# Geometric properties of solutions to the total variation denoising problem Antonin Chambolle

Geometric properties of solutions to the total variation denoising problem Antonin Chambolle∗ Gabriel Peyré†‡§ Vincent Duval†‡ Clarice Poon†‡ January 30, 2016 Abstract This article studies the denoising performance of total variation (TV) image regularization. More precisely, we study geometrical properties of the solution to the so-called Rudin-Osher-Fatemi total variation denoising method. TV denoising is a popular denoising scheme, and is regarded as a baseline for edge-preserving image restoration. A folklore statement is that this method is able to restore sharp edges, but at the same time, might introduce some staircasing (i.e. “fake” edges) in flat areas. Quite surprisingly, put aside numerical evidences, almost no theoretical result are available to backup these claims. The first contribution of this paper is a precise mathematical definition of the “extended support” (associated to the noise-free image) of TV denoising. It is intuitively the region which is unstable and will suffer from the staircasing effect. We highlight in several practical cases, such as the indicator of convex sets, that this region can be determined explicitly. Our second and main contribution is a proof that the TV denoising method indeed restores an image which is exactly constant outside a small tube surrounding the extended support. The radius of this tube shrinks toward zero as the noise level vanishes, and are able to determine, in some cases, an upper bound on the convergence rate. For indicators of so-called “calibrable” sets (such as disks or properly eroded squares), this extended support matches the edges, so that discontinuities produced by TV denoising cluster tightly around the edges. In contrast, for indicators of more general shapes or for complicated images, this extended support can be larger. Beside these main results, our paper also proves several intermediate results about fine properties of TV regularization, in particular for indicators of calibrable and convex sets, which are of independent interest. ∗ CNRS and CMAP, Ecole Polytechnique 91128 Palaiseau Cedex, FRANCE Université Paris-Dauphine, Place du Marechal De Lattre De Tassigny, 75775 Paris 16, FRANCE ‡ INRIA, MOKAPLAN, Domaine de Voluceau Le Chesnay, FRANCE § CNRS † CEREMADE, 1 1 Introduction The total variation (TV) denoising method was introduced by Rudin, Osher and Fatemi in [83]. It is one of the first proposed non-linear image restoration method, and had an enormous impact on shaping modern imaging sciences. Despite being quite old, this method is still routinely used today, and its popularity probably stems from both its simplicity and its ability to restore “cartoonlooking” images. While being far from the state of the art for denoising in terms of performance (see Section 1.2 for some more recent works), it is still featured as a benchmark in most papers being published on image restoration. 1.1 Total Variation Denoising The total variation a function u ∈ L2 R2 is defined as Z Z def. def. J(u) = |Du| = sup u div z ; z ∈ Cc1 (R2 , R2 ), ||z||∞ 6 1 . R2 (1) R2 Given some noisy input function f , following [83], we are interested in the total variation denoising problem min u∈L2 (RN ) 1 λJ(u) + ||u − f ||2L2 . 2 (Pλ (f )) Here, λ > 0 is the regularization parameter, and it should adapted by the user to the noise level. The goal of this paper is to study the ability to restore the geometrical structures (in particular the edges) of some (typically unknown) noise-free function f by solving Pλ (f + w), i.e. by applying TV regularization to the input noisy image f + w. Here w accounts for some additive noise in the image formation process, and is assumed to have a finite L2 norm ||w||L2 . 1.2 Previous Works Image restoration. The TV denoising method, often referred to as the RudinOsher-Fatemi (ROF) model, was introduced in [83]. Its basic properties (including the existence and uniqueness of the solution) are derived in [43]. We refer to [45] for an introduction to this model and an overview of its numerous applications in image processing. A thorough study of its properties can be found in [4, 5]. It is important to realize that TV is far beyond the state of the art in imaging sciences, and we refer to recent works such as [81, 27, 3, 52] that obtain superior denoising performance on natural images by exploiting more complex and involved regularizers and statistical models. Beyond denoising, TV methods have been used successfully to solve a wide range of ill-posed inverse problems, see for instance [2, 47, 48, 75]. Following the work of Meyer [77], TV regularization in conjunction to a norm dual of TV (favoring oscillations) is used to separate texture from structure, see for 2 instance [17]. In a finite dimensional setting (using a discretization of the gradient operator), TV methods have been used to solve compressed sensing, where the linear operator is randomized [78, 80] to obtain accurate reconstructions when the number of random samples is nearly proportional to the number of the discretized edges. Jump sets stability. The use of non-smooth (possibly non-convex) regularizations to restore edges and promote sharp features has been advocated by Mila Nikolova. She provided in a series of papers a detailed analysis of a general class of regularization schemes which admit piecewise smooth solutions, see for instance [79]. In the case of the TV regularization, this analysis can be refined. Explicit solutions are known, mostly in 1-D and for radial 2-D functions (see for instance [86]), as well as for indicators of convex sets in the plane [9, 6]. They suggest that TV methods indeed do maintain sharp features. A landmark result is the proof in [35] that total variation regularization does not introduce jumps, i.e. the “jump set” of the solution of (Pλ (f )) is included in the one of the input f . A review of this result and extensions can be found in [88]. These results are however of little interest when f is replaced by a noisy function f + w (which is the setting of practical use of the method), since the noise w, which is only assumed to be in L2 , might introduce jumps everywhere. It is actually the presence of this noise which is responsible for the “staircasing” effect, which creates spurious edges in flat area. It is the purpose of the present paper to fill this theoretical gap by analyzing the impact of the noise on the jump set of the solution to Pλ (f + w), when both ||w||L2 and λ are not too large. Calibrable and Cheeger sets. Of particular importance for the analysis of TV methods are indicator functions of sets, and their behavior under the regularization. Indicator functions which are invariant (up to a rescaling) under TV denoising define so-called “calibrable” sets. These sets play the role of “stable” sets and one expects the corresponding edges to be well restored by TV denoising, a statement which is made precise in the present paper. We refer to section 2.4 for a detailed description of these sets and their basic properties. An important result is the full characterization of convex calibrable sets in [8]. The notion of a calibrable set is closely related to the one of eigenvectors of the Du ), and is also known as the curvature operator, which informally reads div( |Du| 1-Laplacian, see [68]. Indeed, indicators of calibrable sets are eigenvectors of this operator [22]. These eigenvectors can be used for image processing purposes, as advocated in [73]. The study of fine geometrical properties of TV minimizers is thus deeply linked with geometric measure theory and in particular sets of finite perimeters [13, 74]. In particular, the construction of calibrable sets is related to minimal surface problems [60] and capilarity problems [71]. Calibrable sets are also related to Cheeger sets, which are subsets of a given set minimizing the ratio of perimeter over area. These Cheeger sets are useful to construct the solution of the TV denoising problem. Cheeger sets associated to a given convex sets are unique [36, 7, 66], and can be approximated using either p-Laplacian [67] or 3 strictly convex penalizations [30] to recover an unique maximal set, which can in turn be computed numerically [33]. TV flow. While our paper studies variational problems, a closely related denoising method is obtained by solving the PDE obtained as a gradient flow of J, see [21] for a formal definition. In this setting, the evolution time t plays the role of λ. This PDE corresponds to the celebrated mean-curvature motion of the levelsets, and is a particular example of a general class of non-linear edge-preserving diffusions [39]. This mean curvature flow can be shown in some restricted setting to be equivalent to the TV regularization [26]. All the results available for the variational formulation (Pλ (f )) have equivalent in the PDE setting, such as for instance explicit solutions for the indicators of convex sets [9] and the evolution of the jump-set [38]. Some of these results have been extended to more general PDE’s, see [15]. Links with wavelet regularization. Total variation regularization shares similarities with wavelet-based methods. This stems in most part because the space of bounded variation functions is tightly approximated from the inside and from the outside by Besov spaces, which are characterized by the sparsity of wavelet expansions, see [49]. This means that J is well approximated by a (weighted) `1 norm of wavelet coefficients, so that one can expect that the solution to `1 Wavelet regularization and of TV regularization to be close. For the denoising problem, wavelet regularization corresponds to thresholding operators applied to wavelet coefficients, and have been advocated by Donoho and his collaborators, see for instance [55]. They proved that wavelet thresholding leads to asymptotically optimal denoising in a minimax sense over Besov balls. These thresholding can also be interpreted as solution to sparsity-promoting variational problems [42]. Going beyond orthogonal wavelet bases, translation invariant wavelet thresholding [50] offer an alternative way, not based on PDE’s or optimization schemes, to perform edge-preserving restoration [44]. The connexion between TV and wavelet methods is made more precise in [90] and more recently in [31] who proves a Γ-convergence result. This connexion is however still quite loose, and in particular, does not shed light on the actually ability of both class of methods to recover sharp edges, as we intended to do in the present paper. 1-D setting and statistical estimation. 1-D TV denoising, sometimes referred to as the “taut string method” [76], is a method of choice to perform statistical analysis of time series and in particular to detect jumps and transitions. In the special 1-D case, it is possible to compute exactly the solution on a grid of P points in O(P 2 ) operations using a dynamic programming method [51, 56, 63, 65]. Similarly to wavelet thresholding estimators, 1-D TV denoising is known to achieve asymptotic optimal estimation results [76]. This optimality is however measured in term of L2 error, which does not provide geometric information about the location of jumps. A more precise analysis 4 of the distribution of the jumps is provided in [54]. This analysis is however probabilistic and does not extend to higher dimensions, whereas we targets a deterministic geometric analysis in 2-D (although some of our results cover the general N -dimensional case). Inverse problem and source condition. The systematic study of noise stability of regularization schemes relies on the so-called source-condition [84], which reads in the simple denoising setting that ∂J(f ) should be non-empty (see Section 2.3 for a primer on the total variation sub-differential ∂J). For non-smooth regularizations over Banach spaces, this study started with the seminal paper of Burger and Osher [29] who show that this source condition implies stability of the solution according to the Bregman divergence associated to J. This Bregman measure of stability is however quite weak, and in particular it does not lead to a precise geometric characterization of the restored jump set. Our analysis can be seen as a generalization and refinement of this approach, as highlighted in Section 1.3.1. Note that under a non-degeneracy condition, namely that 0 is in the relative interior of ∂J(f ), it is possible to state much stronger results, as detailed in the book [84] for `1 -based methods. These results however do not cover the TV regularization and can only be applied to discretized versions of TV regularization problems, see [87]. Numerical algorithms. While this is not the topic of this article, let us note that the discretization (often using finite differences) and the numerical resolution of (Pλ (f )) is notoriously difficult, in large part because of the nonsmoothness of the TV functional J. Early algorithms rely on various smooth approximations of J [89, 43]. The dual projected gradient method proposed by [40] started a wave of activity on the use of first order proximal splitting schemes to solve (Pλ (f )) with a provably convergent scheme, see for instance [19] for accelerated first order schemes. Another option is to solve exactly the denoising problem using graph-cuts methods [70], see [53, 41] and the references therein. These algorithms work however only for the anisotropic total variation, and thus do not cover our J functional, which is the isotropic total variation. Let us also recall that TV methods, and their discretizations, are intimately linked with iterative non-linear filterings, and in particular local median filters, see [28]. Extension of the basic TV model. Many results known in the case of the isotropic total variation J have been extended to the more general setting of anisotropic total variation (sometime called “crystalline” total variations). This includes in particular calibrable sets [34], gradient flows [23, 20] and Cheeger sets [37]. Another generalization is to include some spatially varying weights, see [32] for the analysis of Cheeger sets and [38] for a study of the jump set stability. Lastly, it is possible to devise higher order regularization schemes [47, 25] to promote piecewise smooth (e.g. piecewise polynomial) instead of piecewise constant solutions. While our analysis does not cover these more general settings, it could serve as a starting point for the systemic study of non-smooth 5 regularization methods for denoising. 1.3 Contributions Level lines in the low noise regime. Let us first stress the fact that our analysis focusses on regimes where the noise and the regularization parameter are small. It is not very difficult to see that, as λ → 0+ and ||w||L2 → 0+ , the solution uλ,w to Pλ (f + w) converges towards f in the L2 topology. Our goal is to describe this convergence more precisely: is it possible to say that the level lines of uλ,w converge to those of f ? In what sense? Morevoer, does the support of Duλ,w converge towards the support of Df ? Those questions are all the more important as it is widely acknowledged in image analysis theory that the shape information of an image is contained in the level sets of an image [91, 85], determined in particular by their boundary. To assess the support stability of the method with respect to that matter, and in particular to study its ability to restore edges, it is necessary to make stronger assumption on the noise level. More precisely, we say that a property holds in the low noise regime determined by λ0 > 0 and α0 > 0 if (λ, w) ∈ Dλ0 ,α0 , where def. Dλ0 ,α0 = 1.3.1 (λ, w) ∈ R+ × L2 RN ; 0 6 λ 6 λ0 and ||w||L2 6 α0 λ . Our approach – the minimal norm certificate A common approach to studying the stability properties of a variational problems is by analysis of the source condition. To explain our approach, we first recall a result of Burger and Osher [29] which provides a link between the source condition and stability of regularized solutions: Given f ∈ L2 R2 with finite total variation, suppose that the source condition holds. i.e. there exists some v ∈ ∂J(f ) such that v = − div z for some z ∈ L∞ (R2 , R2 ) with ||z||L∞ 6 1. Note that elements in ∂J(f ) are often referred to as dual certificates. Let T ⊂ R2 and δ ∈ (0, 1) be such that |z(x)| < 1−δ for a.e. x 6∈ T . Then, uλ,w , the solution to Pλ (f + w) satisfies Z (1 − δ) |Duλ,w | 6 R2 \T λ||v||2L2 ||w||2L2 + + ||w||L2 ||v||L2 . 2λ 2 While this result informs us that the variation of uλ,w is concentrated in the region T , it does not provide any information on the regions where Duλ,w is identically zero and no information is given about how the support of Duλ,w differs from the support of Df . Instead of studying any v ∈ ∂J(f ), in this paper, we shall study the minimal norm certificate def. v0,0 = argmin {||v||L2 ; v ∈ ∂J(f )} . The minimal norm certificate was first proposed in [57] for studying the support of solutions to the sparse spikes deconvolution problem using total variation of measures regularization, but in this particular framework of denoising, 6 it is also known as the minimal section of ∂J(f ). Although dual certificates have been widely used to derive stability properties of solutions to the sparse spikes deconvolution problem in terms of the L2 norm, see for instance [64], the novelty of the minimal norm certificate (which is itself a dual certificate) is that it additionally addresses support stability questions such as the number and the location of the recovered diracs. In this paper, we follow the same philosophy: Similarly to the problem of sparse spikes deconvolution, we show that the minimal norm certificate naturally gives rise to the notion of an extended support, which in turn, governs the support of the regularized solution in the low noise regime. Unlike previous works, our analysis is carried out for this very specific dual certificate and in doing so, we are able to characterize the support stability of the total variation denoising problem. 1.3.2 Our main contribution The extended support. Based on the minimal norm certificate, we define the extended support of a function f ∈ L2 R2 with bounded variation when the source condition is satisfied. Intuitively, this is the region where, within the low noise regime, any support instabilities which occur within a neighbourhood of this region. The statement is made precise in the main result of this paper, Theorem 3, where we prove that given any tube around the extended support, there exists λ0 , α0 > 0 such that the support of Duλ,w is contained inside this tube for all (λ, w) ∈ Dλ0 ,α0 . Furthermore, the radius of this tube converges to zero as the noise level converges to zero. Explicit examples of the extended support are given for indicator functions on calibrable sets and convex sets with smooth boundaries, and in particular, for these examples, our definition of the extended support is in fact tight. Moreover, when denoising the indicator function of a calibrable set C, the support of regularized solutions to the TV denoising problem will cluster around ∂C. Convergence rates. We stress that via the approach of [29], characterization of the regions where the variation of uλ,w is small is possible only when the source condition holds and there is precise knowledge of the extremal points and decay of some vector field z ∈ L∞ (R2 , R2 ) for which v = − div z ∈ ∂J(f ). In general, this vector field is not unique and such precise characterization is a difficult problem. In contrast, via our approach, explicit knowledge of the vector field associated with the minimal norm certificate is not essential, and in fact, the definition of the extended support is dependent only v0 and not on the vector fields z for which v0 = − div z. Nonetheless, in the special cases where the vector field associated with v0 is known, we provide an explicit upper bound on the rate of shrinkage of the tube around the extended support with respect to the decay of the noise level. For the indicator function on a calibrable set C with C 2 boundary, we describe an explicit construction of the vector field z0 associated with the minimal norm certificate with |z0 | < 1 on all compact subsets of R2 \ ∂C. Therefore, our main 7 result can be seen as a refinement of the work of [29] and can be applied in much greater generality. Stability estimates in the absence of the source condition. The final contribution of this paper is stability analysis of cases where the source condition is not satisfied, i.e. ∂J(f ) = ∅. One important class of functions which this covers are the indicator functions on convex sets with nonsmooth boundary, such as the square. To our knowledge, there were no previous studies on stability analysis in the absence of the source condition and hence, no stability guarantees for even the simple case of denoising the indicator function of a square. Although in this case, the minimal norm certificate is not defined, the techniques developed in the analysis of the minimal norm certificate can be adapted to such special cases to derive stability estimates for general convex sets. 1.4 Outline of the paper Section 2 recalls some essential tools which will be used throughout this paper. Section 3 introduces the dual formulation of (Pλ (f )) and defines the minimal norm certificate. Explicit examples of the minimal norm certificate are also given. Based on the existence of the minimal norm certificate, Section 4 derives some geometric properties of the level sets of solutions of (Pλ (f )) in the low noise regime. The definition and examples of the extended support can be found in Section 5. In Section 6, we describe the behaviour of the vector field associated with the minimal norm certificate of indicator functions on calibrable sets. This information is useful in understanding the convergence of the support of TV regularized solutions to the extended support. The main results of this paper are presented in Section 7. Finally, stability results for denoising general convex sets in the absence of the source condition are given in Section 8. 2 Preliminaries This section recalls some essential results which are applied throughout this paper. 2.1 Set convergence We shall use the notion of Painlevé-Kuratowski set convergence (see [82] for more detail). Given a sequence of sets {Sn }n∈N , Sn ⊆ R2 , let us define the outer (resp. inner) limit of {Sn }n∈N as def. lim sup Sn = x ∈ R2 ; lim inf dist(x, Sn ) = 0 , (2) n→+∞ n→+∞ def. (resp.) lim inf Sn = x ∈ R2 ; lim sup dist(x, Sn ) = 0 . (3) n→+∞ n→+∞ 8 It is clear that lim inf n→+∞ Sn ⊆ lim supn→+∞ Sn . Moreover, those two sets are closed. We say that Sn converges towards S ⊆ R2 , i.e. limn→+∞ Sn = S, if lim inf Sn = S = lim sup Sn . (4) n→+∞ n→+∞ If the sequence Sn is bounded (there exists R > 0 such that S ⊆ B(0, R) and Sn ⊆ B(0, R) for all n large enough), then the Painlevé-Kuratowski convergence is equivalent to the so-called Hausdorff convergence, that is, lim sup |dist(x, Sn ) − dist(x, S)| = 0. n→+∞ x∈S∪Sn 2.2 (5) Functions with bounded variation and sets with finite perimeter We briefly recall some properties of functions of bounded variations and sets of finite perimeter. We refer the reader to [14, 74] for a comprehensive treatment of the subject. Total variation, perimeter. Given u ∈ L1loc (R2 ), its total variation is equal to Z Z def. 1 2 2 |Du| = sup u div z ; z ∈ Cc (R , R ), ||z||∞ 6 1 . R2 R2 If J(u) < +∞, we say that u has bounded variation. The mapping u 7→ J(u) is lower semi-continuous with respect to the L1loc (R2 ) topology (hence for the L2 topology). If E ⊆ R2 is a measurable set, we denote by |E| its 2-dimensional Lebesgue measure. The set E is said to be of finite perimeter if J(1E ) < +∞,Rwhere 1E is the indicator function of E. Its perimeter is defined as P (E) = R2 |D1E |. For a Borel set S ⊆ R2 , H1 xS denotes the 1-dimensional Hausdorff measure restricted to S, namely H1 xS(A) = H1 (A ∩ S). The reduced boundary of E is defined as −D1E (B(x, r) def. def. ∗ ∂ E = x ∈ Supp |D1E | ; νE (x) = lim exists and |νE (x)| = 1 . r→0+ |D1E (B(x, r))| (6) The vector νE (x) is the measure theoretic outer unit normal to E. When the context is clear, we shall write ν instead of νE . Moreover, D1E = νE H1 x∂ ∗ E, and |D1E | (A) = H1 (∂ ∗ E ∩ A) for all open set A ⊆ R2 . In the following, we use the construction in [60, Prop. 3.1] so as to always consider a Lebesgue representative of E such that for all x in the topological boundary ∂E, 0 < |E ∩ B(x, r)| < |B(x, r)|. Then, with this representative, Supp D1E = ∂ ∗ E = ∂E. 9 The area and the perimeter are related by the so-called isoperimetric inequality: for any Lebesgue measurable set E ⊆ R2 , c2 min{|E| , RN \ E } 6 (P (E))2 , where c2 = 4π is the isoperimetric constant. Level sets and the coarea formula. The coarea formula relates the total variation of a function f ∈ L1loc (R2 ) and the perimeter of its level sets. Define the level sets of f as def. F (t) = x ∈ R2 ; f (x) > t for t > 0, (7) def. F (t) = x ∈ R2 ; f (x) 6 t for t < 0. It is clear that F (t) < +∞ except possibly for t = 0. Moreover, the family is monotone on [0, +∞) and (−∞, 0) with \ \ 0 0 F (t) = F (t ) for t > 0, F (t) = F (t ) for t < 0. 0<t0 <t 0>t0 >t 0 We handle 0 as a special case with F (0) = R2 \ t0 <0 F (t ) . Now, given an open set U ⊆ R2 , the coarea formula states that if J(f ) < +∞ then Z Z +∞ |Df | = P (F (t) ; U )dt. S U −∞ def. where P (F (t) ; U ) = |D1E | (U ). 2.3 Subdifferential of J In the following, unless otherwise stated, we use the L2 R2 topology. The functional J : L2 R2 → R ∪ {+∞} is convex, proper lower semi-continuous. It is in fact the support function of the closed convex set div z ; z ∈ X2 R2 , ||z||∞ 6 1 ⊆ L2 R2 , where we defined X 2 RN def. = z ∈ L∞ (R2 , R2 ) ; div z ∈ L2 R2 . As a result, it is possible to prove that ∂J(0) = div z ; z ∈ X2 R2 , ||z||∞ 6 1 , Z ∂J(u) = v ∈ ∂J(0) ; uv = J(u) . (8) (9) RN Provided that J(u) < +∞, Du is a Radon measure, i.e it is possible to evaluate (z, Du) for all vector field z ∈ Cc0 (R2 ; R2 )). Following the construction 10 by Anzellotti (z, Du) for less smooth z, namely [16], it is possible to define z ∈ X2 RN provided that u ∈ L2 R2 . Given ϕ ∈ Cc1 (R2 ), define Z Z h(z, Du), ϕi = − u(x)ϕ(x) div z(x)dx − u(x)z(x) · ∇ϕ(x)dx. R2 R2 Then (z, Du) is a Radon measure which is absolutely continuous with respect to |Du|, with Z |h(z, Du), ϕi| 6 ||ϕ||∞ ||z||L∞ (U ) |Du| , U Cc1 (R2 ) 2 for all ϕ ∈ and U ⊂ R open set such that Supp(ϕ) ⊂ U . Moreover, the following integration by parts holds Z Z u div z = − (z, Du). R2 R2 Remark 1. If u is smooth, then (z, Du) can be interpreted as a (defined almost everywhere) pointwise inner product: Z Z (z, Du) = z(x) · ∇u(x)dx for any Borel set B ⊆ R2 . B B If u is the characteristic function of set with finite perimeter E ⊂ R2 Z Z div z(x)dx = (z, −D1E ). E E The question whether it is possible to give a pointwise meaning to (z, −D1E ) is investigated in [24, 46]. It turns out that (z, −D1E ) = T z · νE H1 x∂ ∗ E, where T z is the full trace of z defined on H1 -a.e. on ∂ ∗ E [24]. Hence, we shall write Z Z div z(x)dx = z · νE dH1 . (10) ∂∗E E R Remark 2. That enables us to interpret the optimality R2 uv = J(u) as an ”optimality |Du|-almost everywhere”: Z Z Z Z u div z = |Du| ⇔ − (z, Du) = |Du| RN RN RN RN Z ⇔0= (1 + θ(z, Du))d |Du| , RN where θ(z, Du) is the Radon-Nikodym derivative of (z, Du) with respect to |Du|. Since |θ(z, Du)| 6 1, this implies that in fact the equality (z, Du) = − |Du| holds |Du|-a.e. Informally, recalling that ||z||∞ 6 1, this means that z=− Du , |Du| |Du| − almost everywhere. In other words z must be orthogonal to the level lines, and its saturation points contains the support of Du (see also [24]). 11 Examples Let us examine two examples which can be found in [77]. Characteristic function of a disc: Given R > 0, consider the vector field ( x if |x| 6 R, (11) z(x) = RR otherwise. |x| x ∞ 2 2 One may check that ), div z = R2 1B(0,R) ∈ L∞ (R2 , R2 ), ||z||∞ 6 1, z ∈ 2L (R , R and z · ν = 1 on x ∈ R ; |x| = R . Hence div z ∈ ∂J(u) for u = 1B(0,R) . Characteristic function of a square : Let u = 1[0,1]2 be the characteristic function of the unit square. It turns out that ∂J(u) = ∅. The argument provided in [77] is the following. Assume that there exists v ∈ ∂J(u) and let z ∈ L∞ (R2 , R2 ) be the corresponding vectorfield. We denote by Tε the triangle x ∈ R2 ; 0 6 x1 6 1, 0 6 x2 6 1, x1 + x2 6 ε and by ν its outer unit normal (defined H1 -a.e.). By the Gauss-Green theorem: Z Z z · νdH1 . div z = Tε ∂Tε qR p Since v = div z ∈ L2 (R2 ), the left term is upper-bounded by (div z)2 |Tε | = Tε √ o(ε) whereas the right term is lower-bounded by (2 − 2)ε. This is a contradiction. Hence ∂J(u) = ∅ 2.4 Calibrable sets in R2 A remarkable family of elements of ∂J(0) is the family of characteristic functions of sets F such that λF 1F ∈ ∂J(1F ) for some λF ∈ R. This family of functions, known as the calibrable sets, will serve as a prime example in the illustration of our theoretical results. In this section, we recall some key results about these functions. 2.4.1 Sets that evolve at constant speed In [21], the authors study on the total variation flow ∂u Du = div . ∂t |Du| ∂u ∂t ∈ −∂J(u), namely: (12) The prove existence and uniqueness of a “strong solution” (see [21]) for all initial data u0 ∈ L2 R2 , and existence and uniqueness of an “entropy solution” for u0 ∈ L1loc (RN ). In the second part of the paper, they characterize the bounded sets of finite perimeter Ω such that u = 1Ω satisfies Du def. P (Ω) − div = λΩ u, where λΩ = . (13) |Du| |Ω| 12 Such sets are exactly the sets which evolve with constant boundary, i.e. such that u(x, t) = λ(t)1Ω (x), with λ > 0. Such sets are called calibrable. They are characterized by the fact that λΩ 1Ω ∈ ∂J(1Ω ): Definition 1 (Calibrable sets). A set of finite perimeter Ω ⊂ R2 is said to be calibrable if, writing v = 1Ω , there exists a vector field z ∈ L∞ (R2 , R2 ) such that kzk∞ 6 1 and Z Z (z, Dv) = |Dv|, RN R2 − div z = λΩ v. In that case, we say that z is a calibration for Ω. Remark 3. If λ1Ω ∈ ∂J(1Ω ) for some λ ∈ R, then necessarily λ = λΩ . 2.4.2 Characterization in R2 The following results characterize convex calibrable sets. Proposition 1 ([21]). Let C ⊂ R2 be a bounded set of finite perimeter, and assume that C is connected. C is calibrable if and only if the following three conditions hold: 1. C is convex; 2. ∂C is of class C 1,1 ; 3. the following inequality holds: ess sup κ∂C (p) 6 p∈∂C P (C) . |C| (14) Proposition 2 ([21]). Let Ω ⊂ R2 be a bounded set of finite perimeter which is calibrable. Then, 1. The following relation holds: P (Ω) P (D) 6 , |Ω| |D ∩ Ω| ∀D ⊆ R2 , D of finite perimeter; 2. each connected component of Ω is convex. 2.5 From the subdifferential to the level sets Let f ∈ L2 R2 , J(f ) < +∞, and v ∈ ∂J(f ). By definition of the subdifferential, Z Z Z Z 2 2 ∀g ∈ L R , |Dg| − vg > |Df | − vf. (15) R2 R2 13 R2 R2 In fact, using the coarea formula, one may reformulate that optimality property (see Proposition 3 below) as an optimality property of the level sets. That result is very similar to [69, Corollary 2.4] but it requires a bit more care in our framework since the domain is R2 and v ∈ L2 R2 . The level sets of f (resp. g), are denoted by {F (t) }t∈R (resp. {G(t) }t∈R ). Proposition 3. Let f ∈ L2 R2 , J(f ) < +∞, and v ∈ L2 R2 . The following conditions are equivalent. (i) v ∈ ∂J(f ), (ii) v ∈ ∂J(0) and the level sets of f satisfy Z ∀t > 0, P (F (t) ) = v, ∀t < 0, P (F (t) ) Z =− F (t) v. (16) F (t) (iii) The level sets of f satisfy ∀t > 0, ∀G ⊂ R2 , |G| < +∞, Z P (G) − v > P (F (t) ) − ∀G ⊂ R2 , |G| < +∞, Z P (G) + v, (17) F (t) G ∀t < 0, Z v > P (F (t) ) + Z v. (18) F (t) G R R∞ Proof. (iii) ⇒ (i) It suffices to use the coarea formula |Dg| = −∞ P (G(t) )dt and Fubini’s theorem in Z Z Z +∞ Z 0 gv = 1g(x)>t v(x)dt − 1g(x)6t v(x)dt dx, (19) R2 R2 −∞ 0 and similarly for the level sets of f . R (i) ⇒ (ii) Using (9), we see that v ∈ ∂J(0) and R2 f v = J(f ). From v ∈ ∂J(0), and choosing ±1F (for anyRF ⊂ R2 with |F | <R+∞) in the subdifferential inequality, we infer that P (F ) ± R2 1F v > 0. Now, R2 f v = J(f ) rewrites Z +∞ Z Z 0 Z (t) (t) 0= P (F ) − v dt + P (F ) + v dt. F (t) 0 F (t) −∞ (t) Since the R integrands are nonnegative, we obtain that for a.e. t ∈ R, P (F ) = sign(t) F (t) v. In fact, the equality holds for all t 6= 0. Indeed, for t > 0, we R (tn ) may find a sequence t % t as n → +∞ such that P (F ) = v. Since n (t) F (tn ) F < +∞ and by monotonicity, 1F (tn ) converges in L2 R2 towards 1F (t) and we have ! Z Z \ (t) (tn ) (tn ) P (F ) = P F 6 lim inf P F = lim inf v= v. n∈N n→+∞ n→+∞ F (tn ) F (t) R The converse inequality holds from the fact that v ∈ ∂J(0) so that F (t) v 6 R P (F (t) ). In a similar way, we may prove that for all t < 0, P (F (t) ) = F (t) v. 14 R (ii) ⇒ (iii) From v ∈ ∂J(0), we infer that P (G) ± G v > 0 for any G ⊂ R2 R with |G| < +∞. Since P (F (t) ) − sign(t) F (t) v = 0, we obtain the claimed result. As a consequence of Proposition 3, if we are given v ∈ ∂J(f ) rather than f , we may control the localization of the support of Df simply by studying the solutions of (16). The following proposition formalizes this idea. Proposition 4. Let f ∈ L2 R2 with J(f ) < +∞, v ∈ ∂J(f ) and let Supp(Df ) denote the support of the Radon measure Df . Then [ ∂ ∗ F (t) ; t ∈ R \ {0} (20) Supp(Df ) = Z [ ⊆ v . ∂ ∗ F ; |F | < +∞ and P (F ) = ± (21) F Proof. Let x ∈ R2 \ Supp(Df ). There exists r > 0 such that |Df | (B(x, r)) = 0, hence f is constant in B(x, r), identically equal to some C ∈ R. Depending the value of t ∈ R, we see that either B(x, r) ⊆ F (t) or B(x, r) ∩ F (t) = ∅. In any S ∗ (t) case, ∂ ∗ F (t) ⊆ R2 \ B(x, r). As a result x ∈ R2 \ ∂ F ; t ∈ R , which S ∗ (t) ∂ F ; t ∈ R ⊆ Supp(Df ). proves that For the converse inclusion, let x ∈ Supp(Df ), so that for all r > 0, |Df | (B(x, r)) > 0. We apply the coarea formula Z ∞ P (F (t) , B(x, r))dt, |Df | (B(x, r)) = −∞ to see that H1 ∂ ∗ F (t) ∩ B(x, r) > 0 for some t ∈ R, hence o [n B(x, r) ∩ ∂ ∗ F (t) ; t ∈ R 6= ∅. S ∗ (t) Since this is true for all r > 0, we see that x ∈ ∂ F ; t∈R . Now, we prove the last inclusion in (21). First, we observe that [ [ ∂ ∗ F (t) ; t ∈ R = ∂ ∗ F (t) ; t 6= 0 ∪ ∂ ∗ F (0) . By Proposition 3, we know that for every t 6= 0, F (t) satisfies F (t) < +∞ and R ± F (t) v = P (F (t) ). Hence Z [ [ ∗ (t) ∗ ∂ F ; t 6= 0 ⊆ ∂ F ; |F | < +∞ and P (F ) = ± v , F and it is sufficient to prove that ∂ ∗ F (0) ⊆ [ ∂ ∗ F (t) ; t 6= 0 . 15 S S (−1/k) Let x ∈ ∂ ∗ F (0) = ∂ ∗ RN \ k∈N∗ F (−1/k) = ∂ ∗ . Then for k∈N∗ F all r > 0, [ [ F (−1/k) > 0 F (−1/k) > 0 and B(x, r) \ B(x, r) ∩ ∗ ∗ k∈N k∈N In there exists k0 such that B(x, r)∩ F (−1/k0 ) > 0, and moreover particular, S (−1/k) B(x, r) \ F (−1/k0 ) > B(x, r) \ > 0. Hence 1 (−1/k0 ) is not ∗ F F k∈N constant in B(x, r), so that ∂ ∗ F (−1/k0 ) ∩ B(x, r) 6= ∅. As a result, B(x, r) ∩ S ∗ (t) S ∗ (t) ∂ F ; t 6= 0 . ∂ F ; t 6= 0 = 6 ∅ for all r > 0, which proves that x ∈ 2.6 The prescribed mean curvature problem As a consequence of Propositions 3 and 4, we are led to study the solutions of the prescribed curvature problem Z min P (X) + H (22) X⊂RN |X|<+∞ X for H = ±v, where v ∈ ∂J(f ) is fixed. Following [18], if E ⊂ R2 is a solution to (22), we say that v is a variational mean curvature 1 for E. Depending on the integrability of H, the solutions of such a problem have the following regularity properties. Proposition 5 ([12]). Assume that H ∈ Lploc (RN ) for some p ∈ (N, +∞], and let E ⊆ RN be a nonempty solution of (22). Then Σ = ∂E \ ∂ ∗ E is a closed set of Haussdorf dimension at most N − 8, and ∂ ∗ E is a C 1,α hypersurface for all α < (p − N )/2p. If p = ∞, then ∂ ∗ E is C 1,α for all α > 0, and if additionally N = 2, then ∂ ∗ E is C 1,1 . Let us comment on the term variational curvature. Let x ∈ ∂ ∗ E. Up to a translation and rotation we may assume that ∂ ∗ F coincides locally with the graph of some C 1,α function ψ : B(0, r) → (−r, r) such that ∇ψ(0) = 0. If H is continuous in an open A, then it is possible to prove [12, Th. 1.1.3] that the “mean” curvature is equal to − N 1−1 H, 1 ∇ψ(z) = 1 H ((z, ψ(z))) , div q N −1 N −1 2 1 + |∇ψ(z)| 1 The careful reader will note that we make a slight abuse in the terminology since in [18, 1] the function H is assumed to be integrable, and the condition |F | < +∞ is not imposed. We make this slight abuse since the local properties of the sets studied in [18, 1] also hold for the solutions of (22). 16 in the sense of distributions. If N = 2, this equation holds in the classical sense and ∂ ∗ E ∩ A is in fact C 2 . The integrability p of H is crucial. For instance, if p = 1, it implies nothing on the regularity of E since every set of finite perimeter has a variational mean curvature in L1 [18]. The case p = N which we are interested in is a limit case, and counterexamples in [1] are provided where the Haussdorff dimension of ∂E \ ∂ ∗ E is more than N − 8. However, we may rely on the weak regularity theorem [1, Th. 3.6] (see also [62]) which ensures that for all x ∈ ∂F , def. 1 > DF (x) = lim+ r→0 |F ∩ B(x, r)| > 0. |B(x, r)| (23) Furthermore, in the case of N < 8, we have that DF (x) = 1/2. In particular, the topological boundary ∂F is equal to the essential boundary ∂M F , def. ∂F = ∂ M F = x ∈ R2 ; DF (x) > 0 and DF (x) > 0 , where DF (x) = lim sup r→0+ |F ∩ B(x, r)| |B(x, r)| and DF (x) = lim inf + r→0 |F ∩ B(x, r)| . |B(x, r)| Thus, in the case where p = N , although Proposition 5 cannot be applied, the boundary ∂F does not contain singular points such as cusps or points of zero density. In Section 4.2, we apply this result, observing that this weak regularity holds uniformly for the boundaries of the level sets of solutions to (Pλ (f + w)) in some low noise regime. 2.7 Decomposition of boundaries into Jordan curves We shall occasionally rely on the results on the decomposition of sets with finite perimeter provided in [13]. Let E be a set of finite perimeter. By [13, Corollary 1], E can be decomposed into an at most countable union of its M -connected components [ X E= Ei where P (E) = P (Ei ), |Ei | > 0. i∈I i∈I⊆N and each M -connected component can be decomposed as [ [ Ei = int(Ji+ ) \ int(Jj− ) and ∂ M Ei = Ji+ ∪ Jj− j∈Li (mod H1 ), j∈Li def. where each Jk± is a rectifiable Jordan curve, Li = j ∈ N ; int(Jj− ) ⊆ int(Ji+ ) . Here int(J) denotes the interior of a Jordan curve (but when the context is clear, we shall also use int to denote the topological interior). Moreover, X X 1 − P (E) = H1 Ji+ + H Jj and P (int Ji± ) = H1 Ji± for all i. i j 17 R Remark 4. Let E ⊂ R2 with |E| < +∞ such that P (E) = E v,S where v ∈ ∂J(0). Let us decompose E into its M -connected components, E = i∈I Ei , where we can assume that I is either N or of the form {0, 1, . . . , n}. We observe that {Ei }i∈I,i>1 yields the decomposition of E \ E0 into its M-connected components. Hence, Z 0 = P (E) − v E Z Z [ = P (E0 ) − v+P Ei − S v. E0 i∈I,i>1 i∈I,i>1 Ei S R R S Since P (E0 ) − E0 v > 0 and P v > 0, we deduce i∈I,i>1 Ei − i∈I,i>1 Ei that those inequalities are in fact equalities. By induction, we deduce that for all i ∈ I, Z P (Ei ) − v = 0. Ei Now decomposing, ∂ M Ei into rectifiable Jordan curves, this equivalent to ! ! Z Z X − + v v + P (int Jj ) + 0 = P (int Ji ) − int Ji+ int Jj− j∈Li R Since each Jordan curve J satisfies P (int J) − J v > 0, we see that for all i and j in the decomposition, Z Z v. v and P (int Jj− ) = − P (int Ji+ ) = Jj− Ji+ R Similarly, we may prove that if P (E) = − E v, Z P (Ei ) = − v, Ei Z Z P (int Ji+ ) = − v and P (int Jj− ) = 3 v. Jj− Ji+ Duality for the study of the low noise regime 3.1 Dual problems and “dual certificates” We are interested in solving: min u∈L2 (R2 ) where J(u) = R R2 J(u) + |Du| ∈ R+ ∪ {+∞}. 18 1 ||f − u||2L2 . 2λ (Pλ (f )) Using the framework and notations of [58], we set V = L2 R2 , Λ = Id, 1 k · −f k2 and we compute the Fenchel-Rockafellar Y = L2 R2 , F = J, G = 2λ dual problem as sup hf, vi − v∈∂J(0) or equivalently || inf v∈∂J(0) 1 ||v||2 2 , 2λ L (Dλ0 (f )) f − v||2L2 λ (Dλ (f )) It is easy to check that Problem (Pλ (f )) is stable in the sense of [58]. In particular, there exists a solution to(Dλ0 (f )) and strong duality holds between (Pλ (f )) and (Dλ (f )), namely inf (Pλ (f )) = sup (Dλ0 (f )). In fact (Dλ (f )) is a projection problem onto a nonempty closed convex set, hence it always has a unique solution. Observe that the limit of (Pλ (f )) as λ → 0+ is the trivial problem min u∈L2 (R2 ) J(u) s.t. u = f, (P0 (f )) having u = f as solution. The dual associated with this “exact reconstruction problem” is sup hf, vi, (D0 (f )) v∈∂J(0) having ∂J(f ) solutions. Here again, strong duality holds, since it is possible to prove that (D0 (f )) is stable. However, a solution to (D0 (f )) does not always exist since it may be that ∂J(f ) = ∅. The main point in studying the dual problems is that their solutions vλ are related to the primal solutions uλ by the extremality relations vλ ∈ ∂J(uλ ) 1 vλ = (f − uλ ), λ which enables to study the support of Duλ (see Section 2). For the noiseless problem, the extremality relation is v ∈ ∂J(f ), for every v solution to (D0 (f )). The term ”certificate” stems from the fact that if u ∈ L2 R2 and v ∈ L2 R2 satisfy the extremality relations, then u is a solution of the primal problem and v is a solution of the dual problem. 3.2 Low noise regimes and the minimal norm certificate We shall often consider noisy observations f + w, where w ∈ L2 R2 , and from now on we denote by uλ,w (resp. vλ,w ) the unique solution to Pλ (f + w) (resp. Dλ (f + w)). Given λ0 > 0, α0 > 0, we consider the low noise regime def. Dλ0 ,α0 = (λ, w) ∈ R+ × L2 R2 ; 0 6 λ 6 λ0 and ||w||L2 6 α0 λ . (24) 19 The dual solution vλ,w being the projection of (f + w)/λ onto a convex set, the non-expansiveness of the projection yieds ∀(λ, w) ∈ R∗+ × L2 R2 , ||vλ,0 − vλ,w ||L2 6 ||w||L2 6 α0 . λ As a result, the properties of vλ,w are governed by those of vλ,0 , and it turns out that the properties of vλ,0 are governed, in the low noise regime, by those of a specific solution to (D0 (f )), as the next result hints. The proof is identical to the one in [57]. Proposition 6. Let f ∈ L2 R2 , J(f ) < +∞, and assume that ∂J(f ) 6= ∅. Let v0,0 ∈ L2 R2 be the solution to (D0 (f )) with minimal L2 norm. Then lim+ vλ,0 = v0,0 λ→0 strongly in L2 R2 , We call v0,0 the minimal norm certificate for f . It is also known as the minimal section in maximal monotone operator theory. The goal of the present paper is to show that v0,0 governs the support of the solutions in the low noise regime. In particular, v0,0 determines whether the support of Duλ,w is close to the support of Df in that regime. In the next paragraphs, we illustrate the minimal norm certificate in simple cases. 3.3 The minimal norm certificate for calibrable sets Proposition 7 (Minimal norm certificates for calibrable sets). Let C ⊆ R2 be a bounded calibrable set and f = 1C . Then the minimal norm certificate is (C) . v0,0 = hC 1C , where hC = P|C| We provide two different proofs of the above result, each highlighting different aspects of the minimal norm certificate. Proof (v0,0 as a limit). From [21], we know that for a calibrable set C ⊆ R2 , the solution to (Pλ (f )) with f = 1C is given by uλ,0 = (1 − λhC )+ 1C . From the optimality conditions, vλ,0 = 1 (f − uλ,0 ), λ we obtain that vλ,0 = hC 1C provided 0 < λ 6 we obtain v0,0 = hC 1C . 1 hC . Taking the limit as λ → 0+ , Another Proof (v0,0 as a minimal norm element). Observe that for all f ∈ L2 R2 2 2 with J(f ) < +∞, and v ∈ L R , v is a solution to (D0 (f )) if and only if v ∈ ∂J(f ). For C ⊂ R2 bounded calibrable, we obtain that hC 1C is a solution to (D0 (f )). It remains to prove that it is the one with minimal norm. 20 Let v ∈ L2 R2 be any solution to (D0 (f )). By the Cauchy-Schwarz inequality p sup D0 (1C ) = hv, 1C i 6 ||v||L2 ||1C ||L2 = |C|||v||L2 . P (C) = But ||hC 1C ||L2 = √ |C| 3.4 sup D0 (1C ) √ |C| , so that hC 1C has minimal norm. The minimal norm certificate for smooth convex sets Let C be a nonempty open bounded convex subset of R2 . Given ρ > 0 we S denote by Cρ the opening of C by open balls with radius ρ, namely Cρ = B(y,ρ)⊆C B(y, ρ). For f = 1C , it is proved in [8, 10, 45] that the solution uλ,0 to (Pλ (f )) is + uλ,0 = (1 + λvC ) 1C , where, by letting R be such that CR function vC : R2 → R is defined by 1/R def. vC (x) = 1/r 0 is the maximal calibrable set in C, the x ∈ CR x ∈ ∂Cr , r ∈ [0, R) otherwise. Since vλ,0 = λ−1 (f − uλ,0 ), it follows that vC (x) x ∈ Cλ vλ,0 (x) = 1/λ x ∈ C \ Cλ 0 otherwise. (25) (26) Now we assume that C ⊂ R2 has C 1,1 boundary, and we let ρ0 > 0 such that κ∂C (x) 6 1 ρ0 for H1 -a.e. x ∈ ∂C, where κ∂C is the curvature of ∂C (defined H1 -almost everywhere on ∂C). We shall need the following lemma. Lemma 1 ([21]). Let C ⊂ R2 be a bounded open convex set. The following conditions are equivalent: • there exists ρ > 0 such that C = Cρ ; • ∂C is of class C 1,1 and ess supp∈∂C κ∂C (p) 6 ρ1 . Since for 0 < r 6 ρ0 , Cρ0 ⊆ Cr ⊆ C, we see that Cr = C for 0 < r 6 ρ0 . As a result, λ 7→ vλ,0 is constant on (0, ρ0 ], and the minimal norm certificate is thus v0,0 = vC . (27) 21 It turns out that v0,0 is precisely the subgradient constructed by Alter et al. [10] for the evolution of convex sets by the total variation flow. It is instructive to look at the associated vector field z0 such that div z0 = v0,0 For every x ∈ int(C) \ CR , there exists a unique r(x) such that x ∈ ∂Cr(x) , and x belongs to an arc of circle of radius r(x). Defining ν(x) as the outer unit normal to this set, define ν(x) if x ∈ int(C) \ CR def. zCR (x) if x ∈ CR z0 (x) = z(x) if x ∈ R2 \ C. where z is a calibration of R2 \ C (see Section 6.2). As for zCR , since CR is calibrable (CR is then the Cheeger set of C) there exists a vector field zCR such that |zCR | 6 1, θ(zCR , −D1CR ) = 1, and div zCR = hCR 1CR with hCR = P (CR )/ |CR |. It is proved in [10] that div z0 = vC (in the sense of distributions). It is notable that the construction is quite similar to the one proposed by Barrozzi et al. in [18] and studied in [1]. In particular, the L2 -minimality (or even Lp minimality) of the above constructions is already noted in [1]. 4 Properties of the level sets in the low noise regime In this section, we rely on the properties of the minimal norm certificate v0,0 to study the solutions of (21) for v = vλ,w in a low noise regime. More precisely we study the elements of Z def. 2 Fλ,w = E ⊂ R ; |E| < +∞, and ± vλ,w = P (E) , (28) E for (λ, w) ∈ Dλ0 ,α0 with λ0 > 0, α0 > 0 small enough. In the following, we denote by E or Eλ,w any nonempty element of Fλ,w . Let us emphasize that we allow the case (λ, w) = (0, 0), in which case vλ,w in (28) is the minimal norm certificate v0,0 . Typically, from Section 2, one may think of Eλ,w as a level set of uλ,w (or f , for (λ, w) = (0, 0)), but additional sets may solve (28). 4.1 Upper and lower bounds In the following lemmas, we prove that there exist uniform upper and lower bounds on the perimeters and the measures of all sets in Fλ,w with (λ, w) ∈ D1,√c2 /4 . √ Lemma 2. Let α0 6 c2 4 , where c2 = 4π is the isoperimetric constant. Then, sup {P (E) ; E ∈ Fλ,w , and (λ, w) ∈ D1,α0 } < +∞, (29) sup {|E| ; E ∈ Fλ,w , and (λ, w) ∈ D1,α0 } < +∞. (30) 22 Proof. First, we prove (29). Since limλ→0+ vλ,0 = v0,0 in L2 R2 , the mapping λ 7→ vλ,0 is continuous on the compact set [0, 1], hence bounded. Moreover, the 2 family {vλ,0 }06λ61 so that given any ε > 0, there exists R is L -equiintegrable 2 R > 0 such that R2 \B(0,R) vλ,0 6 ε2 for all λ ∈ [0, 1]. Let us also assume that ||w|| α0 6 ε (so that ||vλ,w − vλ,0 ||L2 6 λL2 6 ε). To simplify the notation, we denote by E (rather than Eλ,w ) any nonempty R set such that P (E) = ± E vλ,w . Now, the triangle and the Cauchy-Schwarz inequalities yield Z Z P (E) 6 (vλ,w − vλ,0 ) + vλ,0 E E Z Z p vλ,0 6 ε |E| + vλ,0 + E\B(0,R) E∩B(0,R) sZ p p p 2 6 ε |E| + |B(0, R)|||vλ,0 ||L2 + |E \ B(0, R)| vλ,w R2 \B(0,R) ! 6 ε + sup ||vλ,0 ||L2 p |B(0, R)| + 2ε p |E \ B(0, R)| λ∈[0,1] Recalling that P (E \B(0, R)) 6 P (E)+P (B(0, R)), and using the isoperimetric inequality, we obtain p 1 |E \ B(0, R)| 6 √ (P (E) + P (B(0, R))) , c2 √ where is c2 the isoperimetric pconstant. We choose ε = |B(0, R)| so as to get C = ε + supλ∈[0,1] ||vλ,0 ||L2 P (E) − c2 4 and we define 1 (P (E) + P (B(0, R))) 6 C. 2 We obtain that P (E) is uniformly bounded in E ∈ Fλ,w , (λ, w) ∈ D1,α0 . As for (30), the isoperimetric inequality yields |E| 6 1 (P (E))2 c2 hence |E| is uniformly bounded in E ∈ Fλ,w , (λ, w) ∈ D1,α0 . Conversely, the perimeters and areas of the solutions are also lower bounded, as the next result shows. √ √ c Lemma 3. Let α0 6 4 2 = π. Then, inf {P (E) ; E ∈ Fλ,w , E 6= ∅ and (λ, w) ∈ D1,α0 } > 0, (31) inf {|E| ; E ∈ Fλ,w , E 6= ∅ and (λ, w) ∈ D1,α0 } > 0. (32) 23 Moreover, there exists a number N0 ∈ N such that the number of M -connected components E and the number of Jordan curves in the essential boundary ∂ M E is uniformly bounded by N0 for all E ∈ Fλ,w , (λ, w) ∈ D1,α0 . Proof. By the L2 -equiintegrability of the family {vλ,0 }06λ61 , for all ε > 0, there exists δ such that for all E ⊂ R2 , with |E| 6 δ, Z 2 vλ,0 6 ε2 . E √ c We choose ε = 4 2 , 0 < α0 6 ε, and we consider by contradiction a set E ∈ Fλ,w such that 0 < |E| 6 δ. Then, Z Z P (E) 6 (vλ,w − vλ,0 ) + vλ,0 E E sZ p p 2 6 ||vλ,w − vλ,0 ||L2 |E| + vλ,0 |E| E p 1 6 2ε |E| 6 P (E), 2 by the isoperimetric inequality. Dividing by P (E) > 0 yields a contradiction, hence |E| > δ for all E 6= ∅, that is (32). We deduce the uniform lower bound on the perimeter (31) by the isoperimetric inequality. Now, let us decompose the essential boundary of E ∈ Fλ,w into at most countably many non trivial Jordan curves Ji+ , Jj− ; i ∈ I, j ∈ J, I ⊆ N, J ⊆ N . R By Remark 4 we know that for each σ ∈ {−1, 1} and j ∈ N, int J σ vλ,w = j H1 (Jjσ ), that is (int Jjσ ) ∈ Fλ,w . As a result H1 (Jjσ ) > µ, where µ is the infimum defined in (31) (in I, J, we only consider the non-trivial Jordan curves). Expressing the perimeter of E in terms of these Jordan curves, we get X X C > P (Eλ,w ) = H1 (Ji+ ) + H1 (Jj− ) > µ(Card I + Card J), i∈I j∈J where C is the supremum in (29). Hence the number of Jordan curves is at most C/µ, and the same holds for the number of M-connected components. Additionally, the next result shows that the level sets are uniformly contained in some large ball. √ √ c Lemma 4. Let α0 6 4 2 = π. Then, there exists R > 0 such that ∀(λ, w) ∈ D1,α0 , ∀E ∈ Fλ,w , E ⊂ B(0, R). Proof. We begin with the same equiintegrability argument as in Lemma 2, √ c2 choosing again ε = 4 . Now, let E ∈ Fλ,w . By the results of Section 2.7, 24 we may further decompose, up to an H1 -negligible set, its essential boundary ∂ M E into a countable union of Jordan curves J which satisfy Z ± vλ,w = P (int J) = H1 (J) int J Assume by contradiction that J is such that (int J) ∩ B(0, R) = ∅. Then by the isoperimetric inequality, sZ p 2ε 2 vλ,w |int J| 6 √ P (int J). P (int J) 6 c2 2 R \B(0,R) √ c Dividing by P (int J) yields a contradiction for ε = 4 2 if J is not trivial. Hence (int J) ∩ B(0, R) 6= ∅. But the uniform bound (29) also holds for J, hence there is some C > 0 (independent from (λ, w) ∈ D1,α0 ) such that H1 (J) 6 C. As a result, diam(int J) 6 C so that (int J) ⊂ B(0, R + C), and since this holds for any J which is involved in the decomposition of ∂ M E, it also holds for all E ∈ Fλ,w , uniformly in (λ, w) ∈ D1,α0 . Remark 5. LetR us divide Fλ,w into two R classes corresponding respectively to the condition E vλ,w = P (E) and − E vλ,w = P (E) (the empty set being the only element which belongs to both). A consequence of (30) is that each class is stable by finite or countable Runion or intersection. Indeed, if E and F are two elements of Fλ,w such that E vλ,w = P (E) (and similarly for F ), the submodularity of the perimeter yields Z Z Z Z P (E∩F )+P (E∪F ) 6 P (E)+P (F ) = vλ,w + vλ,w = vλ,w + vλ,w . E F E∩F E∪F Using the subdifferential inequality (on vλ,w ∈ ∂J(0)) we obtain that P (E∩F ) = R R v and P (E ∪ F ) = E∩F λ,w R finite union or R E∪F vλ,w . Iterating, Sn Tn we get for intersection P ( k=1 Ek ) = Sn Ek vλ,w and P ( k=1 Ek ) = Tn Ek vλ,w . The k=1 k=1 lower semi-continuity of the perimeter together with |E1 | < +∞ yields ! ! Z Z ∞ n \ \ P Ek 6 lim inf P Ek = lim T vλ,w = T vλ,w , k=1 n→+∞ k=1 n→+∞ n k=1 Ek ∞ k=1 Ek and the converse inequality holds by the subdifferential S∞ Sn inequality. As for the union, we know from (30) that | k=1 Ek | = supn∈N | k=1 Ek | < +∞, hence ! ! Z Z ∞ n [ [ P Ek 6 lim inf P Ek = lim S vλ,w = S vλ,w , k=1 n→+∞ k=1 n→+∞ n k=1 Ek ∞ k=1 Ek and the opposite inequality also holds, for the same reason as above. 25 4.2 Weak regularity In this section, we show that (23) holds uniformly on the boundaries of the sets in Fλ,w with (λ, w) ∈ D1,√c2 /4 . The proof of Proposition 8 is in fact almost identical to the proof of [62, Lem. 1.2], however, it is included for the sake of completeness, and so as to emphasize the uniformity of this estimate with respect to (λ, w). Proposition 8. There exists r0 > 0 such that for all r ∈ (0, r0 ] and Eλ,w ∈ Fλ,w with (λ, w) ∈ D1,√c2 /4 , |B(x, r) ∩ Eλ,w | 1 |B(x, r) \ Eλ,w | 1 > and > . (33) |B(x, r)| 16 |B(x, r)| 16 R Proof. We give the proof for P (Eλ,w ) = Eλ,w vλ,w , the other case being similar. Since {vλ,0 }λ∈[0,1] is equiintegrable, there there exists r0 > 0 such that for all subsets E ⊂ R2 with |E| 6 πr02 , ∀x ∈ ∂Eλ,w , Z |vλ,0 | 2 √ 1/2 c2 . 4 6 E (34) First observe that by optimality of Eλ,w , Z Z P (Eλ,w ) − vλ,w 6 P (E \ B(x, r)) − vλ,w (35) Eλ,w \B(x,r) Eλ,w which leads to 1 Z ∗ vλ,w 6 H1 (∂Br ∩ Eλ,w ). H (∂ Eλ,w ∩ B(x, r)) − Eλ,w ∩Br By adding H1 (∂B(x, r) ∩ Eλ,w ) to both sides, it follows that Z P (Eλ,w ∩ B(x, r)) − vλ,w 6 2H1 (∂B(x, r) ∩ Eλ,w ). Eλ,w ∩B(x,r) By the Cauchy-Schwarz inequality, (34) and since ||vλ,w − vλ,0 ||L2 6 √ P (Eλ,w ∩ B(x, r)) − √ c2 /4 1/2 c2 |Eλ,w ∩ B(x, r)| 2 6 2H1 (∂B(x, r) ∩ Eλ,w ). The isoperimetric inequality then implies that √ 1/2 6 4H1 (∂B(x, r) ∩ Eλ,w ). c2 |Eλ,w ∩ B(x, r)| Let g(r) = |Eλ,w ∩ B(x, r)|. Then g(r) > 0 since x ∈ ∂Eλ,w , and for a.e. r, g 0 (r) = H1 (∂B(x, r) ∩ Eλ,w ). Therefore, for a.e. r ∈ (0, r0 ], √ c2 6 8 dp g(r). dr 26 By integrating on both sides, p √ r c2 6 8 g(r). and the first inequality in (33) follows by recalling that c2 = 4π. The proof of |B(x, r) \ Eλ,w | > |B(x, r)| /16 is similar: instead of comparing Eλ,w with Eλ,w \ B(x, r) in (35), simply compare Eλ,w with Eλ,w ∪ B(x, r) and proceed as before. 5 The extended support Let f ∈ L2 R2 , with J(f ) < +∞, such that ∂J(f ) 6= ∅, or equivalently that (D0 (f )) has a solution (source condition). Let v0,0 be the corresponding minimal norm certificate and let us define the extended support as def. Ext(Df ) = [ {Supp Dg ; v0,0 ∈ ∂J(g)} As we shall see in Section 7, the extended support governs the location of Supp(Duλ,w ) for (λ, w) in some low noise regime. 5.1 Properties of the extended support A first remark in view of Proposition 4 is that we may rewrite the extended support as Ext(Df ) = [ ∂∗E Z ; |E| < +∞ and ± v0,0 = P (E) (36) E The first inclusion is clear by Proposition 4. The converse inclusion is obtained by considering, for any E in the right hand-side, the function g = 1E , so as to have g ∈ L2 and v0,0 ∈ ∂J(g) (since ∂ ∗ E = Supp(Dg)). From the above equalities, we see that all the properties of Section 4 (lower and upper boundedness of the perimeter, uniform boundedness. . . ) hold for the elements of the right hand-side whose union determines the extended support. The rest of the section is devoted to examples of minimal certificates, in the case of indicator function of convex calibrable sets or more general convex sets. 5.2 Convex Calibrable sets Let C ⊂ R2 be a bounded convex calibrable set. We wish to describe the extended support of f = 1C . This may be done by looking at a vector field z with divergence v0,0 (see Section 6), which is more informative, or by the following approach. 27 By Proposition 7, we know that the minimal norm certificate associated to (C) f = 1C is v0,0 = hC 1C , where hC = P|C| . By (36), we are thus led to solve inf P (E) − hC |E ∩ C| , (37) inf P (E) + hC |E ∩ C| . (38) E⊂R2 |E|<+∞ and E⊂R2 |E|<+∞ Problem (38) is trivial and its only solution is ∅, so that we only focus on (37). By Proposition 2, we see that E = C is a minimizer. Moreover, since C is convex, for all E with finite perimeter P (C ∩ E) 6 P (E) with strict inequality whenever |E \ C| > 0. As a result, any other solution must satisfy E ⊆ C. But with this condition, either E = ∅ or E is a solution to the Cheeger problem min E⊆C P (E) . |E| The uniqueness of the solution to the Cheeger problem inside any convex set is proved in [61, 7], and we already know that C is optimal. As a result, either E = ∅ or E = C, and eventually Ext(Df ) = ∂C. 5.3 Smooth convex sets Let C ⊂ R2 be a bounded open convex set with C 1,1 boundary. We describe the extended support of f = 1C by considering the minimal norm certificate v0,0 defined in (25). We need to study the solutions of Z inf 2 P (E) − v0,0 , (39) E⊂R |E|<+∞ E Z and inf E⊂R2 |E|<+∞ P (E) + v0,0 . (40) E Since v0,0 > 0, we see that the only solution to (40) is ∅. As for (39), the same convexity argument as above shows that any solution must be included in C. Now let r ∈ [ρ0 , R], where 1/ρ0 > ess supx∈∂C κ(x) and Cr be the opening of C with radius r as defined in Section 3.4. Denoting by νCr the outer unit normal to ∂Cr , we have Z Z Z P (Cr ) = z0 · νCr dH1 = div z0 = v0,0 , ∂Cr Cr Cr S hence Cr is a solution to (39), hence Ext(Df ) ⊇ {∂Cr ; ρ0 6 r 6 R}. Let us prove that there is no solution E such that the reduced boundary ∂ ∗ E intersects CR . By Remark 5, the solutions to (39) are stable by intersection. 28 If a solution E is such that E ∩ CR 6= ∅, then P (E ∩ CR ) = hCR |E ∩ CR | where hCR = problem P (CR ) |CR | R E∩CR v0,0 = and E ∩ CR is a solution to the Cheeger min F ⊆CR P (F ) . |F | By uniqueness of the Cheeger set of CR , we obtain that E ∩ CR = CR . Eventually, we have proved Ext(Df ) = 1A [ {∂Cr ; ρ0 6 r 6 R}. Ext(D1A ) 1B (41) Ext(D1B ) Figure 1: Examples of the extended support for two indicator functions. 6 Non-Degeneracy for calibrable sets The aim of this section is to show that if C ⊂ R2 is a convex calibrable set, (C) the minimal norm certificate v0,0 = hC 1C (where hC = P|C| ) can be written as v0,0 = div z0 where z0 ∈ X2 R2 , (z, D1C ) = − |D1C | and for every compact set K ⊂ R2 \ ∂C, ess sup |z0 | < 1, K with an estimation on that inequality. We do not aim at full generality, and we assume that ∂C is of class C 2 for the sake of simplicity. Reducing the hypotheses is the subject of future work. The proof relies on the notion of inner and outer calibrations described in [21], which amounts to constructing vector fields ”inside” and ”outside” the studied set, and then “glue” the two constructions. Definition 2. Let C ⊆ R2 be a set of finite perimeter. We say that C is − −calibrable if there exists a vector field zC : R2 → R2 such that − − 1. zC ∈ L2loc (R2 , R2 ) and div zC ∈ L2loc (R2 ); − 2. |zC | 6 1 almost everywhere in C; − 3. div zC is constant on C; 4. θ(z − , −D1C )(x) = −1 for H1 -almost every x ∈ ∂ ∗ C. 29 Similarly C is +calibrable if 1), 2), 3) hold and θ(z + , −D1C )(x) = +1 in 4). The following lemma tells that one may ”glue” calibrations: Lemma 5 ([21]). Let C be a bounded set of finite perimeter. Then v = 1C is − calibrable if and only if C is −calibrable with − div ξC = hC in C, and R2 \ C + 2 is +calibrable with div zR2 = 0 in R \ C, defining − zC on C, def. z = + zC on R2 \ C. 6.1 Inner calibrations Let C ⊂ R2 be a bounded open convex set of class C 2 , and hC = P (C) C . Following [11] in order to build the calibration, we consider the following auxiliary problem: def. ∇u = hC . div q 2 1 + |∇u| and we define ( def. z = √ ∇u 1+|∇u|2 νC (42) on C (43) on ∂C Giusti proved the following result in [61] (see also [11, Prop.6.2]) Theorem 1 ([61]). There exists a solution u ∈ C 2 (C) to (42) if and only if ∀B ( C, B 6= ∅, hC < P (B) . |B| That solution u is unique up to an additive constant, bounded from below in C, and its graph is vertical at the boundary of C, in the sense that ∇u q 2 → νC uniformly on ∂C. 1 + |∇u| The consequence is that z defined in (43) is a C 1 vector field in C, (in fact analytic, see [11]), continuous in C. In fact, Giusti also proved that the condition (1) is equivalent to (14), namely the calibrability of C (this result was extended to RN in [7]). As a result, for a calibrable set C, one may choose the calibration given by the vectorfield z such that ∇u(x) ∀x ∈ C, z(x) = q 2 1 + |∇u(x)| and z|R2 \C is a vectorfield such that ||z||∞ 6 1, div z = hC and θ(z, D1C ) = −1. A first step in proving that |z| < 1 inside C is the following theorem by Giusti. 30 Theorem 2 ([61]). For every compact set K ⊂ int C, there exist exists Q > 0 such that for any solution of (42) in int C, sup |∇u| 6 Q. K This implies that supK √ |∇u| 1+|∇u|2 < 1. In the next proposition, we study further its decay inside C, which yields a non-degenerate inner calibration for C. Proposition 9. Let C ⊂ R2 be a bounded strictly convex calibrable set such that ∂C is of class C 2 and hC > sup∂C |κ∂C |. Assume moreover that the solution to (42) is continuous up to the boundary, i.e. u ∈ C (C). Then, there exists α > 0, there exists a vector field z ∈ C (Ω) ∩ C 1 (Ω) such that div z = hC , z · ν = 1 on ∂C, and α ∀x ∈ C, |z(x)| 6 p , d(x)2 + α2 def. where d(x) = dist(x, ∂C). Proof. By Theorem 1, there exists a C 2 solution u to (42) which is vertical at the boundary, and the inequality hC > ess sup∂C |κ∂C | implies that u is bounded def. (see [61, Th. 3.1]). We define z(x) = √ ∇u(x) 2 for all x ∈ C. 1+|∇u(x)| Let us prove that |∇u(x)| > 0 for a.e. x ∈ C. First, we assume that C is strictly convex. Since u ∈ C 2 (Ω) ∩ C (Ω), by [72, Th. 2.2] u is a convex function. As a result, {x ∈ C ; ∇u(x) = 0} = argminC u, and it is thus a closed convex set. Assume by contradiction that the dimension of argminC u is 2, i.e. argminC u contains an open ball B(x0 , r) ⊂ int(C) for some x0 ∈ int(C), r > 0. Let T denote the operator T : u 7→ √ ∇u 2 , and let w be the constant 1+|∇u| function x 7→ minC u. We have u 6 w in ∂B(x0 , r) (in fact equality holds), and 0 = div T w < div T u = hC in B(x0 , r). By Theorem 2, Problem (42) is locally uniformly elliptic, and the comparison principle [59, Th. 10.1] yields that u < w in B(x0 , r), which is a contradiction. As a result, the dimension of argminC u is strictly less than 2 and argminC u is Lebesgue-negligible. def. ∇u(x) , and we observe Now, for a.e. x ∈ int(C), we may define y = x + d(x) |∇u(x)| that y ∈ C. By convexity of u, u(y) − u(x) > ∇u(x) · (y − x) = d(x) |∇u(x)|. ∞ As a result, |∇u(x)| 6 2||u|| d(x) , and 2||u||∞ |T u(x)| 6 p . d(x)2 + 4||u||2∞ The claimed result holds by a density argument. 6.2 Outer calibrations It is proved in [21, Th. 5] (see also [8, Th. 13] in dimension N ) that sets which satisfy a geometric condition (namely convex sets that are far enough 31 from one another) have a complement which is +calibrable. That condition holds for C 1,1 convex sets. However, it is not clear from the proof that the corresponding vector field has norm < 1 in compact sets of R2 \ C. We provide below an explicit construction when the set has C 2 boundary. Admittedly the hypothesis is quite restrictive but we think that this construction gives some insight on the geometric properties involved. Proposition 10. Let C ⊂ R2 be a nonempty bounded open convex subset with C 2 boundary. There exists a vector field z ∈ L∞ ∩ C (R2 \ C) such that z = ν on ∂C, div z = 0 in the sense of distributions and |z| < 1 on every compact subset of R2 \ C. The decay of z is discussed in Remark 6 below. Proof. We choose an arclength parametrization of ∂C, s 7→ y(s) defined on def. S = R/(P (C)Z), and we consider a basis (τ (s), ν(s)) such that τ (s) = y 0 (s), ν(s) = R−π/2 τ (s), where R−π/2 the rotation with angle −π/2. We assume that the parametrization is such that ν(s) is the outer unit normal to C. The mapping ϕ : (s, d) 7→ y(s) + dν(s) is a C 1 -diffeomorphism from S × R∗+ onto R2 \ C, with ∂ϕ (s, d) = τ (s) + dκ(s)τ (s), ∂s and ∂ϕ (s, d) = ν(s), ∂d where κ(s) > 0 is the curvature of ∂C at y(s). In order to define a vector field z : R2 \ C → R2 such that div z = 0, it is sufficient to define a vector field z : S × R∗+ → R2 such that z(x) = z(ϕ−1 (x)) and Tr(DzDϕ−1 ) = 0. In other words, we shall build a vector field z such that 1 κ(s) ∂s z1 (s, d) + z2 (s, d) + ∂d z2 (s, d) = 0. 1 + κ(s)d 1 + κ(s)d (44) Here, for the sake of brevity, we have denoted by ∂s (resp. ∂d ) the derivatives with respect to s (resp. d), and by (z1 , z2 ) the coordinates of z in the basis (τ (s), ν(s)). Given α > 0 (to be fixed later), and the function η : t 7→ min(t, 2 − t), we define Z s 2π z1 (s, d) = −α (κ(s0 ) − )ds0 η(d), (45) P (C) 0 ! Z d ! 2π 1 z2 (s, d) = 1+α η κ(s) − . (46) 1 + κ(s)d P (C) 0 32 Observe that lim(s,d)→(s0 ,0) z(s, d) = ν(s0 ), and that z is continuous in R2 \ C R P (C) since 0 κ(s0 )ds0 = 2π. Moreover, it is not difficult to check that z satisfies (44) as well. As a result, div z = 0 pointwise in R2 \ C, and since z is continuous we see by approximation that it also holds in the sense of distributions. 2 It remains to prove that |z| − 1 < 0. 2 Z s 2π 2 2 2 (κ − ) η2 z1 + z2 − 1 = α P (C) 0 !2 Z d ! 1 2π 1+α η − 1 − (κd)2 − 2κd + κ− (1 + κd)2 P (C) 0 There is a constant M > 0 which only depends on sup∂C κ and P (C) such that 2 R 2 s 2π 2π ) 6 M and − (κ − κ P (C) P (C) 6 M . 0 The term inside brackets is equal to 2 Z d !2 Z d ! 2π 2π 2 α + 2α − (κd)2 − 2κd η κ− η κ− P (C) P (C) 0 0 ! Z d !2 Z d ! Z d 4πα 2 6α M η + 2κ α η −d − η − (κd)2 P (C) 0 0 0 ! Z d Z d ! 4πα 6 2κ α η − d + α2 M − η − (κd)2 , P (C) 0 0 Rd since 0 η 6 1. Hence, for d 6 1, we obtain that for α small enough (depending on M and P (C)), that term is less than or equal to −κd − 2πα d2 − (κd)2 6 0, P (C) 2 def. which yields (writing K = sup∂C κ) 1 2πα d2 z12 + z22 − 1 6 α2 M d2 + −κd − − (κd)2 1+K P (C) 2 2 1 πα d 6− κd + + (κd)2 < 0, 1+K P (C) 2 (47) for α > 0 small enough (depending on M , K and P (C)). R +∞ As for d > 1, we may assume that α is small enough so that α 0 η 6 R1 Rd 1/2 6 d/2. Moreover, 0 η > 0 η = 1/2, so that the term inside brackets is less than or equal to πα −κd − − (κd)2 . P (C) 33 1 κd + 1 + κd 1 6 α2 M − κd + 1 + κd z12 + z22 − 1 6 α2 M − πα + (κd)2 P (C) πα P (C) def. x+a For a = Pπα (C) < 1, the mapping x 7→ − x+1 is (strictly) decreasing on [0, +∞), hence upper bounded by −a, and we obtain that z12 + z22 − 1 6 α2 M − Pπα (C) < 0 for α small enough. Remark 6. A more straightforward construction would have been to construct z parallel to the normals to C, or equivalently set α = 0 in (45) and (46). However, such a vector field would not decay in front of flat areas, where κ(s) = 0, and we would have |z(s, d)| = 1 for all d > 0. The above construction “twists” the field lines so as to obtain some decay of the norm. Still, the resulting upper bound (47) for small d depends on the local curvature of ∂C. If κ(s) > 0, then, as d → 0+ , 2 |z(s, d)| 6 1 − κ(s) d + o(d) 1+K On the other hand, if κ(s) = 0, then 2 |z(s, d)| 6 1 − 7 d2 πα . (1 + K)P (C) 2 Support stability outside the extended support Let f ∈ L2 R2 with J(f ) < +∞, such that (D0 (f )) has a solution (source condition), and let v0,0 be the corresponding minimal norm certificate. In this section, we prove the main result of this paper, Theorem 3 which shows that, as λ → 0 and ||w||L2 /λ → 0, almost all topological boundaries of the level sets of the solutions to (Pλ (f + w)) converge towards the topological boundaries of the corresponding level sets of f in the sense of Hausdorff convergence, and that the support of Duλ,w is contained in arbitrarily small tubular neighborhoods of the extended support Ext(Df ). Moreover, in Section 7.2, we show that the width of this tube can be further characterized through knowledge of the vector field z0 associated with v0,0 . We also observe that an interesting consequence of our main result is that the minimal norm certificate v0,0 is constant on each connected component of the extended support. Throughout this section, we denote by vλ,w the solution of (Dλ (f + w)) and R let Eλ,w be any set of finite perimeter such that Eλ,w vλ,w = P (Eλ,w ). We also denote by uλ,w the solution of Pλ (f + w). Finally, let the level sets of f be denoted by F (t) (refer to (7) for the definition of level sets). 34 Theorem 3. Let {wn }n∈N , {λn }n∈N be sequences such that wn ∈ L2 R2 , ||wn ||L2 λn λn & 0, and uλn ,wn . Then, (t) → 0 as n → +∞. Let us denote by Un the level sets of Supp(Df ) ⊆ lim inf Supp(Duλn ,wn ) ⊆ lim sup Supp(Duλn ,wn ) ⊆ Ext(Df ). n→+∞ n→+∞ (48) Moreover, for almost every t ∈ R, lim Un(t) ∆F (t) = 0, n→+∞ and lim ∂Un(t) = ∂F (t) , n→+∞ (49) and the last equality holds in the sense of Hausdorff convergence. Remark 7. It is possible to reformulate (48) in the following way. By Lemma 4, there exists R > 0 such that for all n, Supp(Duλn ,wn ) ⊆ B(0, R) and Ext(Df )) ⊆ B(0, R) so that by [82, Thm. 4.10]), (48) is equivalent to • (outer limit inclusion) for all r > 0, there exists n0 ∈ N such that, def. ∀n > n0 , Supp(Duλn ,wn ) ⊆ Tr = x ∈ R2 ; dist(x, Ext(Df )) 6 r . • (inner limit inclusion) for all r > 0, there exists n1 ∈ N such that, ∀n > n1 , Supp(Df ) ⊆ x ∈ R2 ; dist(x, Supp(Duλn ,wn )) 6 r . The second equation of (49) has a similar reformulation. Proof. We begin by proving lim supn→+∞ Supp(Duλn ,wn ) ⊆ Ext(Df ). Let (xn )n∈N such that xn ∈ Supp(Duλn ,wn ) and (up to the extraction of a subsequence) limn→+∞ xn = x for some x ∈ R2 . By Proposition 4, it is not restrictive to assume that xn ∈ ∂En for some En ∈ Fλn ,wn (otherwise we may replace xn with yn ∈ ∂En such that |xn − yn | 6 1/n). By Lemma 2 and 4, the family {En }n∈N is precompact in the L1 topology (see [74, Thm. 12.26]), that is, there exists E ⊆ R2 with finite measure such that, up to the extraction of a subsequence, limn→+∞ |E4En | = 0 (we do not relabel the subsequence). Moreover, up to theR additional extraction of a subsequence, we may assume that either for all n, En vλn ,wn = P (En ), or for R all n, En vλn ,wn = −P (En ). We deal with the first case, the other being similar. Passing to the limit in the optimality equation for En , we get Z Z P (E) 6 lim inf P (En ) = lim vλn ,wn = v0,0 , (50) n→+∞ n→+∞ En E by the lower semi-continuity of the perimeter, and since 1En (resp. vλn ,wn ) converges strongly in L2 R2 towards 1E (resp. v0,0 ). Since v0,0 ∈ ∂J(f ) ⊆ R ∂J(0), the converse inequality also holds, so that P (E) = E v0,0 , and E ∈ F0,0 . By definition of the extended support, this means that ∂ ∗ E ⊆ Ext(Df ), hence ∂E ⊆ Ext(Df ). 35 Then, by Proposition 8, for all r0 6 r0 , and for n large enough so that (λn , wn ) ∈ D1,√c2 /4 , we see that, 1 |B(xn , r0 )| , 16 and |B(xn , r0 ) \ En | > 1 |B(x, r0 )| , 16 and |B(x, r0 ) \ E| > 1 |B(xn , r0 )| . 16 (51) By the dominated convergence theorem, we obtain for n → +∞, |B(xn , r0 ) ∩ En | > |B(x, r0 ) ∩ E| > 1 |B(x, r0 )| . 16 Since this holds for all r0 ∈ (0, r0 ], we see that x ∈ ∂E. As a result we have proved the claimed result. (t) Now, we observe that limn→+∞ Un ∆F (t) = 0 for a.e. t ∈ R is a straight forward consequence of the convergence of uλn ,wn towards f in L2 R2 . Let us (t) fix such t. The same argument as above proves that lim supn→+∞ ∂Un ⊆ ∂F (t) . (t) To prove ∂F (t) ⊆ lim inf ∂Un , we consider x ∈ ∂F (t) . By Proposition 8, for all 0 0 < r 6 r0 , 1 1 |B(x, r0 )| , and B(x, r0 ) \ F (t) > |B(x, r0 )| . B(x, r0 ) ∩ F (t) > 16 16 By L1 convergence of 1U (t) towards F (t) , we see that for each r0 ∈ (0, r0 ], for n (t) (t) all n large enough, B(x, r0 ) ∩ Un > 0 and B(x, r0 ) \ Un > 0. This implies (t) that lim supn→+∞ dist(x, ∂Un ) 6 r0 , and since this is true for all r0 ∈ (0, r0 ], (t) we obtain x ∈ lim inf n→+∞ ∂Un . This immediately yields Supp(Df ) ⊆ lim inf n→+∞ Supp(Duλn ,wn ), observing that in (20) we may remove a set of t with empty interior. Remark 8 (On dimensions N > 3). The are two key elements to the proof of Theorem 3: 1. Compactness. Lemma 2 and 4 which gives that there exists R, L > 0 such that P (En ) < L and the fact that there exists R such that En ⊂ B(0, R). This allows the required compactness result to be applied. 2. Weak Regularity. Proposition 8 which ensures that the boundaries of all level sets are uniformly weakly regular. The difficulty with extending Theorem 3 to higher dimensions is that the second property of weak regularity is no longer true: In dimension N , for weak regularity, we would require that lim λ,||w||L2 /λ→0 ||vλ,w − v0,0 ||LN = 0. However, the natural topology for {vλ,w }λ,w is L2 (RN ) and when N > 3, there is no guarantee that the boundaries of the level sets of uλ,w will have arbitrarily 36 many singular points such as cusps, and it may be the case that there are level sets of uλ,w arbitrarily far out with arbitrarily small measure and perimeter. When N > 3, it is still true that there exists L such that P (E) 6 L for all E ∈ Fλ,w with (λ, w) ∈ D1,√c2 /4 and it is possible to adapt the argument in the proof of Theorem 3 to conclude that for each r > 0, lim sup H1 (∂ ∗ E \ Tr ) ; E ∈ Fλ,w , (λ, α) ∈ Dα0 ,λ0 = 0. (λ0 ,α0 )→(0,0) However, we have no guarantee that there exists λ0 , α0 > 0 such that HN −1 (∂ ∗ E\ Tr ) = 0 for all E ∈ Fλ,w with (λ, w) ∈ Dλ0 ,α0 . 7.1 The minimal norm certificate outside the extended support Corollary 1. Let Ω be any connected component of R2 \ Ext(f ). Then v0,0 is constant on Ω. Proof. For δ > 0, let Ωδ = {x ∈ Ω : dist(∂Ω, x) > δ}. From Theorem 3, we know that for all δ, there exists λδ > 0 such that for all λ ∈ Dλδ ,0 , uλ,0 is constant on Ωδ . Since vλ,0 = (f − uλ,0 )/λ, it follows that vλ,0 is also constant on Ωδ . So, since v0,0 is the L2 limit of vλ,0 , v0,0 must be constant on Ωδ for all δ > 0. Therefore, v0,0 is constant on Ω. 7.2 Support stability and calibrations Theorem 3 shows that as a result of the strong L2 convergence of vλ,w to v0,0 , one is guaranteed support stability outside a small neighbourhood of the extended support. This section upper bounds the rate of convergence in the outer limit inclusion of (48). In particular, we make explicit the relationship between the width of this neighbourhood, the decay of ||vλ,w − v0,0 ||L2 and the nondegeneracy of z0 , the vector field for which v0,0 = − div z0 . . In this section, we define def. Tr = x ∈ R2 ; dist(x, Ext(Df )) 6 r , we make an additional assumption about the decay of z0 away from the extended support: def. δr = 1 − ess sup |z0 (x)| > 0. (52) x∈TrC We also let C be such that |E| 6 C 2 , ∀ E ∈ Fλ,w , (λ, w) ∈ D1,√c2 /4 . Recall that the existence of C is guaranteed by Lemma 2. Examples of vector fields whose decay is known outside the extended support are described in Section 6. 37 Proposition 11. Given any E ∈ Fλ,w with (λ, w) ∈ D1,√c2 /4 , Z δr H1 (∂ ∗ E \ Tr ) 6 (vλ,w − v0,0 ) 6 C||vλ,w − v0,0 ||L2 , E Proof. Comparing the energy of E with that of the empty set we get, Z Z Z P (E) 6 vλ,w = (vλ,w − v0,0 ) + v0,0 E E E Z p z0 · ν. 6 ||vλ,w − v0,0 ||L2 |E| + ∂∗E Recall from Lemma 2 that there exists C > 0 such that |E| 6 C 2 . So, Z Z P (E) − z0 · ν − z0 · ν 6 C||vλ,w − v0,0 ||L2 . ∂ ∗ E∩Tr ∂ ∗ E\Tr Since ess supRN \Tr |z0 | 6 1 − δr , and more generally ||z0 ||L∞ 6 1, Z Z z0 · ν − P (E) − ∂ ∗ E\Tr ∂ ∗ E∩Tr z0 · ν > H1 (∂ ∗ E \ Tr ) + H1 (∂ ∗ E ∩ Tr ) − (1 − δr )H1 (∂ ∗ E \ Tr ) − H1 (∂ ∗ E ∩ Tr ) > δr H1 (∂ ∗ E \ Tr ) , hence the claimed result. Proposition 12. Let λ > 0 and w ∈ L2 R2 be such that ||vλ,w − v0 ||L2 < √ δr/2 c2 . Then, E ∈ Fλ,w and P (E) > 0 implies that H1 ∂ ∗ E ∩ Tr/2 > 0. Proof. Assume by contradiction that P (E) > 0 and H1 ∂ ∗ E ∩ Tr/2 = 0 so c that ∂ ∗ E ⊂ Tr/2 up to an H1 -negligible set. Then, Z P (E) = vλ,w 6 ||E||1/2 ||vλ,w − v0,0 ||L2 + E 6 Z v0,0 E P (E)||vλ,w − v0,0 ||L2 + (1 − δr/2 )P (E) √ c2 where we have applied the isoperimetric inequality and the fact that v0,0 = div z0 c with |z0 | 6 (1 − δr/2 ) on Tr/2 . Since P (E) > 0, this implies that √ δr/2 c2 6 ||vλ,w − v0,0 ||L2 . This contradicts the assumption of this proposition. 38 Theorem 4. Let r > 0. If (λ, w) ∈ D1,√c2 /4 are such that ||vλ,w − v0 ||L2 6 δr/2 min n r √ o , c2 , 2C (53) then for all level sets E of uλ,w , ∂E ⊆ Tr . Proof. It is sufficient to show that H1 (∂ ∗ E \ Tr ) = 0 for all E ∈ Fλ,w with λ > 0 2 2 and w ∈ L R satisfying (53). For, if we have H1 (∂ ∗ E \ Tr ) = 0, this means that 1E is constant on every connected component of the open set R2 \ Tr , hence the topological boundary satisfies ∂E ⊆ Tr . Furthermore, by Section 2.7, we may assume that up to an H1 -negligible set, ∂ ∗ E is equivalent to a Jordan curve J. First observe that by Proposition 12, H1 (∂ ∗ E ∩ Tr/2 ) > 0. Now, for a contradiction, suppose that H1 (∂ ∗ E \ Tr ) > 0. Then since this implies that H1 (J ∩ Tr/2 ) > 0, H1 (J \ Tr ) > 0 and J is a continuous curve, it follows that H1 (J \ Tr/2 ) = H1 (∂ ∗ E \ Tr/2 ) > r/2. However, this is a contradiction Proposition 11 implies that lim sup H1 (∂ ∗ E \ Tr ) ; E ∈ Fλ,w , (λ, α) ∈ Dα0 ,λ0 = 0. (λ0 ,α0 )→(0,0) Indeed, by our choice of (λ, w) in (53), if H1 (∂ ∗ E \Tr ) > 0, then the combination of Proposition 11 and Proposition 12 yields rδr/2 rδr/2 < H1 ∂ ∗ E \ Tr/2 6 C||vλ,w − v0,0 ||L2 6 . 2 2 Example In the case where f = 1B(0,R) , by the construction of z0 from (11), δr 6 r/R. Furthermore, since vλ,w = R2 f , for each E ∈ Fλ,w , Z √ 1/2 1/2 2 π |E| 6 P (E) 6 vλ,w 6 2πR + ||w||L2 |E| , E √ √ 1/2 and |E| 6 2πR/(2 π − ||w||L2 ) provided that 2 π > ||w||L2 . So, Theorem 4 implies that for all λ > 0, and w ∈ L2 R2 such that √ λr2 ||w||L2 6 min 2 π − π, , 8R2 any level set E of uλ,w satisfies ∂ ∗ E ⊂ Tr up to an H1 -negligible set. 39 8 Support stablity for nonsmooth convex sets The theory developed in Sections 4, 5 and 7 relies on the existence of a subgradient of the total variation for f in the L2 topology (source condition). As natural as it may seem, this hypothesis does not always hold even for simple signals (like the indicator function of a square). In some cases, however, there is a natural limit for the dual certificates vλ,0 when considering another topology. This section studies the case of a union Ω of disjoint convex subsets of R2 which are sufficiently far apart. If their boundary is not smooth enough, the source condition is not satisfied. Still, we shall prove that one can guarantee support stability for the solutions of (Pλ (f )). A notable example is the unit square where Ω = [0, 1]2 . As usual, througout this section, we let uλ,w be the solution of (Pλ (f + w)) and vλ,w be the solution of (Dλ (f + w)). We also recall the notation from Section 3 where given any bounded open convex set C, Cρ is the opening of C by open balls of radius ρ and there exists a unique function r(x) such that x ∈ ∂Cr(x) and x belongs to an arc of a circle of radius r(x). 8.1 Dual certificates for unions of convex sets Let C be a bounded open convex subset of R2 . The dual certificate vλ,0 associated with f = 1C is given in (26). (j) Now, more generally, if f = 1Ω , where Ω = ∪M and {C (j) }M j=1 C j=1 are bounded open convex sets such that given any 0 6 k 6 M and any permutation {i1 , . . . , iM } of {1, . . . , M }, M k [ [ C (ij ) Ei1 ,...,ik ∈ argmin P (E) ; P (E) < ∞, C (ij ) ⊂ E ⊂ R2 \ j=1 implies that P (Ei1 ,...,ik ) > uλ,0 to (Pλ (f )) is Pk j=1 uλ,0 = j=k+1 P (C (ij ) ), then, as proved in [8, 10], the solution M X + (1 + λvC (j) ) 1C (j) , j=1 and consequently, (j) vC (j) (x) x ∈ Cλ , j = 1, . . . , M (j) vλ,0 (x) = 1/λ x ∈ C (j) \ Cλ , j = 1, . . . , M 0 otherwise. While limλ→0 ||vλ,0 ||L2 = +∞, we observe that the function def. v0,0 = M X vC (j) ∈ L1 (R2 ). j=1 40 Indeed, for each j, by the monotone convergence theorem Z Z (j) ||vC (j) ||L1 = vC (j) = lim vC (j) 1C (j) = lim P (C1/n ) = P (C (j) ) < +∞, R2 n→+∞ 1/n R2 n→+∞ (j) where C1/n denotes the opening of C (j) with radius 1/n. Moreover, since vλ,0 > vµ,0 for µ > λ and since for a.e. x ∈ R2 , lim vλ,0 (x) = v0,0 (x), λ→0 if follows by the monotone convergence theorem that ||vλ,0 − v0,0 ||L1 → 0, λ → 0. As before, we may define the extended support of f via v0,0 as Z def. Ext(Df ) = ∂ ∗ E ; ± v0,0 = P (E), |E| < ∞ E M [ (j) =Ω\ int CRj , (54) j=1 (j) where CRj is the maximal calibrable set inside C (j) for each j = 1, . . . , M . We remark that vλ,0 = v0,0 on R2 \ Ext(Df ). 8.2 Support stability In this section, we prove that the support of uλ,w is stable around the extended support (54), i.e. its support is contained inside some neighborhood of Ext(Df ), whenever λ and ||w||L2 /λ are sufficiently small. We begin by proving some properties of the level sets. Proposition 13. The following statements are true. (i) There exists α0 , λ0 , L > 0 such that P (E) 6 L for all E ∈ Fλ,w and (λ, w) ∈ Dλ0 ,α0 . (ii) There exists R > 0 such that Eλ,w ⊂ BR for all Eλ,w ∈ Fλ,w with (λ, w) ∈ D1,1 . Proof. To prove (i), let λ0 , α0 be such that ||vλ,0 − v0,0 ||L1 6 1/2 and α0 6 √ c2 /2. Then, for all E ∈ Fλ,w with (λ, w) ∈ Dλ0 ,α0 , Z 1/2 ||w||L2 |E| P (E) = ± vλ,w 6 + ||vλ,0 − v0,0 ||L1 + ||v0,0 ||L1 λ E α0 P (E) 6 √ + ||vλ,0 − v0,0 ||L1 + ||v0,0 ||L1 , c2 41 and so, P (E) 6 1 + 2||v0,0 ||L1 . For the proof of (ii) is very similar to the proof of Lemma 4. We first show that there exists R > 0 such that Eλ,w ∩ BR 6= ∅ for all Eλ,w ∈ Fλ,w with (λ, w) ∈ D1,1 : let R be such that BR ⊃ Ω. For a contradiction, suppose that c ∅= 6 Eλ,w ⊂ BR . Then, since vλ,0 = 0 on Ωc , we have that Z (vλ,w − vλ,0 ) 6 P (Eλ,w ) = Eλ ,w ||w||L2 |Eλ,w | λ 1/2 < P (Eλ,w ) , √ c2 which is impossible if P (Eλ,w ) > 0. So, Eλ,w ∩BR 6= ∅ if Eλ,w 6= ∅. Finally, since by (i), there exists L > 0, λ0 , α0 such that P (Eλ,w ) 6 L for all (λ, w) ∈ Dλ0 ,α0 , it follows that Eλ,w ⊂ B(0, R + L). When the source condition is not satisfied, Proposition 8 cannot be applied directly to the level sets of uλ,w . However, even if the source condition does not hold, there may still be a subset of U of R2 for which lim ||vλ,0 − v0,0 ||L2 (U ) = 0. λ→0 In this case, one can argue along the lines of Proposition 8 to deduce that there is still weak regularity on a subset of ∂E. Note that for characteristic functions on unions of convex sets as described in Section 8.1, we can let U = R2 \Ext(Df ) since vλ,0 = v0,0 on R2 \ Ext(Df ). The precise regularity statement is given in the following proposition. Proposition 14. Let U ⊂ R2 be an open set. Suppose that lim ||vλ,0 − v0,0 ||L2 (U ) = 0. λ→0 Then, there exists r0 > 0 such that for all r ∈ [0, r0 ] and Eλ,w ∈ Fλ,w with (λ, w) ∈ D1,√c2 /4 , if x ∈ ∂Eλ,w is such that B(x, r0 ) ⊆ U , then |B(x, r) ∩ Eλ,w | 1 > |B(x, r)| 16 and |B(x, r) \ Eλ,w | 1 > . |B(x, r)| 16 Theorem 5. Let {wn }n∈N , {λn }n∈N be sequences such that wn ∈ L2 R2 , ||w || (t) λn & 0, and λnnL2 → 0 as n → +∞. Let us denote by Un the level sets of uλn ,wn and by F (t) the level sets of f . Then, lim sup Supp(Duλn ,wn ) ⊆ Ext(Df ), n→+∞ and for almost every t ∈ R, lim Un(t) ∆F (t) = 0, n→+∞ and lim sup ∂Un(t) ⊆ ∂F (t) . n→+∞ 42 Proof. It is sufficient to prove that for all r > 0 def. lim sup Supp(Duλn ,wn ) ⊆ Ext(Df ) ⊆ Tr = x ∈ R2 ; dist(x, Ext(Df )) 6 r . n→+∞ The proof is almost identical to the proof of Theorem 3 and we simply highlight the required modifications here: (a) The proof of Theorem 3 relies on the compactness property provided by Lemma 2 and 4. However, analogous statement are proved in (i) and (ii) of Proposition 13. (b) L2 convergence was applied to prove equation (50), but in fact, L1 convergence is sufficient: Let En ∈ Fλn ,wn be such that limn→+∞ (λn + ||wn ||L2 /λn ) = 0 and limn→+∞ (|En \ E| + |E \ En |) = 0. Note that by (i) and the isoperimetric inequality, there exists L such that |En | 6 L2 for all n. Then by letting En 4E = (En \ E) ∪ (E \ En ), Z Z Z Z Z Z v0,0 − v0,0 vλn ,wn − v0,0 + vλn ,wn − v0,0 6 En E En En En E Z L||wn ||L2 6 ||vλn ,0 − v0,0 ||L1 + + v0,0 → 0, λn En 4E by (i) and the absolute continuity of the Lebesgue integral. (c) The final part of the proof of Theorem 3 requires the weak regularity result from Proposition 8. However, it is only required that (51) is satisfied for x ∈ R2 \ Tr and since vλ,0 = v0,0 on R2 \ Ext(Df ) ⊂ R2 \ Tr , one can still apply Proposition 14 (with U = R2 \ Tr ) to reach the same conclusion. Conclusion In this paper, we have characterized the regions in which the solutions to the two-dimensional TV denoising problem are geometrically stable under L2 additive noise. In particular, via the minimal norm certificate, we introduced the notion of an extended support and although the support of TV regularized solutions are in general not stable, we have proved that the support instabilities are confined to a neighbourhood of the extended support. We have also provided explicit examples of the extended support in the case of indicators of convex sets. Within the low noise regime, for the indicator set of a calibrable set C, the support of the solutions was shown to cluster around ∂C. While for indicator functions of general convex sets (including convex sets for which the source condition is not satisfied), the support of the solutions was shown to cluster around the domain C \ int(CR∗ ), where CR∗ is the maximal calibrable set inside C. 43 Acknowledgements Vincent Duval acknowledges support from the CNRS (Défi Imag’in de la Mission pour l’Interdisciplinarité, project CAVALIERI). 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