# CAHN-HILLIARD INPAINTING AND A GENERALIZATION FOR GRAYVALUE IMAGES

CAHN-HILLIARD INPAINTING AND A GENERALIZATION FOR GRAYVALUE IMAGES MARTIN BURGER∗ , LIN HE† , AND CAROLA-BIBIANE SCHÖNLIEB‡ Abstract. 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. The inpainting of binary images using the Cahn-Hilliard equation is a new approach in image processing. In this paper we discuss the stationary state of the proposed model and introduce a generalization for grayvalue images of bounded variation. This is realized by using subgradients of the total variation functional within the flow, which leads to structure inpainting with smooth curvature of level sets. Key words. Cahn-Hilliard equation, TV minimization, image inpainting AMS subject classifications. 49J40 1. Introduction. 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 referred to as inpainting. Given an image f in a suitable Banach space of functions defined on Ω ⊂ R2 , an open and bounded domain, the problem is to reconstruct the original image u in the damaged domain D ⊂ Ω, called inpainting domain. In the following we are especially interested in so called non-texture inpainting, i.e., the inpainting of structures, like edges and uniformly colored areas in the image, rather than texture. In the pioneering works of Caselles et al. [14] (with the term disocclusion instead of inpainting) and Bertalmio et al. [5] partial differential equations have been first proposed for digital nontexture inpainting. The inpainting algorithm in [5] extends the graylevels at the boundary of the damaged domain continuously in the direction of the isophote lines to the interior. The resulting scheme is a discrete model based on the nonlinear partial differential equation ut = ∇⊥ u · ∇∆u, to be solved inside the inpainting domain D using image information from a small strip around D. The operator ∇⊥ denotes the perpendicular gradient (−∂y , ∂x ). To avoid the crossing of level lines inside the inpainting domain, intermediate steps of anisotropic diffusion are implemented into the model. In subsequent works variational models, originally derived for the tasks of image denoising, deblurring and segmentation, have been adopted to inpainting. In contrast to former approaches (like [5]) the proposed variational algorithms are applied to the image on the whole domain Ω. This procedure has the advantage that inpainting can be carried out for several damaged domains in the image simultaneously and that possible noise outside the inpainting domain is removed at the same time. The general form of such a variational inpainting approach is 1 2 û(x) = argminu∈H1 J(u) = R(u) + kλ(f (x) − u(x))kH2 , 2 where f ∈ H2 (or f ∈ H1 depending on the approach) is the given damaged image and û ∈ H1 is the restored image. H1 , H2 are Banach spaces on Ω and R(u) is the so called regularizing term R : H1 → R. The function λ is the characteristic function of Ω \ D multiplied by a (large) constant, i.e., λ(x) = λ0 >> 1 in Ω \ D and 0 in D. Depending on the choice of the regularizing ∗ Institut für Numerische und Angewandte Mathematik, Fachbereich Mathematik und Informatik, Westfälische Wilhelms Universität (WWU) Münster, Einsteinstrasse 62, D 48149 Münster, Germany. Email: [email protected] † Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69 A-4040, Linz, Austria. Email: [email protected] ‡ DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. Email: [email protected] 1 2 M. Burger, L. He, and C.-B. Schönlieb term R(u) and the Banach spaces H1 , H2 various approaches have been developed. The most famous model is the total variation (TV) model, where R(u) = |Du| (Ω), is the total variation of u, H1 = BV (Ω) the space of functions of bounded variation (see Appendix A for the definition of functions of bounded variation) and H2 = L2 (Ω), cf. [19, 17, 34, 35]. A variational model with 2 R ∇u a regularizing term of higher order derivatives, i.e., R(u) = Ω (1 + ∇ · ( |∇u| ) )|∇u| dx, is the Euler elastica model [16, 28]. Other examples are the active contour model based on the Mumford and Shah segmentation [36], and the inpainting scheme based on the Mumford-Shah-Euler image model [22]. 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 (cf. [35], [17], [18]) have drawbacks as in the connection of edges over large distances or the smooth propagation of level lines (sets of image points with constant grayvalue) into the damaged domain. In an attempt to solve both the connectivity principle and the so called staircasing effect resulting from second order image diffusions, a number of third and fourth order diffusions have been suggested for image inpainting. A variational third order approach to image inpainting is the CDD (Curvature Driven Diffusion) method [18]. While realizing the Connectivity Principle in visual perception, (i.e., level lines are connected also across large inpainting domains) the level lines are still interpolated linearly (which may result in corners in the level lines along the boundary of the inpainting domain). This has driven Chan, Kang and Shen [16] to a reinvestigation of the earlier proposal of Masnou and Morel [28] on image interpolation based on Euler´s elastica energy. In their work the authors present the fourth order elastica inpainting PDE which combines CDD and the transport process of Bertalmio et. al [5]. The level lines are connected by minimizing the integral over their length and their squared curvature within the inpainting domain. This leads to a smooth connection of level lines also over large distances. This can also be interpreted via a second boundary condition, necessary for an equation of fourth order. 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 directions of the level lines are given. Further, also combinations of second and higher order methods exist, e.g. [27]. The combined technique is able to preserve edges due to the second order part and at the same time avoids the staircasing effect in smooth regions. A weighting function is used for this combination. The main challenge in inpainting with higher order flows is to find simple but effective models and to propose stable and fast discrete schemes to solve them numerically. To do so also the mathematical analysis of these approaches is an important point, telling us about solvability and convergence of the corresponding equations. This analysis can be very hard because often these equations do not admit a maximum or comparison principle and sometimes do not even have a variational formulation. 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 [7], [8] 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 = ∆(−∆u + F 0 (u)) + λ(f − u) in Ω, (1.1) 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 is the characteristic function of Ω \ D multiplied by a constant λ0 >> 1. The Cahn-Hilliard equation is a relatively simple fourth-order PDE used for this task rather than more complex models involving curvature terms. Still the Cahn-Hilliard inpainting approach has many of the desirable properties of curvature based inpainting models such as the smooth continuation of level Cahn-Hilliard and BV-H −1 inpainting 3 lines into the missing domain. In fact in [8] the authors prove that in the limit λ0 → ∞ a stationary solution of (1.1) solves ∆(∆u − 1 F 0 (u)) = 0 u = f ∇u = ∇f in D on ∂D on ∂D, (1.2) for f regular enough (f ∈ C 2 ). Moreover its numerical solution was shown to be of at least an order of magnitude faster than competing inpainting models, cf. [7]. In [8] the authors prove global existence and uniqueness of a weak solution of (1.1). Moreover, they derive properties of possible stationary solutions in the limit λ0 → ∞. Nevertheless the existence of a solution of the stationary equation 1 ∆(−∆u + F 0 (u)) + λ(f − u) = 0 in Ω, (1.3) remains unaddressed. The difficulty in dealing with the stationary equation is the lack of an energy functional for (1.1), i.e., the modified Cahn-Hilliard equation (1.1) 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) isnt symmetric with respect to the H −1 inner product. In fact the most evident variational approach would be to minimize the functional Z 1 1 2 (1.4) ( |∇u|2 + F (u)) dx + kλ(u − f )k−1 . 2 2 Ω This minimization problem exhibits the optimality condition 1 0 = −∆u + F 0 (u) + λ∆−1 (λ(u − f )) , which splits into 0 = −∆u + 1 F 0 (u) in D 0 = −∆u + 1 F 0 (u) + λ20 ∆−1 (u − f ) 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 0 = ∆(−∆u + 1 F 0 (u)) in D 0 = ∆(−∆u + 1 F 0 (u)) + λ0 (f − u) in Ω \ D. One challenge of this paper is to extend the analysis from [8] by partial answers to questions concerning the stationary equation (1.3) using alternative methods, namely by fixed point arguments. We shall prove Theorem 1.1. Equation (1.3) admits a weak solution in H 1 (Ω) provided λ0 ≥ C 13 , for a positive constant C depending on |Ω|, |D|, and F only. We will see in our numerical examples that the condition λ0 ≥ C 13 in Theorem 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. Note that the same condition also appears in [8] where it is needed to prove the global existence of solutions of (1.1). The second goal of this paper is to generalize the Cahn-Hilliard inpainting 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. 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 time-discrete Cahn-Hilliard inpainting approach Γ-converges to a functional regularized with the total variation for binary arguments u = χE , where E is some Borel measurable subset of Ω. This is stated in more detail in the following Theorem. 4 M. Burger, L. He, and C.-B. Schönlieb Theorem 1.2. 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 Notation section, and |Du| (Ω) denote the total variation of the distributional derivative Du (cf. Appendix B). Then the sequence of functionals 2 Z λ 1 1 λ0 λ 2 2 u− f − 1− J (u, v) = |∇u| + F (u) dx + ku − vk−1 + v 2 2τ 2 λ0 λ0 Ω −1 Γ−converges for → 0 in the topology of L1 (Ω) to 2 λ 1 λ0 λ 2 u− f − 1− J(u, v) = T V (u) + ku − vk−1 + v , 2τ 2 λ0 λ0 −1 where ( C0 |Du| (Ω) T V (u) = +∞ if u = χE for some Borel measurable subset E ⊂ Ω otherwise, R1p with χE denotes the characteristic function of E and C0 = 2 0 F (s) ds. Remark 1.3. Setting v = uk and a minimizer u of the functional J (u, v) to be u = uk+1 , the minimization of J can be seen as an iterative approach with stepsize τ to solve (1.1). 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 T V − H −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), p ∈ ∂T V (u), (1.5) with ( |Du| (Ω) if |u(x)| ≤ 1 a.e. in Ω T V (u) = +∞ otherwise. (1.6) The inpainting domain D and the characteristic function λ(x) are defined as before for the CahnHilliard inpainting approach. The space BV (Ω) is the space of functions of bounded variation on Ω (cf. Appendix B). Further ∂T V (u) denotes the subdifferential of the functional T V (u) (cf. Appendix B for the definition). The L∞ bound in the definition of the TV functional (1.6) 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 1.2. 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 [38], [33], and [26]. Further, in Bertalmio at al. [6] an application of the model from [38] to image inpainting has been proposed. In contrast to the inpainting approach (1.5) the authors in [6] only use a different form of the TVH −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 [5]. Using the same methodology as in the proof of Theorem 1.1 we obtain the following existence theorem, Theorem 1.4. Let f ∈ L2 (Ω). The stationary equation ∆p + λ(f − u) = 0, p ∈ ∂T V (u) (1.7) 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 (1.6), i.e., T V (u) = |Du| (Ω) + χ1 (u), where ( 0 if |u| ≤ 1 a.e. in Ω χ1 (u) = +∞ otherwise. Cahn-Hilliard and BV-H −1 inpainting Namely, we shall prove the following theorem. Theorem 1.5. Let p̃ ∈ ∂χ1 (u). An element following set of equations ∇u p = −∇ · |∇u| ∇u p = −∇ · |∇u| + p̃, p̃ ≤ 0 ∇u p = −∇ · |∇u| + p̃, p̃ ≥ 0 5 p ∈ ∂T V (u) with |u(x)| ≤ 1 in Ω, fulfills the a.e. on supp ({|u| < 1}) a.e. on supp ({|u| = −1}) a.e. on supp ({|u| = 1}) . For (1.5) 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 (1.5). In our considerations we use the symmetric Bregman distance defined as DTsymm (û, ftrue ) = hû − ftrue , p̂ − ξi , V p̂ ∈ ∂T V (û), ξ ∈ ∂T V (ftrue ). We prove the following result Theorem 1.6. Let ftrue fulfill the so called source condition, i.e., 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 (1.5) 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 2 2 (r−2)/r kû − ftrue k−1 ≤ kξk1 + Cλ0 |D| errinpaint , 2 λ0 with constant C > 0 and errinpaint := K1 + K2 |D| C (M (us ), β) + 2 R(ud ) , where K1 , K2 , and C are appropriate positive constants, 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 . Finally we present numerical results for the proposed binary- and grayvalue inpainting approaches and briefly explain the numerical implementation using convexity splitting methods. Organization of the paper. In Section 2 a fixed point approach is proposed to prove Theorem 1.1, i.e., the existence of a stationary solution for the modified Cahn-Hilliard equation (1.1) with Dirichlet boundary conditions. In Section 3 we discuss the new T V − H −1 inpainting approach. We prove Theorem 1.2 in Subsection 3.1. This Γ−limit is generalized to an inpainting approach for grayvalue images, called T V − H −1 inpainting (cf. (1.5)). Similarly to the existence proof in Section 2 we prove in Subsection 3.2 the existence of a stationary solution of this new inpainting approach for grayvalue images, i.e., Theorem 1.4. In Section 3.3 we additionally give a characterization of elements in the subdifferential of the corresponding regularizing functional (1.6). In addition we present error estimates for both the error in inpainting the image by means of (1.5) and for the stability of solutions of (1.5) in terms of the Bregman distance, i.e., the proof of Theorem 1.6, in Subsection 3.4. Section 4 is dedicated to the numerical solution of Cahn-Hilliard- and T V − H −1 −inpainting and the presentation of numerical examples. Finally in Appendix A the space H∂−1 is defined and its elements are characterized in order to analyze (1.1) for Neumann boundary conditions. In Appendix B basic facts about functions of bounded variation are presented. Notation. Before we begin with the discussion of our results let us introduce some notations. By k.k we in L2 (Ω) with corresponding inner product h., .i always denote the norm −1 −1 and by k.k−1 := ∇∆ . the norm in H (Ω) = (H01 (Ω))∗ with corresponding inner product h., .i−1 := ∇∆−1 ., ∇∆−1 . where ∆−1 denotes the inverse of the negative Laplacian −∆ on H01 , i.e., ∆−1 : H −1 (Ω) → H01 (Ω). 6 M. Burger, L. He, and C.-B. Schönlieb 2. Cahn-Hilliard inpainting - proof of Theorem 1.1. In this chapter we prove the existence of a weak solution of the stationary equation (1.3). Let Ω ⊂ R2 be a bounded Lipschitz domain and 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. [8]) we use Dirichlet boundary conditions for our analysis, i.e., we consider ut = ∆ −∆u + 1 F 0 (u) + λ(f − u) in Ω (2.1) u = f, −∆u + 1 F 0 (u) = 0 on ∂Ω. 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 A we nevertheless propose a setting to extend the presented analysis for (1.1) to the originally proposed model with Neumann boundary data. In our new set ting we define a weak solution of equation (1.3) as a function u ∈ H = u ∈ H 1 (Ω), u|∂Ω = f |∂Ω that fulfills 1 0 F (u), φ − hλ(f − u), φi−1 = 0, ∀φ ∈ H01 (Ω). (2.2) h∇u, ∇φi + Remark 2.1. 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 (2.2) defines a weak formulation for (1.3) with Dirichlet boundary conditions we integrate by parts in (2.2) and get R (−∆u + 1 F 0 (u) − ∆−1 (λ(f − u))) φ dx ΩR − ∂Ω ∆−1 (λ(f − u)) ∇∆−1 φ · ν dH1 = 0, ∀φ ∈ H01 (Ω), where H1 denotes the one dimensional Hausdorff measure. This yields ∆u − 1 F 0 (u) + ∆−1 (λ(f − u)) = 0 in Ω ∆−1 (λ(f − u)) = 0 on ∂Ω. (2.3) Assuming sufficient regularity on u we can use the definition of ∆−1 to see that u solves in Ω −∆∆u + 1 ∆F 0 (u) + λ(f − u) = 0 ∆−1 (λ(f − u)) = −∆u + 1 F 0 (u) = 0 on ∂Ω, Since additionally u|∂Ω = f |∂Ω , the function u solves (1.3) with Dirichlet boundary conditions. For the proof of existence of a solution to (2.2) we follow the subsequent 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 (u − v) = ∆u − 1 F 0 (u) + ∆−1 [λ(f − u) + (λ0 − λ)(v − u)] in Ω, τ∆ (2.4) u = f, ∆−1 τ1 (u − v) + λ(f − u) + (λ0 − λ)(v − u) = 0 on ∂Ω, where τ > 0 is a parameter. The boundary conditions of A are given by the second equation in (2.4). 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 (2.4) as before by a function u ∈ H = u ∈ H 1 (Ω), u|∂Ω = f |∂Ω that fulfills 1 1 0 τ (u − v), φ −1 + h∇u, ∇φi + F (u), φ (2.5) − hλ(f − u) + (λ0 − λ)(v − u), φi−1 = 0 ∀φ ∈ H01 (Ω). Cahn-Hilliard and BV-H −1 inpainting 7 A fixed point of the operator A, provided it exists, then solves the stationary equation with Dirichlet boundary conditions as in (2.3). Note that in (2.4) the characteristic 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 (2.4), i.e., (2.5), we can therefore state a variational formulation. This is, for a given v ∈ L2 (Ω) equation (2.4) is the Euler-Lagrange equation of the minimization problem u∗ = argminu∈H 1 (Ω),u|∂Ω =f |∂Ω J (u, v) (2.6) with J (u, v) = Z Ω 2 1 1 λ0 2 2 u − λ f − 1 − λ v . (2.7) |∇u| + F (u) dx + ku − vk−1 + 2 2τ 2 λ0 λ0 −1 We are going to use the variational formulation (2.7) to prove that (2.4) admits a weak solution in H 1 (Ω). This solution is unique under additional conditions. Proposition 2.2. Equation (2.4) admits a weak solution in H 1 (Ω) in the sense of (2.5). For τ ≤ C3 , where C is a positive constant depending on |Ω|, |D|, and F only, the weak solution of (2.4) is unique. Further we prove that the operator A admits a fixed point under certain conditions. Proposition 2.3. Set A : L2 (Ω) → L2 (Ω), A(v) = u, where u ∈ H 1 (Ω) is the unique weak solution of (2.4). Then A admits a fixed point û ∈ H 1 (Ω) if τ ≤ C3 and λ0 ≥ C 13 for a positive constant C depending on |Ω|, |D|, and F only. Hence the existence of a stationary solution of (1.1) follows under the condition λ0 ≥ C/3 . We begin with considering the fixed point equation (2.4), i.e., the minimization problem (2.6). In the following we prove the existence of a unique weak solution of (2.4) by showing the existence of a unique minimizer for (2.7). (Proof of Proposition 2.2) We want to show that J (u, v) has a minimizer in H = Proof. 1 u ∈ H (Ω), u|∂Ω = f |∂Ω . For this we consider a minimizing sequence un ∈ H of J (u, v). To see that un is uniformly bounded in H 1 (Ω) we show that J (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 C1 C2 1 1 2 2 2 2 J (u, v) ≥ k∇uk2 + kuk2 − + kuk−1 − kvk−1 2 2τ 2 2 ! λ λ0 1 λ 2 + kuk−1 − f + 1 − v 2 2 λ0 λ0 −1 C1 2 2 kuk2 + ≥ k∇uk2 + 2 λ0 1 + 4 4τ 2 kuk−1 − C3 (v, f, λ0 , , Ω, D). Therefore a minimizing sequence un is bounded in H 1 (Ω) and it follows that un * u∗ in H (Ω). To finish the proof of existence for (2.4) we have to show that J (u, v) is weakly lower semicontinuous in H 1 (Ω). For this we divide the sequence J (un , v) of (2.7) in two parts. We denote the first term Z 1 n n 2 n a = |∇u | + F (u ) dx 2 |Ω {z } 1 CH(un ) and the second term 1 λ0 2 b = kun − vk−1 + |2τ {z } |2 n D(un ,v) 2 n u − λ f − 1 − λ v . λ0 λ0 −1 {z } F IT (un ,v) 8 M. Burger, L. He, and C.-B. Schönlieb 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 . (2.8) We begin with the consideration of the last term in (2.7). We denote f˜ := We want to show 2 n ˜2 −→ u∗ − f˜ u − f −1 −1 ⇐⇒ h∆−1 (un − f˜), un − f˜i −→ h∆−1 (u∗ − f˜), u∗ − f˜i. λ λ0 f + (1 − λ λ0 )v. For this we consider the absolute difference of the two terms, |h∆−1 (un − f˜), un − f˜i − h∆−1 (u∗ − f˜), u∗ − f˜i| = |h∆−1 (un − u∗ ), un − f˜i − h∆−1 (u∗ − f˜), un − u∗ i| ≤ |hun − u∗ , ∆−1 (un − f˜)i| + |h∆−1 (u∗ − f˜), u∗ − un i| ≤ kun − u∗ k · ∆−1 (un − f˜) + kun − u∗ k · ∆−1 (u∗ − f˜) | {z } | {z } →0 →0 Since the operator ∆−1 : H −1 (Ω) → H01 (Ω) is a linear and continuous operator it follows that −1 −1 ∆ F ≤ ∆ · kF k for all F ∈ H −1 (Ω). Thus |h∆−1 (un − f˜), un − f˜i − h∆−1 (u∗ − f˜), u∗ − f˜i| ≤ kun − u∗ k ∆−1 un − f˜ + kun − u∗ k ∆−1 u∗ − f˜ | {z } | {z } | {z } | {z } | {z } | {z } →0 const bounded →0 const const −→ 0 as n → ∞, 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 in sum 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 J (u∗ , v) ≤ lim inf J (un , v). Therefore J has a minimizer in H 1 , i.e., ∃u∗ with u∗ = argminu∈H 1 (Ω) J (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. [23] § 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 (2.4) 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 J , i.e., d J (u + δφ, v) , dδ δ=0 Cahn-Hilliard and BV-H −1 inpainting 9 which has to be zero for a minimizer u. Thus we have 1 λ λ 1 h∇u, ∇φi + hF 0 (u), φi + (u − v) + λ0 u − f − 1 − v ,φ = 0. τ λ0 λ0 −1 Integrating by parts in both terms we get 1 1 λ λ −∆u + F 0 (u) − ∆−1 (u − v) + λ0 u − f − 1 − v ,φ τ λ0 λ0 Z Z 1 λ λ + ∇u · νφ ds + ∆−1 (u − v) + λ0 u − f − 1 − v ∇∆−1 φ · ν ds = 0. τ λ λ 0 0 ∂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 (2.5) of (2.4). For the uniqueness of the minimizer, we need to prove that J is strictly convex. To do so, we prove that for any u1 , u2 ∈ H 1 (Ω), u1 + u2 , v > 0, (2.9) J (u1 , v) + J (u2 , v) − 2J 2 2 based on an assumption that F (.) satisfies F (u1 ) + F (u2 ) − 2F ( u1 +u ) > −C(u1 − u2 )2 , for a 2 1 1 2 2 constant C > 0. For example, when F (u) = 8 (u − 1) , C = 8 . Denote u = u1 − u2 , we have 1 λ0 C u1 + u2 2 2 2 J (u1 , v) + J (u2 , v) − 2J , v > kukH 1 + + kuk−1 − kuk2 2 4 4τ 4 By using the inequality 2 kuk2 ≤ kukH 1 kuk−1 , (2.10) and the Cauchy-Schwarz inequality, for (2.9) to be fulfilled, we need s 1 λ0 C 2 + ≥ . 4 4τ 4 i.e., 3 1 λ0 + τ ≥ C 2. Therefore J (u, v) is strictly convex in u and our minimization problem has a unique minimizer if τ is chosen smaller than C3 for a constant C depending on |Ω|, |D|, and F only. Because of the convexity of J in ∇u and u, every weak solution of the Euler-Lagrange equation (2.4) is in fact a minimizer of J . This proves the uniqueness of a weak solution of (2.4) provided τ << C3 . Next we want to prove Proposition 2.3, i.e., the existence of a fixed point of (2.4) and with this the existence of a stationary solution of (1.1). To do so we are going to apply Schauder’s fixed point theorem. Proof. (Proof of Proposition 2.3) We consider a solution A(v) = u of (2.4) 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 2 2 2 kA(v)k = kuk ≤ β kvk + α, (2.11) for a constant β < 1. Having this we have shown that A is a map from the closed ball K = B(0, M ) = u ∈ L2 (Ω) : kuk ≤ M into itself for an appropriate constant M > 0. We conclude 10 M. Burger, L. He, and C.-B. Schönlieb 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. Let us, for the time being, additionally assume that ∇u and ∆u are bounded in L2 (Ω). Hence we can multiply (2.4) with −∆u and integrate over Ω to obtain Z Z 1 1 − ∆u∆−1 (u − v) − λ(f − u) − (λ0 − λ)(v − u) dx = − h∆u, ∆ui + F 0 (u)∆u dx τ Ω Ω After integration by parts we find with the short-hand notation w := 1 (u − v) − λ(f − u) − (λ0 − λ)(v − u) τ that Z Z uw dx − Z 1 1 2 2 ∇u · ν(∆−1 w + F 0 (u)) − u∇(∆−1 w) · ν dH1 = − k∆uk − F 00 (u) |∇u| dx. Ω ∂Ω Ω Now we insert the boundary conditions ∆−1 w = 0, u = f =: f1 and F 0 (u) = F 0 (f ) = f2 on ∂Ω with constants f1 and f2 on the left-hand side, i.e. Z Z Z f2 1 2 2 uw dx − ∇u · ν − f1 ∇(∆−1 w) · ν dH1 = − k∆uk − F 00 (u) |∇u| dx. Ω ∂Ω Ω An application of Gauss’ Theorem to the boundary integral implies Z Z Z f2 f2 ∇u · ν − f1 ∇(∆−1 w) · ν dH1 = ∆u dx + f1 w dx, Ω ∂Ω Ω and we get Z 2 uw dx = − k∆uk − Ω 1 Z 2 F 00 (u) |∇u| dx + Ω f2 Z Z ∆u dx + f1 Ω w dx. Ω By further applying Young’s inequality to the last two terms we get Z Z f2 δ 1 f1 δ 2 2 2 uw dx ≤ − k∆uk − F 00 (u) |∇u| dx + kwk + C(f1 , f2 , |Ω|, , δ). 2 Ω 2 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 1 f2 δ 1 f1 δ 2 2 2 u · (u − v) ≤ − k∆uk − F 00 (u) |∇u| dx + kwk τ 2 2 Ω Ω ! Z Z +λ0 u(f − u) dx + u(v − u) dx + C(f1 , f2 , |Ω|, , δ). Ω\D D 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 that (1 − λ/λ0 ) ≤ 1 in the resulting L2 norm of v we get Z 1 u · (u − v) ≤ τ Ω 2 Z f2 δ 1 1 2 2 2 00 − k∆uk − F (u) |∇u| dx + f1 δ + λ0 kuk 2 Ω τ ! 2 Z Z 1 2 +2f1 δ + λ0 kvk + λ0 u(f − u) dx + u(v − u) dx τ Ω\D D +C(f, f1 , f2 , |Ω|, , δ, λ0 ). Cahn-Hilliard and BV-H −1 inpainting 11 With F 00 (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 2 Z 1 f2 δ C1 C2 1 2 2 2 2 u · (u − v) ≤ − k∆uk − ku |∇u|k + k∇uk + f1 δ + λ0 kuk τ 2 τ Ω " 2 Z Z δ2 1 δ1 2 2 +2f1 δ + λ0 kvk + λ0 − 1 − u dx + −1 u2 dx τ 2 2 Ω\D D Z 1 2 + v dx + C(f, f1 , f2 , |Ω|, |D|, , λ0 , δ, δ2 ). 2δ1 D Setting δ2 = 1 and δ1 = 2 we see that 2 Z 1 f2 δ C1 C2 1 2 2 2 2 u · (u − v) ≤ − k∆uk − ku |∇u|k + k∇uk + f1 δ + λ0 kuk τ 2 τ Ω " # 2 Z Z 1 1 1 2 2 2 + λ0 kvk + λ0 − u dx + v dx +2f1 δ τ 2 Ω\D 4 D +C(f, f1 , f2 , |Ω|, |D|, , δ, λ0 ). We follow the argumentation of the proof of existence for (1.1) in [8] by observing the following property: A standard interpolation inequality for ∇u reads 2 2 k∇uk ≤ δ3 k∆uk + C3 2 kuk . δ3 (2.12) 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 2.4. (Poincare’s inequality in L1 ). Assume that Ω is precompact open subset of n-dimensional Euclidean space Rn having Lipschitz boundary (i.e., Ω is 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| u(y) dy is the average value of u over Ω, with |Ω| denoting the Lebesgue measure Ω of the domain Ω. Then, assuming that u 6= 0 in Ω \ D, we choose the constant C4 (which depends on the size of D compared to Ω) large enough such that Z Z Z Z 2 2 u − u2Ω dx ≤ C4 ∇u dx. u2 dx − C4 u2 dx ≤ Ω Ω\D Ω Ω or in other words 2 kuk ≤ C4 ∇u2 L1 (Ω) + C4 Z u2 dx. (2.13) Ω\D By Hölders inequality we also have that 2 C5 α 2 ∇u 1 . ≤ ku |∇u|k + L (Ω) 2 2α Putting the last three inequalities (2.12)-(2.14) together we obtain Z C3 C4 α C3 C4 C3 C4 C5 2 2 2 k∇uk ≤ δ3 k∆uk + ku |∇u|k + u2 dx + . 2δ3 δ3 2αδ3 Ω\D (2.14) 12 M. Burger, L. He, and C.-B. Schönlieb We now use the last inequality to bound the gradient term in our estimates from above to get R 2 2 C 3 C4 α 2 δ3 u · τ1 (u − v) ≤ ( f2 δ+2C − ) k∆uk + ( C2 2δ − C1 ) ku|∇u|k 2 Ω 3 2 2 2 2 kvk +(f1 δ τ1 + λ0 + C2δC33C4 − C42λ0 ) kuk + λ40 + 2f2 δ τ1 + λ0 +C(f, f1 , f2 , |Ω|, |D|, , δ, λ0 ). (2.15) 22 −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 ! 2 2 ! λ0 C4 λ0 1 C2 C3 C4 1 1 1 2 2 kuk ≤ + − f1 δ − λ0 − + + 2f2 δ + λ0 kvk 2τ 2 τ δ3 4 2τ τ +C(f, f1 , f2 , |Ω|, |D|, , δ, λ0 ). Choosing δ small enough, C4 large enough, and λ0 ≥ CC4 13 the solution u and v fulfill 2 2 kuk ≤ β kvk + C, (2.16) 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 (2.15) with appropriate constants δ3 , δ, and α as specified in the paragraph below (2.15). 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 C2 C3 C4 2 2 C4 λ0 1 1 2 δ3 + − f δ − λ kuk + ( − f2 δ+2C ) k∆uk − δ3 1 0 2 τ 2 2τ 2 2 λ0 1 1 ≤ kvk + C(f, f1 , f2 , |Ω|, |D|, , δ, λ0 ), 4 + 2τ + 2f2 δ τ + λ0 2 δ3 with the coefficient − f2 δ+2C > 0 due to our choice of δ3 . Therefore not only the L2 − norm of 2 u is uniformly bounded but also the L2 − norm of ∆u. By the standard interpolation inequality (2.12) 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 (2.4) weakly converges to a weak solution u of 1 1 (−∆−1 )(u − v) = ∆u − F 0 (u) − ∆−1 [λ(f − u) + (λ0 − λ)(v − u)] , τ where u is the weak limit of A(vk ) as k → ∞. Because the solution of (2.4) is unique provided τ ≤ C3 (cf. Proposition 2.2), 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 0 F (û), φ −hλ(f − û), φi−1 + h∇û, ∇φi+ ∆−1 (λ(f − û)) ∇∆−1 φ·ν dH1 = 0, ∀φ ∈ H01 (Ω). ∂Ω Because the solution of (2.4) is an element of H = u ∈ H 1 (Ω), u|∂Ω = f |∂Ω also the fixed point û ∈ H. Following the arguments from the beginning of this section we conclude with the existence of a stationary solution for (1.1). Cahn-Hilliard and BV-H −1 inpainting 13 By modifying the setting and the above proof in an appropriate way one can prove the existence of a stationary solution for (1.1) also under Neumann boundary conditions, i.e., ∇u · ν = ∇∆u · ν = 0, on ∂Ω. A corresponding reformulation of the problem is given in Appendix A. 3. Total Variation - H −1 inpainting. In this section we discuss our newly proposed inpainting scheme (1.5), i.e., the inpainted image u of f ∈ L2 (Ω) evolves via ut = ∆p + λ(f − u), p ∈ ∂T V (u), with ( |Du| (Ω) if |u(x)| ≤ 1 a.e. in Ω T V (u) = +∞ otherwise. Before starting this section we suggest readers who are unfamiliar with the space BV (Ω) to first read Appendix B and maybe recall the definition of the subdifferential of a function in Definition B.9. 3.1. Γ-Convergence of the Cahn-Hilliard energy - proof of Theorem 1.2. In the following we want to motivate our new inpainting approach (1.5) by considering the Γ−limit for → 0 of an appropriate time-discrete Cahn-Hilliard inpainting approach, i.e., the Γ− limit of the functionals from our fixed point approach in (2.7). More precisely we want to prove Theorem 1.2 stated in the Introduction of this paper. Before starting our discussion lets recall the definition of Γ-convergence and its impact within the study of optimization problems. For more details on Γ-convergence we refer to [29]. Definition 3.1. 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 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 )). We write F (x) = Γ − limh Fh (x), x ∈ X, is the Γ-limit of (Fh ) in X. The formulation of the Γ-limit for → 0 is analogous by defining a sequence h with h → 0 as h → ∞. The important property of Γ-convergent sequences of functions Fh is that its minima converge to minima of the Γ-limit F . In fact we have the following theorem Theorem 3.2. Let (Fh ) be like in Definition 3.1 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 1.2. 14 M. Burger, L. He, and C.-B. Schönlieb Proof. Modica and Mortola have shown in [31] and [32] that the sequence of Cahn-Hilliard functionals Z 1 2 CH(u) = |∇u| + F (u) dx 2 Ω Γ-converges in the topology L1 (Ω) to ( C0 |Du| (Ω) if u = χE for some Borel measurable subset E ⊂ Ω T V (u) = +∞ otherwise R1p as → 0, where C0 = 2 0 F (s) ds. (The space BV (Ω) and the total variation |Du| (Ω) are defined in Appendix B.) Now, for a given function v ∈ L2 (Ω) the functional J from our fixed point approach (2.4), i.e., 2 Z λ 1 λ0 1 λ 2 2 J (u, v) = u − f − (1 − )v , |∇u| + F (u) dx + ku − vk + 2 2τ {z −1} 2 λ0 λ0 −1 Ω | {z } | | {z } :=D(u,v) :=CH(u) :=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 3.3. [Dal Maso, [29], Prop. 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 J that are independent from , 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 Cahn-Hilliard energy, we have proven that the modified Cahn-Hilliard functional J 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) = χE 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 ) to a sharp interface limit in which the image function now jumps from 0 to 1. This property 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 T V − H −1 inpainting equation (1.5). 3.2. Existence of a stationary solution - proof of Theorem 1.4. Our strategy for proving the existence of a stationary solution for T V − H −1 inpainting (1.5) is similar to our existence proof for a stationary solution of the modified Cahn-Hilliard equation (1.1) in Section 2. Similarly as in our analysis for (1.1) in Section 2 we consider equation (1.5) 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 J(u, v), u∈BV (Ω) with functionals J(u, v) := T V (u) + λ0 λ λ 1 ||u − v||2−1 + ||u − f − (1 − )v||2−1 , 2τ 2 λ λ 0 0 | {z } | {z } D(u,v) F IT (u,v) (3.1) Cahn-Hilliard and BV-H −1 inpainting 15 with T V (u) defined as in (1.6), i.e., ( |Du| (Ω) if |u(x)| ≤ 1 a.e. in Ω T V (u) = +∞ otherwise. Note that Lr (Ω) can be continuously embedded in H −1 (Ω). Hence the functionals in (3.1) 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 3.4. 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., J(un , v) → inf J(u, v). u∈BV (Ω) 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, and, because of the uniform boundedness of |u(x)| for every point x ∈ Ω, kun kLp (Ω) ≤ M̃ , for an M ≥ 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 2.2) 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 ! u1 + u2 2 = + − 2 2 −1 u1 + u2 +T V (u1 ) + T V (u2 ) − 2T V 2 1 λ0 2 ≥ + ku1 − u2 k−1 > 0. 4τ 4 1 λ0 + 2τ 2 2 ku1 k−1 2 ku2 k−1 This finishes the proof. Next we shall prove the existence of stationary solution for (1.5). For this sake we consider the corresponding Euler-Lagrange equation to (3.1), i.e., u−v ∆−1 + p − ∆−1 (λ(f − u) + (λ0 − λ)(v − u)) = 0, τ 16 M. Burger, L. He, and C.-B. Schönlieb with the weak formulation 1 τ (u − v), φ −1 + hp, φi − hλ(f − u) + (λ0 − λ)(v − u), φi−1 ∀φ ∈ H01 (Ω). A fixed point of the above equation, i.e., a solution u = v, is then a stationary solution for (1.5). Thus, to prove the existence of a stationary solution of (1.5), i.e., to prove Theorem 1.4, we as before 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 (3.1). The choice of the fixed point operator A over Lr (Ω) was made in order to obtain the necessary compactness properties for the application of Schauder’s theorem. Since here the treatment of the boundary conditions is similar as in Section 2 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. 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 (3.1). Existence and uniqueness follow from Theorem 3.4. 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 2τ ||u − v||2−1 + λ0 2 ||u − λ λ0 f − (1 − λ 2 λ0 )v||−1 + T V (u) ≤ ≤ 1 2 2τ ||v||−1 |Ω| 2τ + λ0 λ 2 || λ0 f + (1 − λ 2 λ0 )v||−1 + λ0 (|Ω| + |D|). (3.2) Here the last inequality was obtained since Lr (Ω) ,→ H −1 (Ω) and hence ||v||−1 ≤ C and ||λv||−1 ≤ 0 C for a C > 0. (In fact, since H 1 (Ω) ,→ Lr (Ω) for all 1 ≤ r0 < ∞ from duality it follows 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 B.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 uk − vk ∆pk = − (λ(f − uk ) + (λ0 − λ)(vk − uk )) , τ 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 u−v ∆p = − (λ(f − u) + (λ0 − λ)(v − u)) . (3.3) τ If we additionally apply Poincare’s inequality to ∆pk we conclude k∇pk − (∇pk )Ω kLr (Ω) ≤ C k∇ · (∇pk − (∇pk )Ω )kLr (Ω) , R 1 where (∇pk )Ω = |Ω| ∇pk dx. In addition, since pk ∈ ∂T V (uk ), it follows that (pk )Ω = 0 and Ω kpk kBV ∗ (Ω) ≤ 1. Thus (∇pk )Ω < ∞ and pk is uniformly bounded in W 1,r (Ω). Thus there exists a 0 subsequence pkl such that pkl * p in W 1,r (Ω). In addition Lr (Ω) ,→ BV ∗ (Ω) for 2 < r0 < ∞ (this 2r follows again from Theorem B.7 by a duality argument) and W 1,r (Ω) ,→,→ Lq (Ω) for 1 ≤ q < 2−r 2r (Rellich-Kondrachov Compactness Theorem, cf. [2], Theorem 8.7). By choosing 2 < q < 2−r we have in sum W 1,r (Ω) ,→,→ BV ∗ (Ω). Thus pkl → p strongly in BV ∗ (Ω). Hence the element p in (3.3) is an element in ∂T V (u). Because the minimizer of (3.1) is unique, u = A(v), and therefore A is continuous in Lr (Ω). From Schauder’s fixed point theorem the existence of a stationary solution follows. Cahn-Hilliard and BV-H −1 inpainting 17 3.3. Characterization of Solutions - proof of Theorem 1.5. Finally we want to compute elements p̂ ∈ ∂T V (û). Like in [13] the model for the regularizing functional is the sum of a standard regularizer plus the indicator function of the L∞ constraint. Especially 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 (3.4) χ1 (u) = +∞ otherwise. We want to compute the subgradients of T V by pretending ∂T V (u) = ∂ |Du| (Ω) + ∂χ1 (u). This means we can separately compute the subgradients of χ1 . To guarantee that the splitting above is R q 2 allowed we have to consider a regularized functional of the total variation, like Ω |∇u| + δ dx. This is sufficient because both |D.| (Ω) and χ1 are convex and |D.| (Ω) is continuous (compare [21] Proposition 5.6., pp. 26). The subgradient ∂ |Du| (Ω) is already well described, as, for instance, in [4] or [37]. We will just shortly recall its characterization. Thereby we do not insist on the details of the rigorous derivation of these conditions, and we limit ourself to mention the main facts. It is well known [37, Proposition 4.1] that p ∈ ∂|Du|(Ω) implies ( ∇u p = −∇ · ( |∇u| ) in Ω ∇u · ν = 0 on ∂Ω. |∇u| The previous conditions do not fully characterize p ∈ ∂|Du|(Ω), additional conditions would be required [4, 37], 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 = −∇ · ( √ ∇u2 ) in Ω |∇u| +δ √ ∇u |∇u|2 +δ ·ν =0 on ∂Ω. The subgradient of χ1 is computed like in the following Lemma. Lemma 3.5. Let χ1 : Lr (Ω) → R ∪ {∞} be defined by (3.4), and let 1 ≤ r ≤ ∞. Then ∗ r p ∈ Lr (Ω), for r∗ = r−1 , is a subgradient p ∈ ∂χ1 (u) for u ∈ Lr (Ω) with 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 + w for w being any bounded function supported in {|u| < 1 − α} for arbitrary 0 < α < 1. If is sufficiently small we have |v| ≤ 1. Hence Z 0 ≥ hv − u, pi = wp dx. {|u|<1−α} Since we can choose both positive and negative, we obtain Z wp dx = 0. {|u|<1−α} Because 0 < α < 1 and w are arbitrary we conclude p = 0 on the support of {|u| < 1}. If we choose v = u + w with w is an arbitrary bounded function with ( 0 ≤ w ≤ 1 on supp({−1 ≤ u ≤ 0}) w=0 on supp({0 < u ≤ 1}). 18 M. Burger, L. He, and C.-B. Schönlieb Then v is still between −1 and 1 and Z Z 0 ≥ hv − u, pi = wp dx + {u=−1} wp dx {u=1} Z = wp dx. {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 0 ≥ hv − u, pi = wp dx + {u=−1} wp dx {u=1} Z = wp dx. {u=1} Analogue 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, pi ≤ 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 hv − u, pi = p(v − u) dx + p(v − u) dx {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, pi ≤ 0 and we are done. 3.4. Error estimation and stability analysis with the Bregman distance - proof of Theorem 1.6. In the following analysis we want to present estimates for both the error we actually make in inpainting an image with our T V − H −1 approach (1.5) (see (3.12)) and for the stability of solutions for this problem (see (3.13)) in terms of the Bregman distance. This section is motivated by the error analysis for variational models in image restoration with Bregman iterations in [13], and the error estimates for inpainting models developed in [15]. In [13] the authors consider among other things the general optimality condition p + λ0 A∗ (Au − fdam ) = 0, (3.5) 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 (3.5). Considering this equation one realizes that the smoothing consists of two steps. The first Cahn-Hilliard and BV-H −1 inpainting 19 smoothing is created by the operator A which depends on the image restoration task at hand, and is actually a dual one that 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. Let ftrue be the original image then the source condition for ftrue reads: There exists ξ ∈ ∂R(ftrue ) such that ξ = A∗ q for a source element q ∈ D(A∗ ), (3.6) where D(A∗ ) is the domain of the operator A∗ . It can be shown (cf. [12]) that this is equivalent to require from ftrue to be a minimizer of R(u) + λ0 2 kAu − fdam k , 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 the subsequent error analysis. To be more precise, the Bregman distance is defined as p DR (v, u) = R(v) − R(u) − hv − u, pi , 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 , ξi = R(u) − R(ftrue ) − hq, Au − Aftrue i , and thus the Bregman distance can be both related to 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 symm p1 p2 DR (u1 , u2 ) = DR (u2 , u1 ) + DR (u1 , u2 ) = hu1 − u2 , p1 − p2 i , 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 [15]. 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., [13] and [15], in order to proof Theorem 1.6. Proof. 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 (1.5), i.e., −∆p + λ(u − fdam ) = 0, p ∈ ∂T V (u), (3.7) where we define T V (u) as a functional over L2 (Ω) as ( |Du| (Ω) if u ∈ BV (Ω), kukL∞ ≤ 1 T V (u) = +∞ otherwise. In the subsequent we want to characterize the error we make by solving (3.7) 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 (3.5) is the embedding operator from H01 (Ω) into H −1 (Ω) and stands in front of the whole term A(u − fdam ), cf. (3.7). 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 fdam = 0 in Ω \ D in D. (3.8) 20 M. Burger, L. He, and C.-B. Schönlieb Further we assume that ftrue satisfies the source condition (3.6), i.e., There exists ξ ∈ ∂T V (ftrue ) such that ξ = A∗ q = ∆−1 q for a source element q ∈ H −1 (Ω). (3.9) For the following analysis we first rewrite (3.7). For û, a solution of (3.7), we get p̂ + λ0 ∆−1 (û − ftrue ) = ∆−1 [(λ0 − λ)(û − ftrue )] , p̂ ∈ ∂T V (û). Here we replaced fdam by ftrue using assumption (3.8). By adding a ξ ∈ ∂T V (ftrue ) from (3.9) to the above equation we obtain λ −1 −1 p̂ − ξ + λ0 ∆ (û − ftrue ) = −ξ + λ0 ∆ 1− (û − ftrue ) λ0 Taking the duality product with û − ftrue (which is just the inner product in L2 (Ω) in our case) we get λ 2 −1 (û − f ), û − f , DTsymm (û, f ) + λ kû − f k = ∇ξ, ∇∆ (û − f ) + λ 1 − true true true 0 true −1 true 0 V λ0 −1 where DTsymm (û, ftrue ) = hû − ftrue , p̂ − ξi , V p̂ ∈ ∂T V (û), ξ ∈ ∂T V (ftrue ). An application of Young’s inequality yields DTsymm (û, ftrue ) V 2 λ0 1 λ 2 2 + kû − ftrue k−1 ≤ kξk1 + λ0 1 − (û − ftrue ) 2 λ0 λ0 −1 (3.10) For the last term we obtain E D E D = supφ,kφk−1 =1 − ∆−1 φ, 1 − λλ0 v 1 − λλ0 v = supφ,kφk−1 =1 φ, 1 − λλ0 v −1 D −1 E = supφ,kφk−1 =1 − 1 − λλ0 ∆−1 φ, v ≤Hölder kvk2 · supφ,kφk−1 =1 1 − λλ0 ∆−1 φ . 2 With ∆−1 : H −1 → H 1 ,→ Lr , 2 < r < ∞ we get R Ω 1− λ λ0 =choose q= r2 2 R 2 2q q1 R 1 ∆−1 φ dx = D ∆−1 φ dx ≤Hölder |D| q0 · Ω ∆−1 φ 2 r−2 r−2 r−2 2 |D| r · ∆−1 φ ≤H 1 ,→Lr C|D| r kφk = C|D| r , −1 p i.e., 2 2 1 − λ v ≤ C|D| r−2 r kvk . λ0 −1 (3.11) Applying (3.11) to (3.10) we see that DTsymm (û, ftrue ) + V λ0 1 2 2 (r−2)/r 2 kû − ftrue k−1 ≤ kξk1 + Cλ0 |D| kû − ftrue k 2 λ0 To estimate the last term we use some error estimates for T V − inpainting computed in [15]. First we have Z Z 2 kû − ftrue k = (û − ftrue )2 dx + (û − ftrue )2 dx. Ω\D D Cahn-Hilliard and BV-H −1 inpainting 21 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 2 kû − ftrue k ≤ K1 + K2 |û − ftrue | dx. 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 [15] (Theorem 8.) for functions û ∈ BV (Ω) we have kû − ftrue k 2 ≤ 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 T V seminorm is smaller than that from T V − L2 inpainting (cf. Section 3.2. in [15]). Finally we end up with (û, ftrue ) + DTsymm V 1 λ0 2 2 (r−2)/r kû − ftrue k−1 ≤ kξk1 + Cλ0 |D| errinpaint , 2 λ0 (3.12) with errinpaint := K1 + K2 |D| C (M (us ), β) + 2 R(ud ) . The first term in (3.12) depends on the regularizer T V , and the second term on the size of the inpainting domain D. Remark 3.6. From inequality (3.12) we derive an optimal scaling for λ0 , i.e., a scaling which minimizes the inpainting error. It reads λ20 |D| r−2 r ∼1 − r−2 2r λ0 ∼ |D| . 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 (3.7) can also be derived with an analogous technique. For ui being the solution of (3.7) with fdam = fi (again assuming that fi = ftrue in Ω \ D), the estimate Z λ0 λ0 2 DJsymm (u1 , u2 ) + ku1 − u2 k−1 ≤ (f1 − f2 )2 dx (3.13) 2 2 D holds. 4. Numerics. In the following numerical results for the two inpainting approaches (1.1) and (1.5) are presented. For both approaches we used convexity splitting algorithms, proposed by Eyre in [24], for the discretization in time. For more details to the application of convexity splitting algorithms in higher order inpainting compare [10]. For the space discretization we used the cosine transform to compute the finite differences for the derivatives in a fast way and to preserve the Neumann boundary conditions in our inpainting approaches (also cf. [10] for a detailed description). 4.1. Convexity splitting scheme for Cahn-Hilliard inpainting. For the discretization in time we use a convexity splitting scheme applied by Bertozzi et al. [8] to Cahn-Hilliard inpainting. The original Cahn-Hilliard equation is a gradient flow in H −1 for the energy Z 1 2 E1 [u] = |∇u| + F (u) dx, 2 Ω 22 M. Burger, L. He, and C.-B. Schönlieb Fig. 4.1. Destroyed binary image and the solution of Cahn-Hilliard inpainting with switching value: u(1200) with = 0.1, u(2400) with = 0.01 Fig. 4.2. Destroyed binary image and the solution of Cahn-Hilliard inpainting with switching value: u(800) with = 0.8, u(1600) with = 0.01 Fig. 4.3. Destroyed binary image and the solution of Cahn-Hilliard inpainting with switching value: u(800) with = 0.8, u(1600) with = 0.01 while the fitting term in (1.1) can be derived from a gradient flow in L2 for the energy Z 1 E2 [u] = λ(f − u)2 dx. 2 Ω We apply convexity splitting for both E1 and E2 separately. Namely we split E1 as E1 = E11 −E12 with Z C1 2 2 E11 = |∇u| + |u| dx, 2 2 Ω and Z E12 = 1 C1 2 − F (u) + |u| dx. 2 Ω A possible splitting for E2 is E2 = E21 − E22 with Z 1 C2 2 E21 = |u| dx, 2 Ω 2 Cahn-Hilliard and BV-H −1 inpainting 23 and E22 1 = 2 Z −λ(f − u)2 + Ω C2 2 |u| dx. 2 For the splittings discussed above the resulting time-stepping scheme is uk+1 − uk k+1 k+1 k k − E12 ) − ∇L2 (E12 − E22 ), = −∇H −1 (E11 τ where ∇H −1 and ∇L2 represent gradient descent with respect to the H −1 inner product and the L2 inner product respectively. This translates to a numerical scheme of the form uk+1 − uk 1 + ∆∆uk+1 − C1 ∆uk+1 + C2 uk+1 = ∆F 0 (uk ) − C1 ∆uk + λ(f − uk ) + C2 uk(. 4.1) τ To make sure that E11 , E12 and E21 , E22 are convex the constants C1 > 1 , C2 > λ0 . For the discretization in space we used finite differences and spectral methods, i.e., the discrete cosine transform, to symplify the inversion of the Laplacian ∆ for the computation of uk+1 . In [10] the authors prove that the above timestepping scheme is unconditionally stable in the sense that the numerical solution uk is uniformly bounded on a finite time interval. Moreover the discrete solution converges to the exact solution of (1.1) as τ → 0. These properties make (4.1) a stable and reliable discrete approximation of the continuous equation (1.1). Concerning the performance of the scheme in terms of computational speed its already remarked in [7] and [8] that (4.1) is certainly faster than numerical solutions for competing curvature based models. Nevertheless the convexity conditions on the constants C1 and C2 damp the convergence of the scheme and hence the solution of the scheme to steady state may require a lot of iterations (cf. also the number of iterations needed to compute the examples in Figures 4.1-4.3). An investigation of fast numerical solvers for (1.1) is a matter of future research. A very recent approach in this direction was made in [11], where the authors propose a multigrid approach for inpainting with CDD. In Figures 4.1-4.3 Cahn-Hilliard inpainting was applied to three different binary images. In all of the examples we follow the procedure of [7], i.e., the inpainted image is computed in a two step process. In the first step Cahn-Hilliard inpainting is solved with a rather large value of , e.g., = 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 (4.1) for a small , e.g., = 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 [8] the authors give numerical evidence that the steady state of (2.4) 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 only, might not prolongate the level lines into the missing domain as desired. See also [8] 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 (4.1) for, e.g., = 0.1 is faster than solving it for = 0.01. Again, this is because of the damping in (4.1) introduced by the constant C1 . 4.2. Convexity splitting scheme for T V − H −1 inpainting. We consider equation (1.5) ∇u where p ∈ ∂T V (u) is replaced by the formal expression ∇ · ( |∇u| ), namely ut = −∆(∇ · ( ∇u )) + λ(f − u). |∇u| (4.2) Similar to the convexity splitting for the Cahn-Hilliard inpainting we propose the following splitting for the TV-H −1 inpainting equation. The regularizing term in (4.2) can be modeled by a gradient flow in H −1 of the energy Z E1 = |∇u| dx. Ω 24 M. Burger, L. He, and C.-B. Schönlieb We split E1 in E11 − E12 with Z C1 |∇u|2 dx Ω 2 Z C1 |∇u|2 dx. = −|∇u| + 2 Ω E11 = E12 The fitting term is a gradient flow in L2 of the energy Z 1 E2 = λ(f − u)2 dx 2 Ω and is splitted into E2 = E21 − E22 with Z C2 2 |u| dx E21 = Ω 2 Z 1 E22 = −λ(f − u)2 + C2 |u|2 dx. 2 Ω Analogous to above the resulting time-stepping scheme is ∇uk uk+1 − uk + C1 ∆∆uk+1 + C2 uk+1 = C1 ∆∆uk − ∆(∇ · ( )) + C2 uk + λ(f − uk ). (4.3) τ |∇uk | In order to make the scheme unconditionally stable, the constants C1 and C2 have to be chosen so that E11 , E12 , E21 , E22 are all convex. The choice of C1 depends on the regularization of the q 2 total variation we are using. Using the square regularization |∇u| is replaced by |∇u| + δ 2 the condition turns out to be C1 > 1δ and C2 > λ0 . The discrete scheme (4.3) was analyzed in [10]. As for Cahn-Hilliard inpainting, therein the authors prove that the scheme is unconditionally stable. For this case the convergence of the discrete solution to p the exact solution of (1.5) (where the subgradient p was replaced by its relaxed version ∇ · (∇u/ |∇u|2 + δ 2 )) only holds under additional assumptions on the regularity of the exact solution. Results for the one dimensional case developed in [9] suggest that these regularity assumptions also hold in two dimensions with a sufficiently regular initial condition for the equation. However a rigorous proof of this fact is currently missing and is a challenge of additional research. As in the case of Cahn-Hilliard inpainting the convexity splitting scheme (4.3) for TV-H −1 inpainting converges rather slow due to the damping induced by C1 and C2 . Nevertheless its solution is numerically stable and approximates the exact solution accurately (which was rigorously proven for smooth solutions of (1.5) in [10]). In Figures 4.4-4.8 examples for the application of TV-H −1 inpainting to grayvalue images are shown. In Figure 4.5 a comparison of the TV-H −1 inpainting result with the result obtained by the second order TV-L2 inpainting model for a crop of the image in Figure 4.4 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 Figure 4.6 and 4.7 where the line is connected by the TV-H −1 inpainting model but clearly splitted 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 [8] for Cahn-Hilliard inpainting, cf. (1.2). The authors consider this as another important contribution of future research. Acknowledgments. This work was partially supported by KAUST (King Abdullah University of Science and Technology), by the WWTF (Wiener Wissenschafts-, Forschungs- und Technologiefonds) project nr.CI06 003, by the FFG project Erarbeitung neuer Algorithmen zum Image Inpainting project nr. 813610, and the PhD program Wissenschaftskolleg taking place at the University of Vienna. Cahn-Hilliard and BV-H −1 inpainting 25 Fig. 4.4. TV-H −1 inpainting: u(1000) with λ0 = 103 Fig. 4.5. (l.) u(1000) with TV-H −1 inpainting, (r.) u(5000) with TV-L2 inpainting Fig. 4.6. TV-H −1 inpainting compared to TV-L2 inpainting: u(5000) with λ0 = 10 The authors further would like to thank Andrea Bertozzi for her suggestions concerning the fixed point approach for the stationary equation, Massimo Fornasier for several discussions on the topic and Peter Markowich for remarks on the manuscript. We would also like to thank the editor and the referees for useful comments. 26 M. Burger, L. He, and C.-B. Schönlieb Fig. 4.7. TV-H −1 inpainting compared to TV-L2 inpainting: u(5000) with λ0 = 10 Fig. 4.8. T V − H −1 inpainting: u(1000) with λ = 103 Appendix A. 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 can be carried out in a similar way. Namely we consider ut = ∆(−∆u + 1 F 0 (u)) + λ(f − u) in Ω, ∂u ∂∆u on ∂Ω, ∂ν = ∂ν = 0 For the existence of a stationary solution of this equation we consider again a fixed point approach similar to (2.4) in the case of Dirichlet boundary conditions, i.e., ( u−v 1 0 τ = ∆(−∆u + F (u)) + λ(f − u) + (λ0 − λ)(v − u) in Ω, 1 0 (A.1) ∂ (∆u− F (u)) ∂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. Cahn-Hilliard and BV-H −1 inpainting 27 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 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 Hφ1 (Ω) = ψ ∈ H 1 (Ω) : ψ dx = 0 , Ω with norm kukH 1 := k∇ukL2 and inner product hu, viH 1 := h∇u, ∇viL2 . This is a Hilbert space φ φ and the norms k.kH 1 and k.kH 1 are equivalent on Hφ1 (Ω). Let (Hφ1 (Ω))∗ denote the dual of Hφ1 (Ω). φ 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, ∇ψiL2 φ φ φ ∀ψ ∈ Hφ1 (Ω). Lets now define a norm and an inner product on H∂−1 (Ω). Definition A.1. n o 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 iL2 , ∂ H∂−1 (Ω) where F1 , F2 ∈ and where u1 , u2 ∈ Hφ1 (Ω) are the associates of F1 | Hφ1 , F2 | Hφ1 ∈ 1 ∗ (Hφ (Ω)) . At this point it is not entirely obvious that for a given F ∈ H∂−1 (Ω) we have F | Hφ1 ∈ (Hφ1 (Ω))∗ . That this is the case though is explained in the following theorem. Theorem A.2. 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 A.2 can be easily checked just by the application of the definitions and the fact that the norms k.kH 1 and k.kH 1 are equivalent on Hφ1 (Ω). From point 1. of the theorem we have that φ 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). Therefore the norm in Definition A.1 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), there exists a unique element u ∈ Hφ1 (Ω) such that Z hF, ψi(H 1 )∗ ,H 1 = ∇u · ∇ψ dx, ∀ψ ∈ Hφ1 (Ω). (A.2) 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 constants K ∈ R and therefore (A.2) extends to all ψ ∈ H 1 (Ω). We define ∆−1 F := u (A.3) the unique solution to (A.2). R Now suppose F ∈ L2 (Ω) and assume u ∈ H 2 (Ω). Set hF, ψi := Ω F ψ 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 Z (−∆u − F )ψ dx + ∇u · νψ ds = 0, ∀ψ ∈ Hφ1 (Ω). Ω ∂Ω 28 M. Burger, L. He, and C.-B. Schönlieb Therefore u ∈ Hφ1 (Ω) is the unique weak solution of the following problem: −∆u − F = 0 in Ω ∇u · ν = 0 on ∂Ω. (A.4) Remark A.3. With the above characterization of elements F ∈ H∂−1 (Ω) and the notation (A.3) for its associates the inner product and the norm can be written as Z hF1 , F2 iH −1 := ∇∆−1 F1 · ∇∆−1 F2 dx, ∀F1 , F2 ∈ H∂−1 (Ω), ∂ Ω and norm sZ kF kH −1 := ∂ (∇∆−1 F )2 dx. Ω Throughout the rest of this appendix 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. Now its important to notice that in order to rewrite (A.1) in terms of ∆−1 we require the ”right hand side” of the equation, i.e., u−v τ + λ(u − f ) + (λ0 − λ)(u − v) to be an element of our new space H∂−1 (Ω) (cf. Definition A.1). 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 (A.1) we are going to modify the right hand side by subtracting its mean. Let FΩ = τ1 FΩR1 + λ0 FΩ2 1 FΩ1 = |Ω| (u − v) dx Ω R λ 1 2 FΩ = |Ω| Ω λ0 (u − f ) + 1 − λ λ0 (u − v) dx, and consider instead of (A.1) the equation ∆u − 1 F 0 (u) = ∆−1 u−v τ − λ(f − u) − (λ0 − λ)(v − u) − FΩ ∂u ∂ν = 0 in Ω, on ∂Ω, ∂ (∆u− 1 F 0 (u)) where the second Neumann boundary condition = 0 on ∂Ω is included in the ∂ν definition of ∆−1 . The functional of the corresponding variational formulation then reads 2 R 2 1 J (u, v) = Ω 2 |∇u| + 1 F (u) dx + 2τ (u − v) − FΩ1 −1 2 + λ20 u − λλ0 f − 1 − λλ0 v − FΩ2 . −1 With these definitions the proof for the existence of a stationary solution for the modified CahnHilliard equation with Neumann boundary conditions Rcan be carried out similarly to the proof in R d Section 2. Note that every solution of (1.1) fulfills u dx = λ(f − u) dx. This means that dt Ω Ω R for a stationary solution û the integral 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). Appendix B. Functions of bounded variation. The following results can be found in [3]. Let Ω ⊂ R2 be an open and bounded Lipschitz domain. As in [3] the space of functions of bounded variation BV (Ω) in two space dimensions is defined as follows: Definition B.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 u dx = − φdDi u ∀φ ∈ Cc∞ (Ω), i = 1, 2, ∂x i Ω Ω Cahn-Hilliard and BV-H −1 inpainting 29 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 B.2. (Variation) Let u ∈ L1loc (Ω). The variation V (u, Ω) of u in Ω is defined by Z 2 V (u, Ω) := sup u divφ dx : φ ∈ Cc1 (Ω) , kφk∞ ≤ 1 . Ω A simple integration by parts proves that Z |∇u| dx, V (u, Ω) = 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: Definition B.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 [ |µ| (E) := sup |µ(Eh )| : Eh ∈ E pairwise disjoint , E = Eh , ∀E ⊂ E. h=0 h=0 With Definition B.2 the space BV (Ω) can be characterized as follows Theorem B.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 B.5. (Weak∗ convergence) Let u, uh ∈ BV (Ω). We say that (uh ) weakly∗ ∗ converges in BV (Ω) to u (in signs uh * u) if (uh ) converges to u in L1 (Ω) and (Duh ) weakly∗ converges to Du in all (Ω), i.e., Z Z lim φ dDuh = φ dDu ∀φ ∈ C0 (Ω). h→∞ Ω Ω A simple criterion for weak∗ convergence is the following: Theorem B.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 B.7. (Compactness for BV (Ω)) • Let Ω be a bounded domain with compact Lipschitz boundary. Every sequence (uh ) ⊂ BVloc (Ω) satisfying Z sup |uh | dx + |Duh | (A) : h ∈ N < ∞ ∀A ⊂⊂ Ω open, 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. • Let Ω be a bounded domain in Rd with Lipschitz boundary. 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