. Then D′ (Ω) denotes the set of real continuous linear functionals on D(Ω) which we call the space of distributions on Ω. We will denote the pairing of D′ (Ω) and D(Ω) by h., .iD′ ,D. In the following we deﬁne partial derivatives of a distribution. Definition A.1.1. (∂ α u) For u ∈ D′ (Ω) and for any multi-index α we define ∂ α u by h∂ α u, φiD′ ,D := (−1)|α| hu, ∂ α φiD′ ,D for all φ ∈ D(Ω). Compare [Fo99] for more details on distribution theory. A.2 Subgradients and Subdifferentials Let H be a real Hilbert space with norm k.k and inner product (., .). Given J : H → R and u ∈ H we say that J is Fréchet-differentiable at u ∈ H if ∇J(u) = lim t=0 d J(u + tv) < ∞. dt Then ∇J(u) is called the Fréchet derivative (or ﬁrst variation) of J. In some cases the above limit does not exist, i.e., the function J is not diﬀerentiable. Then we introduce the notion of the subdiﬀerential of a function (cf. [Ev98]). 217 A.3 Functional Analysis Definition A.2.1. Let X be a locally convex space, X ′ its dual, h., .i the bilinear pairing over X × X ′ and J a mapping of X into R. The subdifferential of J at u ∈ X is defined as ∂J(u) = p ∈ X ′ | hv − u, pi ≤ J(v) − J(u), ∀v ∈ X . A.3 Functional Analysis Theorem A.3.1. (Rellich-Kondrachov Compactness Theorem) cf. [Al92], Theorem 8.7, p. 243 Assume that Ω is a bounded and open subset of Rd with Lipschitz boundary. Suppose that 1 ≤ r < d. Then W 1,r (Ω) ֒→֒→ Lq (Ω), for each 1 ≤ q < dr d−r . Theorem A.3.2. (Fatou’s Lemma) If f1 , f2 , . . . is a sequence of non-negative measurable functions defined on a measure space (S, Σ, µ), then Z Z fn dµ . lim inf fn dµ ≤ lim inf S n→∞ n→∞ S Let X denote a real Banach space. Definition A.3.3. A set K ⊂ X is convex if for all u, v ∈ K and constants 0 ≤ λ ≤ 1, λu + (1 − λ)v ∈ K. Theorem A.3.4. (Characterization of compact sets) A closed subset K of a Banach space X is compact if and only if there is a sequence (xn ) in X such that ||xn || → 0 and K is a subset of the closed convex hull of (xn ). Theorem A.3.5. (Schauder’s fixed point theorem). Suppose that K ⊂ X is compact and convex, and assume also A : K → K is continuous. Then A has a fixed point. Proof. cf. [Ev98] Section 9.2.2. page 502-507. A.4 The Space H −1 and the Inverse Laplacian ∆−1 We denote by H −1 (Ω) the dual space of H01 (Ω) with corresponding norm k.k−1 . For a function f ∈ H −1 (Ω) the norm is deﬁned as kf k2−1 2 = ∇∆−1 f 2 = 218 Z Ω ∇∆−1 f 2 dx. A.5 Functions of Bounded Variation Thereafter the operator ∆−1 denotes the inverse to the negative Dirichlet Laplacian, i.e., u = ∆−1 f is the unique solution to A.5 − ∆u = f in Ω u=0 on ∂Ω. Functions of Bounded Variation The following results can be found in [AFP00], see also [Gi84], and [EG92]. Let Ω ⊂ R2 be an open and bounded Lipschitz domain. As in [AFP00] the space of functions of bounded variation BV (Ω) in two space dimensions is deﬁned as follows: Definition A.5.1. (BV (Ω)) Let u ∈ L1 (Ω). We say that u is a function of bounded variation in Ω if the distributional derivative of u is representable by a finite Radon measure in Ω, i.e., if Z Z ∂φ dx = − φdDi u ∀φ ∈ Cc∞ (Ω), i = 1, 2, u ∂x i Ω Ω for some R2 - valued measure Du = (D1 u, D2 u) in Ω. The vector space of all functions of bounded variation in Ω is denoted by BV (Ω). Further, the space BV (Ω) can be characterized by the total variation of Du. For this we ﬁrst deﬁne the so-called variation V (u, Ω) of a function u ∈ L1loc (Ω). Definition A.5.2. (Variation) Let u ∈ L1loc (Ω). The variation V (u, Ω) of u in Ω is defined by Z 2 1 V (u, Ω) := sup u divφ dx : φ ∈ Cc (Ω) , kφkL∞ (Ω) ≤ 1 . Ω A simple integration by parts proves that Z V (u, Ω) = |∇u| dx Ω if u ∈ C 1 (Ω). By a standard density argument this is also true for functions u ∈ W 1,1 (Ω). Before we proceed with the characterization of BV (Ω) let us recall the deﬁnition of the total variation of a measure: 219 A.5 Functions of Bounded Variation Definition A.5.3. (Total variation of a measure) Let (X, E) be a measure space. If µ is a measure, we define its total variation |µ| as follows: ) (∞ ∞ [ X Eh , ∀E ⊂ E. |µ(Eh )| : Eh ∈ E pairwise disjoint , E = |µ| (E) := sup h=0 h=0 With Deﬁnition A.5.2 the space BV (Ω) can be characterized as follows Theorem A.5.4. Let u ∈ L1 (Ω). Then u belongs to BV (Ω) if and only if V (u, Ω) < ∞. In addition, V (u, Ω) coincides with |Du| (Ω), the total variation of Du, for any u ∈ BV (Ω) and u 7→ |Du| (Ω) is lower semicontinuous in BV (Ω) with respect to the L1loc (Ω) topology. Note that BV (Ω) is a Banach space with respect to the norm kukBV (Ω) = kukL1 (Ω) + |Du| (Ω). Now we introduce so-called weak∗ convergence in BV (Ω) which is useful for its compactness properties. Note that this convergence is much weaker than the norm convergence. Definition A.5.5. (Weak∗ convergence) Let u, uh ∈ BV (Ω). We say that (uh ) weakly∗ ∗ converges in BV (Ω) to u (denoted by uh ⇀ u) if (uh ) converges to u in L1 (Ω) and (Duh ) weakly∗ converges to Du in all (Ω), i.e., Z Z φ dDu ∀φ ∈ C0 (Ω). lim φ dDuh = h→∞ Ω A simple criterion for weak∗ Ω convergence is the following: Theorem A.5.6. Let (uh ) ⊂ BV (Ω). Then (uh ) weakly∗ converges to u in BV (Ω) if and only if (uh ) is bounded in BV (Ω) and converges to u in L1 (Ω). Further we have the following compactness theorem: Theorem A.5.7. (Compactness for BV (Ω)) • Let Ω be a bounded domain with compact Lipschitz boundary. Every sequence (uh ) ⊂ BVloc (Ω) satisfying Z |uh | dx + |Duh | (A) : h ∈ N < ∞ ∀A ⊂⊂ Ω open, sup A admits a subsequence (uhk ) converging in L1loc (Ω) to u ∈ BVloc (Ω). If the sequence is further bounded in BV (Ω) then u ∈ BV (Ω) and a subsequence converges weakly∗ to u. 220 A.5 Functions of Bounded Variation • Let Ω be a bounded domain in Rd with Lipschitz boundary. Then every uniformly bounded sequence (uk )k≥0 in BV (Ω) is relatively compact in Lr (Ω) for 1 ≤ r < d d−1 , d ≥ 1. Moreover, there exists a subsequence ukj and u in BV (Ω) such that ukj ⇀ u weakly∗ in BV (Ω). In particular for d = 2 this compact embedding holds for 1 ≤ r < 2. Let u ∈ L1 (Ω). We introduce the mean value uΩ of u as Z 1 u(x) dx. uΩ := |Ω| Ω A generalization of the Poincare inequality gives the so-called Poincare-Wirtinger inequality for functions in BV (Ω). Theorem A.5.8. (Poincare-Wirtinger inequality) If Ω ⊂ R2 is a bounded, open and connected domain with compact Lipschitz boundary, we have ku − uΩ kLp (Ω) ≤ C |Du| (Ω) ∀u ∈ BV (Ω), 1≤p≤2 for some constant C depending only on Ω. Finally, since every normed vector space is a locally convex space, the theory of subdiﬀerentials from Section A.2 applies to the framework where X = BV (Ω). For a characterization of elements in the subdiﬀerential ∂ |Du| (Ω) we refer to the very detailed analysis of L. 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