A Posteriori Error Control for the Binary Mumford-Shah Model

Transcription

1 A Posteriori Error Control for te Binary Mumford-Sa Model Benjamin Berkels, Alexander Effland, and Martin Rumpf AICES Graduate Scool, RWTH Aacen University, Institute for Numerical Simulation, University of Bonn, {alexander.effland, January 3, 06 Abstract Te binary Mumford-Sa model is a widespread tool for image segmentation and can be considered as a basic model in sape optimization wit a broad range of applications in computer vision, ranging from basic segmentation and labeling to object reconstruction. Tis paper presents robust a posteriori error estimates for a natural error quantity, namely te area of te non-properly segmented region. To tis end, a suitable uniformly convex and non-constrained relaxation of te originally non-convex functional is investigated and Repin s functional approac for a posteriori error estimation is used to control te numerical error for te relaxed problem in te L -norm. In combination wit a suitable cut out argument, fully practical estimates for te area mismatc are derived. Tis estimate is incorporated in an adaptive mes refinement strategy. Two different adaptive primal-dual finite element scemes, a dual gradient descent sceme, and te most frequently used finite difference discretization are investigated and compared. Numerical experiments sow qualitative and quantitative properties of te estimates and demonstrate teir usefulness in practical applications. Introduction Since te introduction of te image denoising and edge segmentation model by Mumford and Sa in te late 80 s [35], tere as been muc effort to find effective and efficient numerical algoritms to compute minimizers of different variants of tis variational problem. Te original model is based on te functional E MS [u, K] = Ω/K u + α(u u 0 ) dx + βh n (K) wit α, β > 0, were u 0 L (Ω, [0, ]) is a scalar image intensity on te bounded image domain Ω R n, u te reconstructed image intensity and K te associated set of edges, on wic te image intensity u jumps. Here, H n denotes te (n )-dimensional Hausdorff measure. Te space of functions of bounded variation BV(Ω) turned out to be te proper space to formulate te problem in a matematical rigorous way. Indeed, existence in te context of te space of special functions of bounded variation SBV(Ω) was proved by Ambrosio (see [, Teorem 4.]). For details on tese spaces we refer to [3]. Restricting u to be piecewise constant instead of piecewise smoot, one is lead to a basic and widespread image segmentation model. Tis model is discussed from a geometric perspective in te book by Morel and Solimini [34]. In te case of just two intensity values c, c [0, ], te associated energy can be rewritten in terms of a caracteristic function χ BV(Ω, {0, }) as E[χ] = Ω θ χ + θ ( χ) dx + Dχ (Ω). (.)

2 Here, θ i = ν (c i u 0 ) for i =,, te new weigt ν = β/α and te resulting binary model is given by u = c χ + c ( χ). For fixed χ, one immediately obtains te optimal constants c [χ] = ( χ dx) Ω χu 0 dx Ω and c [χ] = ( χ dx) Ω ( χ)u 0 dx. Ω (.) For fixed c and c one aims at minimizing te energy over te non-convex set of caracteristic functions χ BV(Ω, {0, }). In te general case, one is interested in a triple (χ, c, c ) as a minimizer of E w.r.t. te set BV(Ω) [0, ]. Hencefort, if not oterwise stated we assume te intensity values to be fixed. Nikolova, Esedoglu and Can [36] sowed tat te non-convex minimization problem for χ can be solved via relaxation and tresolding a breaktroug for bot reliable and fast algoritms in computer vision [40, 7]. Here, at first one asks for a minimizer of E over all u BV(Ω, [0, ]) and ten tresolds u for any tresold value s [0, ) to obtain te solution χ = χ [u>s] of te original minimization problem. Te relaxed problem coincides wit a constrained version of te classical image denoising model by Rudin, Oser and Fatemi (ROF) [45]. Numerical scemes for an effective and efficient minimization of tis model ave extensively been studied. Making use of a dual formulation, Cambolle [4] introduced an iterative finite difference sceme and proved its convergence. Hintermüller and Kunisc [30] proposed a predual formulation for a generalized ROF model and applied a semismoot Newton metod for a regularized variant. Cambolle and Pock [0] deduced a primal-dual algoritm wit guaranteed first order convergence and applied teir approac to different variational models in BV suc as image denoising, deblurring and interpolation. Te sceme is based on an alternating discrete gradient sceme for te discrete primal and te discrete dual problem. Bartels [6] used te embedding BV(Ω) L (Ω) H (Ω) to improve te step-size restriction for BV functionals. Wang and Lucier [47] employed a finite difference approximation of te ROF model and derived an a priori error estimate for te discrete solution based on suitable projection operators. Following Dobson and Vogel [3], te total variation regularization can be approximated smootly via u + ɛ. In [6], te convergence of te L -gradient flow of tis smoot approximation to te TV flow in L is sown under strong regularity assumptions on te solution. Furtermore, approximations of te original Mumford-Sa model ave been studied extensively. An early overview of different approximation and discretization strategies was given by Cambolle in [3]. Ambrosio and Tortorelli [4] proposed a pase field approximation of tis functional and proved its Γ-convergence. Cambolle and Dal Maso [8] proposed a discrete finite element approximation and establised its Γ-convergence. Bourdin and Cambolle [9] picked up tis approac and studied te generation of adaptive meses iteratively adapted in accordance to an anisotropic metric depending on te current approximate solution. In [46], Sen introduced a Γ-converging approximation of te piecewise constant Mumford-Sa segmentation, were te lengt term in te Mumford-Sa model is approximated via an approac originating from te pase field model by Modica and Mortola [33]. A simple and widespread level set approac was proposed by Can and Vese []. Te goal of tis paper is to derive a posteriori error estimates for te caracteristic function χ. To tis end, we proceed as follows: We take into account a suitable uniformly convex relaxation of te binary Mumford-Sa functional already studied in [8], wic is related to more general relaxation approaces suggested by Cambolle [5] and does not require any constraint in te minimization. For tis relaxation, we consider its predual and set up a corresponding primal-dual algoritm [5, 0, 9]. Ten, following Bartels [5], we use Repin s primal-dual approac [4, 43] to derive functional a posteriori error estimates for te relaxed solution based on upper bounds of te duality gap (cf. also te book by Han [8] wit respect to mecanical applications) (see Section ). Tese estimates can be used togeter wit a suitable cut out argument to derive an a posteriori estimate for te caracteristic function χ minimizing te original functional (.). In addition, a sensitivity analysis of χ depending on c, c and of c, c depending on χ is studied (see Section 3). Moreover, two adaptive finite element discretization scemes and one conventional, non adaptive

3 finite difference sceme are investigated (see Section 4). Based on tese discretization scemes, a primal-dual algoritm and a dual gradient descent are introduced in Section 5. Finally, we apply te resulting estimate to tese scemes incorporating an appropriate post smooting and present te numerical results (see Section 6). Uniformly Convex Relaxation and Functional Error Estimates Hencefort, we use te notation χ A to denote te indicator function of a measurable set A Ω and define [u > c] = {x Ω u(x) > c}. We use generic constants c and C trougout tis paper. Furtermore, if not stated oterwise, we assume te intensity values c and c to be fixed wit c, c [0, ] and c c. Rewriting te binary Mumford-Sa functional (.) as E[χ] = Ω (θ θ )χ dx + Dχ (Ω) + Ω θ dx, (.) one observes tat adding a constant to θ and θ leaves te minimizers χ uncanged. Tus, we may assume tat θ, θ c > 0. Let us introduce te following relaxed functional E rel [u] = Ω u θ + ( u) θ dx + Du (Ω), (.) wic is supposed to be minimized over all u BV(Ω, R). Indeed, E rel [χ] = E[χ] for caracteristic functions χ and one retrieves te original binary Mumford-Sa model. Proving existence of minimizers of (.) via te direct metod in te calculus of variations is straigtforward, for details we refer to [3, 5]. Furtermore, (.) is uniformly convex by our above assumptions on θ and θ. Loosely speaking, a preference for te values 0 and for u is encoded in te quadratically growing data term. Te minimizers of bot functionals (.) and (.) are related in te following sense (cf. [8]): Proposition. (Convex relaxation and tresolding). Under te above assumptions, a minimizer u BV(Ω) of te functional E rel exists, u(x) [0, ] for a.e. x Ω and χ [u>0.5] argmin χ BV(Ω,{0,}) E[χ]. Proposition. is an instance of a more general result, wic can be found in [5, 9, 3]. In fact, let Ψ Ω R R be measurable, Ψ(, t) L (Ω) for a.e. t R, Ψ(x, ) C (R) be strictly convex for a.e. x Ω, Ψ(x, t) c t C and E Ψ [u] = Ω Ψ(x, u) dx + Du (Ω). Ten, tere exists a minimizer u argminũ BV(Ω) E Ψ [ũ] and for s R χ s = χ [u>s] argmin χ BV(Ω,{0,}) Ω t Ψ(x, s)χ dx + Dχ (Ω). For Ψ(x, t) = t θ (x) + ( t) θ (x) te property u(x) [0, ] follows directly wen comparing wit te energy of te function min(, max(0, u)). Furtermore, coosing s = allows to verify te main claim of Proposition.. Indeed, for t R and χ BV(Ω, {0, }), let Et rel [χ] = Ω t Ψ(x, t)χ dx + Dχ (Ω). For our specific coice of Ψ, Et rel [χ] = (t(θ (x) + θ (x)) θ (x)) χ dx + Dχ (Ω) Ω implies tat minimizing te functional E rel E rel [χ] = E[χ] Ω θ dx. is equivalent to minimizing te functional E because 3

4 Remark. Te particular advantage of our model compared to te relaxation approac by Nikolova, Esedoglu and Can [36] is tat te relaxed problem as not to be constrained to functions u wit values in [0, ]. One could also consider a ROF type functional coosing Ψ(x, t) = (t (θ (x) θ (x))) (cf. [7]) and obtain te functional E0 rel [χ] for te tresold value s = 0, but in tis case te L bound of te relaxed solution depends on te L bounds of ν (c i u 0 ) (cf. Proposition.) and requires a more involved cutoff sceme (see Section 3). In wat follows, we make use of convex analysis to derive a duality formulation for te minimization problem of te relaxed functional (.) and derive functional a posteriori estimates for tis problem. Primal and dual formulation will later be used in te a posteriori estimates. Te dual of BV(Ω) is very difficult to caracterize and not suitable for computational purposes. Tus, for a generalized ROF model Hintermüller and Kunisc [9] proposed to consider te corresponding BV functional as te dual of anoter functional, wic we refer to as te predual functional. Bartels [5] made use of tis approac in te context of a posteriori estimates for te ROF model. Here, we follow tis procedure and investigate te predual of (.). Recall tat te Fencel conjugate J of a functional J X R on a Banac space X wit R = R { } is a functional on te dual space X wit values in R, defined as J [x ] = sup x X { x, x J[x]}, were, denotes te duality pairing. Furtermore, we denote by Λ L(Y, X ) te adjoint operator of Λ L(X, Y ) and by J te subgradient of J (cf. [4]). Now, we investigate an energy functional D rel [q] = F [q] + G[Λq] q Q (.3) wit F Q R and G V R being proper, convex and lower semicontinuous functionals, V and Q being reflexive Banac spaces and Λ L(Q, V). In our case, te predual of te convex relaxed binary Mumford-Sa model is given by F [q] = I B [q] = { 0 if q a.e. + else, G[v] = Ω 4 v + vθ θ θ θ + θ dx, wit Λ = div, Q = H N (div, Ω) and V = L (Ω). Recall te definition of te spaces H(div, Ω) = {q L (Ω, R n ) div q L (Ω)}, endowed wit te norm q H(div,Ω) = q L (Ω) + div q L (Ω), and H N (div, Ω) = H(div, Ω) {q ν = 0 on Ω}, were ν is te outer normal on Ω and te operator div is understood in te weak sense. Moreover, Λ = olds in te sense (Λ v, q) L (Ω) = (v, div q) L (Ω) v V, q Q. Based on tis duality and for te particular coice of D rel, we easily verify tat (D rel ) = E rel. Indeed, from te general teory in [4, pp. 58 ff.], we can deduce (D rel ) [v] = F [ Λ v]+g [v]. As a result of te denseness of C c (Ω) in H N (div, Ω) wit respect to te norm H(div,Ω), we can infer for any v BV(Ω) wic leads to Dv (Ω) = sup q Q, q Ω v div q dx = sup ( v div q dx I B [q]), q Q Ω F [ Λ v] = sup ( v div q dx I B [q]) = Dv (Ω). q Q Ω On te oter and, te Fencel conjugate of G can be computed as follows: G [v] = sup w L (Ω) ((v, w) L (Ω) G[w]) = Ω v θ + ( v) θ dx, were te supremum is attained for w = v(θ + θ ) θ. Tis verifies te assertion. 4

5 Te central insigt is tat D rel [p] = (D rel ) [u] (.4) for a minimizer p of D rel and a minimizer u of (D rel ). A rigorous verification can be found in [4, Capter III.4] (see also [44, 4, 5]). Furtermore, one obtains tat q Q and v V are optimal if and only if Λ v F [ q] and v G[Λ q], wic can be deduced from te equivalence J[x] + J [x ] = x, x x J[x] (see [4, Proposition I.5.]). In wat follows, we investigate a posteriori error estimates associated wit te energy D rel [q] = F [q] + G[Λq] and its dual E rel [v] = F [ Λ v] + G [v] (for fixed intensity values c and c ). A crucial prerequisite is te uniform convexity of G, wic is linked to te specific coice of te relaxed Model E rel. Recall tat a functional J X R is uniformly convex, if tere exists a continuous functional Φ J X [0, ) suc tat J[ x+x ] + Φ J (x x ) (J[x ] + J[x ]) for all x, x X and Φ J (x) = 0 if and only if x = 0. Furtermore, we denote by Ψ J a non-negative functional suc tat x, x x + Ψ J (x x ) J[x ] J[x ] for all x J[x ]. Hence, Ψ J allows a quantification of te strict monotonicity of J. If J C and λ min denotes te smallest eigenvalue of D J, ten Φ J and Ψ J admit te representation Φ J (x) = 8 λ min(d J) x and Ψ J (x) = λ min(d J) x, wic follows readily via a Taylor expansion. Remark. Te optimal Ψ J coincides wit te Bregman distance (cf. []), i.e. Ψ J (x x ) Ψ J(x x ) = J[x ] J[x ] for any oter Ψ J and x, x X. sup x J[x ] x, x x Now, te a posteriori error estimate is based on te following direct application of a general result by Repin [4]: Let u argminṽ V E rel [ṽ] and q Q, v V = V = L (Ω). Ten, Φ G (v u) + Φ F ( Λ (v u)) + Ψ E rel( v u ) (Erel [v] + D rel [q]). (.5) Te proof of (.5) relies on te above strict convexity estimates and te fundamental relation E rel [u] D rel [q] known as te weak complementarity principle [4], and can be found in [4, 5]. In te case of te binary Mumford-Sa model, we easily verify tat Φ F 0, Φ G (v) = 4 Ω v (θ + θ ) dx, Ψ E rel(v) = Ω v (θ + θ ) dx, and te estimate (.5) implies for any v V and q Q Ω (u v) (θ + θ ) dx E rel [v] + D rel [q]. Finally, (a b) a + b wit a = c u 0 and b = c u 0 yields ν (c c ) θ + θ. Tus, we obtain te following teorem: Teorem.. Let u V be te minimizer of E rel. Ten, for any v V and q Q it olds tat u v L (Ω) err u[v, q, c, c ] = ν (c c ) (Erel [v] + D rel [q]). (.6) In te application, one asks for te (post-processed) discrete primal v and dual solution q wic ensure a small rigt and side. Additionally, te estimator err u is consistent, i.e. err u[v, q, c, c ] 0 provided v and q converge to te extrema of te corresponding energy functionals w.r.t. te topology of te associated Banac spaces. 5

6 3 A Posteriori Error Estimates for te Binary Mumford-Sa Model In te sequel, we expand te a posteriori teory to te binary Mumford-Sa model. Te key observation is tat for many images approximate solutions u L (Ω) of te relaxed model are caracterized by steep profiles, were te actual solution of te original binary Mumford-Sa model jumps. We proceed as follows. We define a[v, η] = χ [ η v +η] L (Ω) for η (0, ), wic measures te area of te preimage of te interval of size η centered at te tresold value s = (cf. Section ). Based on te above observation, te set S η = [ η v + η] can be regarded as te set of non-properly identified regions. Combining tis definition wit te tresolding argument presented in Proposition., we obtain te subsequent teorem: Teorem 3. (A posteriori error estimator for χ). For fixed c and c, let u BV(Ω) and χ = χ [u> ] BV(Ω, {0, }) be a minimizer of te relaxed functional E rel (see (.)) and te binary Mumford-Sa functional E (see (.)), respectively. Ten for all v V = L (Ω) and q Q = H N (div, Ω) we ave tat χ χ [v> ] L (Ω) err χ[v, q] = inf η (0, ) (a[v, η] + η err u[v, q, c, c ]). (3.) Recall tat χ [v> ] is te indicator function of te set [v > ]. Let us remark tat χ [v> ] is te result of te same tresolding, wic relates χ to te solution u of te relaxed problem (.), i.e. χ = χ [u> ], tis time applied to v. Proof. Any minimizer u of E rel fulfills 0 u and χ [u> ] minimizes E (see Proposition.). For all η (0, ), we obtain te following set relation for te symmetric difference of te sets [u > ] and [v > ] ( denoting te symmetric difference of two sets): [u > ] [v > ] {x Ω η v(x) + η} {x Ω u(x) v(x) > η}. (3.) Now, using Teorem. te Lebesgue measure of te rigtmost set can be estimated as follows L n ( u v > η) { u v >η} u v η dx η err u[v, q, c, c ], were η (0, ). Finally, taking te infimum for all η (0, ) concludes te proof. In te application, te computational cost to find te optimal η is of te order of te degrees of freedom for te discrete solution and tus affordable. Let us empasize tat te error estimator err χ is not tailored to a specific finite element approac. Indeed, we can project any primal and dual solution onto te spaces V = L (Ω) and Q = H N (div, Ω), respectively. Remark. (i) We can obtain an a posteriori error estimate for te segmentation also for intensity values c i, wic are only known to be in intervals [c i ɛ, c i + ɛ] around some value c i for i =, and ɛ > 0. Indeed, applying straigtforward monotonicity arguments, we obtain te estimate sup ci B ɛ(c i), i=, err u[v, q, c, c ] err,ɛ u [v, q, c, c ] for err,ɛ u [v, q, c, c ] = Ω ν ( c c ɛ) (v θ max + ( v) θ max + v + 4 (div q) θ min +θ min + max { 6 (div q)θmax θ min +θ min, (div q)θmin θ max +θ max } θmin θ min θ max +θ max ) dx

7 provided tat ɛ < c c. Here, θi max (x) = ν max{(c i + ɛ u 0 ), (c i ɛ u 0 ) } and θi min (x) = ν min{(c i + ɛ u 0 ), (c i ɛ u 0 ) } for i =,. Tus, for te minimizer χ BV(Ω, {0, }) of E[, c, c ] te a posteriori estimate χ χ [v> ] L (Ω) errɛ χ[v, q, c, c ] = inf η (0, ) (a[v, η] + η err,ɛ u [v, q, c, c ]) (3.3) olds for all v V = L (Ω) and q Q = H N (div, Ω). (ii) Te optimal intensity values for a given caracteristic function χ are given in (.). For te sensitivity of teses values on χ, we straigtforwardly obtain te estimates c [χ] c [ χ] χ χ L (Ω) χ L (Ω) χ χ L (Ω), c [χ] c [ χ] ( χ χ) L (Ω) χ L (Ω) ( χ χ) L (Ω) (3.4) assuming ( χ χ) L (Ω) < min{ χ L (Ω), χ L (Ω)}. (iii) Given te sensitivity results from (i) and (ii) one migt ask for an a posteriori error estimate bot for χ and te intensity values c, c. In fact, if (χ, c, c ) is a minimizer of te in general non-convex energy E = Ω ν (c u 0 ) χ + ν (c u 0 ) ( χ) dx + Dχ (Ω) and one assumes a priori tat eac of te initially cosen intensity values is already in some ɛ neigborood of te corresponding c i value, ten te estimates (3.3) and (3.4) can be combined to obtain an a posteriori error estimate for te numerical approximation of (χ, c, c ). Unfortunately, te estimate (3.3) is not sufficiently sarp to ensure tat te resulting estimated error in te intensities does actually improve compared to te a priori assumption and a bootstrapping argument could not be applied to furter improve te resulting estimates. We refer to Section 6 for an explicit evaluation of te sensitivity of te relaxed solution. 4 Finite Element and Finite Difference Discretization In tis section, we investigate different numerical approximation scemes for te primal and te dual solution of te relaxed problem (.) on adaptive meses and te refinement of te meses based on te a posteriori error estimates derived in Section 3. In te context of image processing applications wit input images usually given on a regular rectangular mes, an adaptive quadtree for n = (or octree for n = 3) turned out to be an effective coice for an adaptive mes data structure. In wat follows, we pick up te finite element approac for a variational problem on BV proposed by Bartels [6] and a simplified version of te latter. Furtermore, we consider te widespread finite different sceme proposed by Cambolle [4]. In all numerical experiments in tis paper, we coose Ω = [0, ]. (FE) Finite element sceme on an induced adaptive triangular grid. On te domain Ω, we consider an adaptive mes M described by a quadtree wit cells C M being squares, wic are recursively refined into four squares via an edge bisection. We suppose tat te level of refinement between cells at edges differs at most by one. Tus, on a single edge at most one anging node appears. Let indicate te spatially varying mes size function on Ω, were on a grid cell C ranges from an initial mes size L init to a finest mes size L0 (usually determined by te image resolution). For all discretization approaces investigated ere, te degrees of freedom are associated wit te non-anging nodes. Let us denote by N v te number of tese nodes, wic will coincide wit te number of degrees of freedom of discrete primal functions. Te finite element discretization is based on a triangular mes S spread over te adaptive quadtree mes via a splitting of eac quadratic leaf cell into simplices T ( cross subdivision ). We ask for discrete primal functions u in te space of piecewise affine and globally continuous functions on S denoted by V. Tus, for functions v V te values at anging nodes are interpolated based on te 7

8 values at adjacent non-anging nodes, wic are associated wit te actual degrees of freedom. By Q = {q V n q ν = 0 on Ω} we denote te discrete counterpart of Q. To accommodate tis boundary condition, te boundary nodes are modified after eac update of te dual solution in a post-processing step. On V, we define discrete counterparts of te continuous functionals F and G as follows: G [v ] = Ω 4 v + v θ, θ, θ, θ, + θ, dx, F [q ] = I B [q ], were θ i, = I (θ i ) = I ( ν (c i u 0 ) ) for i =, wit I denoting te Lagrange interpolation. In te application on images, we suppose tat u 0 V 0, were V 0 is te simplicial finite element space corresponding to te full resolution image on te finest grid level L 0 representing te full image resolution. Furtermore, we consider two different scalar products. On V, we take into account te L -product and on Q te lumped mass product (q, p ) Ω I (q p ) dx and identify V and Q wit teir dual spaces wit respect to te L - and te lumped mass product, respectively. Ten, te associated dual operators are G [v ] = Ω v θ, + ( v ) θ, dx, F [q ] = Ω I ( q ) dx. Finally, we define te discrete divergence Λ Q V, q P div q, were P denotes te L -projection P L (Ω) V. Following Bartels [5] and taking into account te above scalar products on V and on Q, we obtain for te discrete gradient Λ V Q, v Λ v, te defining duality for all q Q and v V. Ω I ( Λ v q ) dx = Ω v P div q dx (4.) (FE ) Finite element sceme based on a simple gradient operator. Instead of te above defined discrete gradient operator Λ, we alternatively consider te piecewise constant gradient v on te simplices T of te simplicial mes for functions v V. To tis end, we coose Q as te space of piecewise constant functions on te simplicial mes, and take into account te standard L -product on bot spaces. Te above definitions of te functionals G and F are still valid. Moreover, G remains te same, only F canges to F [q ] = Ω q dx. Te discrete divergence Λ Q V is defined via duality starting from te preset discrete gradient as Ω I (Λ q v ) dx = Ω q v dx, wic indeed ensures tat Λ v = v. Tis simplified ansatz leads to a non-conforming iterative solution sceme (see Section 5), since te space of piecewise constant finite elements is not contained in H N (div, Ω) (cf. [6]). After eac modification of te (piecewise constant) dual solution te values on te corresponding boundary cells are set to 0 to satisfy te boundary condition. To apply te derived a posteriori error estimates a projection onto te space H N (div, Ω) is required. To tis end, we replace te solution p Q by its L -projection onto te space V n after eac execution of te algoritm. (FD) Finite difference sceme on a regular mes. Te finite difference sceme for te numerical solution of functionals on BV proposed by Cambolle [4] is extensively used in many computer vision applications and applies to image data defined on a structured non adaptive mes. We compare te a posteriori error estimator for tis sceme on non-adaptive meses wit te above finite element scemes on adaptive meses. To tis end, we denote by V R Nv and Q R nnv nodal vectors on te regular lattice for primal and dual solutions, respectively. Here, 8

9 N v = ( +) n, were denotes te fixed grid size of te finite difference lattice. Integration is replaced by summation and we obtain te following discrete analogues G and F of te continuous functionals F and G as functions on R Nv and R nnv, respectively: N v G [V ] = i= 4 (Vi ) + V i Θi, Θi, Θi, Θ i, + Θi,, F [Q ] = max I B [Q i i=,...,n v ] wit Θ i,, Θi, denoting te pointwise evaluation of θ and θ, respectively, and I B [Q i ] = 0 for Q i and + oterwise. Te associated dual operators for te standard Euclidean product as te duality pairing are N v G [V ] = (V) i Θ i, + ( V) i Θ i,, F [Q ] = Q i. i= Finally, we take into account periodic boundary conditions (by identifying degrees of freedom on opposite boundary segments) and use forward difference quotients to define te discrete gradient operator Λ RNv R nnv, i.e. (( Λ )V ) i = V N (i,j) V i j=,...,n, were N (i, j) is te index of te neigboring node in direction of te jt coordinate vector. As a consequence, te matrix representing te discrete divergence operator Λ R nnv R Nv is just te negative transpose of te matrix representing te discrete gradient and tus corresponds to a discrete divergence based on backward difference quotients. To use te a posteriori error estimate in te finite difference context, we consider as a simplest coice te piecewise bilinear functions u and p uniquely defined by te solution vectors U and P, respectively. Te boundary condition for p is taken care of in exactly te same way as in te case (FE). N v i= 5 Duality-Based Algoritms For te numerical solution of te different discrete variational problems, we primarily use te primal-dual algoritm proposed by Cambolle and Pock [0, Algoritm ], wic computes bot a discrete primal and a discrete dual solution to be used in te a posteriori error estimates. Note tat we use [0, Algoritm ] instead of [0, Algoritm ] even toug G is uniformly convex. As we will see later, evaluating (Id + τ G ) requires te inversion of a matrix depending on τ. In Algoritm, τ is fixed and te inverse can be computed once using a Colesky decomposition for te sparse, symmetric and positive-definite matrix (for details see []), wereas in Algoritm te decomposition of te linear system as to be performed in eac iteration. Moreover, using diagonal preconditioning one can improve te convergence speed of Algoritm witout any furter step size control (see [38]). Tere is a variety of alternative algoritms to solve tis convex minimization problem, e.g. te split Bregman metod [7], te semi-implicit dual gradient descent [4], te alternating descent metod for te Lagrangian [5, Algoritm A ] or te alternating direction metod of multipliers (ADMM) (see [0] and te references terein). For distinct images we compare below te aforementioned algoritm by Cambolle and Pock wit a dual gradient descent in terms of te quantity of te error estimator. Before we discuss te algoritm due to Cambolle and Pock in te more conventional matrixvector notation, let us rewrite te finite element approaces correspondingly. Let N v = dim V (te number of non-anging nodes) and N q = dim Q (for (FE) N q = nn v and for (FE ) N q is n times te number of simplices). In wat follows, we will use uppercase letters to denote a vector 9

10 of nodal values, e.g. V i = v (X i ) if X i is te it non-anging node. Te two scalar products are encoded via mass matrices. Here, M R Nv,Nv represents te standard L -product on V and is given by M V U = Ω v u dx for all v, u V. Furtermore, M R Nq,Nq is te mass matrix associated wit te space Q. For te approac (FE) tis is given as te lumped mass matrix wit M P Q = Ω I (p q ) dx for all p, q Q, wereas for te discretization (FE ) M P Q = Ω p q dx for all p, q Q defines a classical (diagonal) mass matrix. For te matrix representations Λ and Λ of te discrete divergence and te discrete gradient, respectively, we obtain te relation (cf. [5]) Λ = M Λ T M. (5.) Moreover, for te discretization (FD) we ave Λ = ΛT. Altogeter, te discrete predual energy D rel RNq R and te discrete energy E rel RNv R are defined as follows: D rel [Q ] = F [Q ] + G [Λ Q ], E rel [V ] = F [ Λ V ] + G [V ]. In te case of all finite element discretizations, te functionals G, F, G and F are defined using te corresponding functions on te finite element spaces, e.g. G [V ] = G [v ]. Now, we are in te position to formulate te primal-dual algoritm. For a fixed mes, fixed intensity values c, c and initial data (U 0, P0 ) R RNv Nq, te Algoritm proposed by Cambolle and Pock [0, Algoritm ] computes a sequence (U k, Pk ), wic converges to te tuple (U, P ) of te discrete primal and dual solution provided τσ Λ <. Indeed, by using k = 0; wile U k+ U k >THRESHOLD do P k+ = (Id + σ F ) [P k σλ Ūk ]; U k+ = (Id + τ G ) [U k + τλ P k+ ]; Ū k+ = U k+ U k ; k = k + ; end Algoritm : Te primal-dual algoritm used to minimize E rel. inverse estimates for finite elements (see [37] for a computation of te constants) te operator norm can be bounded in te case (FE ) for n = as follows: Λ 48(3 + ) min 79.8 min, were min denotes te minimal mes size occurring in M. Moreover, to estimate te operator norm for te case (FE) we use (4.) and obtain Λ = ( max v V, v L = max I ( Λ v q ) dx) q Q, q L = Ω = ( max max v P div q dx) v V, v L = q Q, q L = Ω max P div q L q Q, (Ω) q L = max q Q, q L = div q L (Ω) 96(3 + ) min. Finally, following [4] we can estimate Λ 8 min for te discretization (FD). Suitable stopping criteria are a tresold on te maximum norm of te difference of successive solutions U k+ U k (wic we apply ere) or on te primal-dual gap Erel [Uk ] + Drel [Pk ]. To compute te resolvents (Id + F ) [Q ] and (Id + G ) [V ] we use a variational ansatz (for 0

11 details see [44]), i.e. for te resolvent of a subdifferentiable functional J wit an underlying scalar product (, ) we ave tat (Id + τ J) [x] = argmin(x y, x y) + τj(y). y Te resolvent of F for te approaces (FE) and (FD) is given by (Id + σ F ) [Q ] = ( Q i max{ Q i,}) (5.) i=,...n v wit Q i = q (X i ). In te case (FE ), te above evaluation is performed on eac cell. For te discretizations (FE) and (FE ), we denote by M [W ]U V = Ω w u v dx te weigted mass matrix for functions u, v V and a weigt w V. Ten, te resolvent of G reads as (Id + τ G ) [V ] = (M [ + τ(θ, + Θ, )]) M (V + τθ, ). In te case (FD), te resolvent is given by (Id + τ G ) [V i ] = V i + τθi, + τ (Θ i, + Θi, ) for i N v. In our numerical experiments, we ave cosen THRESHOLD = 0 8. In te sequel, we consider a projected dual gradient descent for te minimization of E rel using te discretization (FE), and picking up Cambolle s semi-implicit gradient descent w.r.t. te dual variable for te ROF model using a finite difference sceme [4]. We remark tat te algoritm can analogously be derived for te discretization sceme (FE ). Starting from te first order condition κ = (uθ + (u )θ ) TV[u] for E rel wit TV[u] = Du (Ω), we can infer u TV [κ] due to [4, Capter I, Corollary 5.], wic is te first order condition of te functional Ω 4(θ + θ ) (κ θ ) dx + TV [κ]. Tus, te unique minimizer is given by κ = P S [θ ], were P S denotes te ortogonal projection onto te set S = {div p p H N (div, Ω), p L (Ω) } w.r.t. te weigted L -space wit weigt w = (4(θ + θ )), denoted by L (Ω, w). Note tat w L (Ω, R >0 ) due to te assumptions regarding θ i. Te primal solution is given by u = θ P S[θ ] (θ +θ ). For te computation of P S [θ ] we ave to solve argmin ι S ι θ L (Ω,w), wic is computed by alternatingly performing a gradient descent for te unconstrained problem and a projection onto S. To be precise, for a fixed step size τ > 0 te usual gradient descent update sceme reads p k+ = p k + τ (w(div p k θ )). After a multiplication by a test function ϑ Q, applying Ω I ( ) dx on bot sides, replacing and div by te differential operators Λ and Λ as introduced in te discretization (FE), respectively, and using (4.) one obtains Ω I (p k+ ϑ) dx = Ω I (p k ϑ) dx + τ Ω (w(λ p k θ ))P div ϑ dx. (5.3) Let M be te lumped mass matrix, L j U V = Ω u j v dx, j =,, for all u, v V, and W i = w(xi ). Ten (5.3) implies [P k+ ] j = [P k ] j + τ M L j (W (Λ P k Θ, )) for j =,. (5.4) After eac update of te dual variable (5.4) a projection onto S is performed (cf. (5.)). Te step size is taken as τ = γ min C M (C ). Te stopping criterion for te resulting algoritm relies on te L -distance of two successive dual solutions (in our case, THRESHOLD = 0 8 ). For a

12 converge analysis of tis approac we refer to []. In te sequel, we will denote results obtained wit te discretization sceme (FE) and tis algoritm by (FE D ). Te adaptive mes refinement is implemented as follows. Given a mes, fixed intensity values and initial data for te primal and dual solution, we run one of te above algoritms and compute te relaxed discrete primal-dual solution pair (u, p ). In te case of te finite difference approac (FD), we define tem as te multilinear interpolation on te cells C of te regular mes. Te corresponding discrete solution of te original problem (.) is ten given as χ = χ [u > ]. Based on u and p, we evaluate te local error estimator for every cell C 0 of te full resolution image grid as follows: err u,c 0 [u, p ] = ν (c c ) ( C0 v θ + ( v ) θ + v + 4 (div q ) + (div q )θ θ θ θ + θ dx). To tis end, a iger order Gaussian quadrature is used. In fact, for (FE), (FE D ) and (FE ) we use a Gaussian quadrature of order 4 on te simplices T 0 composing te cell C 0 on te finest mes wit full image resolution, were te θ i (i =, ) are originally defined, and for (FD) a Gaussian quadrature of order 5 directly on te cells C 0. Te resulting local error estimator for a cell C M and te global estimator are given as err u,c [u, p ] = err u,c 0 [u, p ] and err u[u, p ] = err u,c [u, p ], C 0 C C M respectively. We mark tose cells C for refinement for wic err u,c [u, p ] α max C M err u,c [u, p ], (5.5) were α is a fixed tresold in (0, ). Since tis metod is prone to outliers, we additionally sort all local estimators err u,c according to teir size (starting wit te smallest) and mark te cells in te upper decile for refinement as well. For te input data from Figure we refine up to te resolution of te initial image. 6 Numerical Results In wat follows, we sow numerical results for four different input images sown in Figure. Prior to executing te primal-dual sceme (Algoritm ) or te dual gradient descent sceme, we coose suitable values for c and c by applying Lloyd s Algoritm (see [3]) for te computation of a -means clustering (wit initial values 0 and ). Te resulting initial values are given in Figure togeter wit te values for ν. Te pixels of te input images are interpreted as nodal values of te function u 0 on a uniform mes wit mes size = L0 (L 0 = 9 for (d), L 0 = else). Te algoritm is ten started on a uniform mes of mes size = L init (L init = 3 for (d), L init = 5 else). In all computations we use τ = 0 5 and σ = (Algoritm ), γ = 0.05 for te dual gradient descent and α = 0.. We perform 0 cycles of te adaptive algoritm and refine cells until te dept L 0 of te input image is reaced. We observe sligt local oscillations for te finite element approaces (FE), (FE D ) and (FE ), wic deteriorate te result of te a posteriori estimator (cf. te numerical results in [5]). Tus, in a post-processing step, we compensate tese oscillations prior to te evaluation of te estimator by an application of a smooting filter. Te filter is defined via an implicit time step of te discrete eat equation using affine finite elements on te underlying adaptive mes, i.e. we apply te operator (M + ιs ) M to te solutions, were S denotes te stiffness matrix. For te discretizations

13 image (a) (b) (c) (d) resolution c c ν Figure : Input images togeter wit te corresponding image resolution and te model parameters c, c, and ν (flower image: poto by Derek Ramsey, Canticleer Garden, cameraman image: copyrigt by Massacusetts Institute of Tecnology). (FE) and (FE D ), we coose ι = c min, were min denotes te minimal mes size of te current adaptive grid, wit c = 3 and c = 6 for te primal and te dual solution, respectively. Moreover, in te case (FE ) te smooting is only applied to te dual solution wit parameter ι = a, were a denotes te average cell size on te adaptive mes. In our experiments, we observed tat tese smooting metods and parameters outperformed oter tested coices for te corresponding discretizations. We call te resulting post-processed functions ū and p, respectively, and replace te local error estimator by err u,c [ū, p ]. Table lists (scaled) primal and dual energies, err u, η optimal (te η value corresponding to te optimal a posteriori error bound for given err u) as well as err χ for all input images after te 0 t refinement step of te adaptive algoritm. Te value of err χ peaks for te application (d) due to te relatively low image resolution. Figure plots te error estimator err u after eac refinement step for all input images and all finite element discretizations. In most of our numerical experiments, te sceme (FE) performs comparably to te discretization (FE D ), but sligtly better tan te sceme (FE ). For te flower image, te sequence of adaptive meses and solutions resulting from te adaptive algoritm for te discretization (FE ) is depicted in Figure 3. Figure 4 displays solutions for te discretization (FE ) and te corresponding adaptive meses togeter wit color coded deciles of u, and te graps of η a[u, η] and η err χ. Note tat te displayed deciles explicitly indicate te sets S η for η = 0., 0., 0.3, 0.4. Figures 5 and 6 sow te relaxed solution u and te tresolded solution χ for te input images (b) to (d) using te discretization scemes (FE) and (FD), te corresponding results for (FE D ) are not depicted since tey are almost not distinguisable from te results for (FE). It is known tat solutions of te Mumford-Sa problem are in general not unique. We pick up a classical example for tis non-uniqueness in Figure 7 wit alternating intensity values 0 and on quadrants. We demonstrate te sensitivity of our adaptive sceme wit respect to te topology of te segmentation. In fact, we compute a segmentation using te discretization (FE) (wit initial intensity values c =, c = 0 and ν = 0.0) for sligtly perturbed versions of te original image. I.e. solely te four pixels in te center of te image of resolution are eiter set black or wite, respectively. Te resulting segmentations are sown in Figure 7 along wit te adaptive mes and te decile plots. Te adaptive algoritm is capable to detect properly te decision for one of te two segmentation solutions. Te issue of non-uniqueness is closely related to te flatness of te relaxed solution in te center, in particular leading to an increase of te a posteriori error contribution a[v, η]. Moreover, we applied te above metods to an analytic function consisting of a weigted sum of two Gaussian kernels. To tis end, in eac step te functionals and te error estimator are evaluated on te current adaptive grid and not on a prefixed full resolution grid. Te results are sown in Figure 8 (wit parameters c = , c = and ν = ). 3

14 ν (c c ) E[u ] ν (c c ) D[p ] err u (a) (b) (c) (d) (FE) (FE D ) (FE ) (FD) (FE) (FE D ) (FE ) (FD) (FE) 6.86e (FE D ) 6.63e (FE ) (FD) 9.73e η optimal (FE D ) (FE ) (FE) (FD) err χ (FE D ) (FE ) (FE) (FD) Table : Rescaled dual and primal energy evaluated on te discrete solution (u, p ), error estimator for te relaxed solution, optimal tresold η optimal computed for u and te resulting a posteriori estimator err χ for te L -error of te caracteristic function χ (after 0 cycles of te adaptive algoritm). In addition, we evaluated te sensitivity of te error estimates wit respect to variations in te intensity values. Figure 9 sows te function plot of ɛ err,ɛ u [u, p, c, c ] wit fixed primal u and dual solution p obtained wit te discretization (FE) after te 0 t iteration for te images (b) and (c). Te error estimator for image (b) is less sensitive to small fluctuations in te intensity values compared to image (c) due to te stronger variation of intensity values along te boundary of te segmentation in image (b). Finally, te proposed discretization scemes were compared in terms of te relative CPU time for te images images (b) and (d) in te last iteration. To enforce comparable conditions te stopping criterion was set to P k+ Pk < 0 6 and te primal and dual solution were initialized wit constant values. In comparison wit te discretization sceme (FE), te sceme (FE ) required comparable CPU time (image (b): 6.7%, image (d): +6.5%), wereas (FE D ) performed slower for larger images (image (b): +30.4%, image (d): 0.%). 7 Conclusions We ave investigated te a posteriori error estimation for te binary Mumford-Sa model and applied tis estimate to tree different adaptive finite element discretizations in comparison to a non-adaptive finite difference sceme on a regular grid. Te proposed finite element discretizations 4

15 degrees of freedom degrees of freedom degrees of freedom degrees of freedom (FE) (FE D ) (FE ) err u black ( ) purple (+) red ( ) err χ brown ( ) gray ( ) blue ( ) Figure : Te values of err u and err χ are displayed in relation to te number of degrees of freedom in a log-log plot for te applications (a) (upper left), (b) (upper rigt), (c) (lower left) and (d) (lower rigt). in combination wit te adaptive mes refinement strategy lead to a substantial reduction of te required degrees of freedom wit error values err u and err χ of about te same magnitude as for a standard finite difference sceme on a non-adaptive mes wit mes size equal to te finest mes size of te adaptive meses. To improve te resulting estimate of te duality gap E rel [v] + D rel [q], te finite element scemes require some oscillation damping smooting in a post-processing step. Te proposed approac to a posteriori estimates for te binary Mumford-Sa model derived in tis paper can be applied to more general problems in computer vision. In fact, te calibration metod developed by Alberti, Boucitté and Dal Maso [] provides a convex relaxation of non-convex functionals of Mumford-Sa type via te lifting of a variational problem on a n- dimensional domain to a minimization problem over caracteristic functions of subgraps in n + dimensions. In te context of non-convex functionals in vision, tis approac was studied by Pock et al. [39, 40]. Applications of suc functionals include te computation of minimal partitions [6, 4], te dept map identification from stereo images or te robust extraction of optimal flow fields [40]. Here, an adaptive mes strategy is expected to ave an even larger pay-off due to te increased dimension. Acknowledgements A. Effland and M. Rumpf acknowledge support of te Collaborative Researc Centre 060 and te Hausdorff Center for Matematics, bot funded by te German Science foundation. B. Berkels 5

16 Figure 3: Te sequence of solutions u and a color coding of te corresponding fineness of te adaptive meses at te st, nd, 3rd, 4t and 5t iteration of te adaptive algoritm applied to te input image (c) and computed using te (FE ) discretization. was funded in part by te Excellence Initiative of te German Federal and State Governments. Furtermore, we would like to tank te anonymous referees for teir valuable comments to improve tis article. References [] G. Alberti, G. Boucitte, and G. Dal Maso. Te calibration metod for te Mumford-Sa functional and free-discontinuity problems. Calc. Var. Partial Differential Equations, 6 (3):99 333, 003. [] L. Ambrosio. Variational problems in SBV and image segmentation. Acta Appl. Mat., 7: 40, 989. [3] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Matematical Monograps. Oxford University Press, New York, 000. [4] L. Ambrosio and V. M. Tortorelli. On te approximation of free discontinuity problems. Bollettino dell Unione Matematica Italiana, Sezione B, 6(7):05 3, 99. [5] S. Bartels. Error control and adaptivity for a variational model problem defined on functions of bounded variation. Mat. Comp. (online first), 04. [6] S. Bartels. Broken Sobolev space iteration for total variation regularized minimization problems. IMA Journal of Numerical Analysis, 05. [7] B. Berkels. An unconstrained multipase tresolding approac for image segmentation. In Proceedings of te Second International Conference on Scale Space Metods and Variational Metods in Computer Vision (SSVM 009), volume 5567 of Lecture Notes in Computer Science, pages Springer, 009. [8] B. Berkels. Joint metods in imaging based on diffuse image representations. Dissertation, University of Bonn, 00. [9] B. Bourdin and A. Cambolle. Implementation of an adaptive Finite-Element approximation of te Mumford-Sa functional. Numer. Mat., 85(4): , 000. [0] S. Boyd, N. Parik, E. Cu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via te alternating direction metod of multipliers. Foundations and Trends in Macine Learning, 3:, 00. [] L. M. Bregman. Te relaxation metod of finding te common point of convex sets and its application to te solution of problems in convex programming. USSR Computational Matematics and Matematical Pysics, 7:00 7, 967. [] P. H. Calamai and J. J. More. Projected gradient metods for linearly constrained problems. Matematical Programming, 39:93 6,

17 η η Figure 4: For images (b) (first and second row) and (c) (last two rows) and te discretization (FE ), te components of te relaxed solution u, (p ), (p ) and te resulting solution χ are sown after te 0t iteration of te adaptive sceme. In te second and fourt row, te adaptive grid (after te 6t refinement step), deciles of te discrete solution u encoded wit different colors, and te functions η a[u, η] (red solid line) and η errχ (blue dased line) are rendered. [3] A. Cambolle. Image segmentation by variatonal metods: Mumford-Sa functional and te discrete approximations. SIAM J. Appl. Mat., 55(3):87 863, 995. [4] A. Cambolle. An algoritm for total variation minimization and applications. Journal of Matematical Imaging and Vision, 0(-):89 97, November 004. [5] A. Cambolle. Total variation minimization and a class of binary MRF models. In A. Rangarajan et al., editors, Energy Minimization Metods in Computer Vision and Pattern Recognition, volume 3757 of LNCS, pages Springer, 005. [6] A. Cambolle, D. Cremers, and T. Pock. A convex approac for computing minimal partitions. Tecnical Report 649, Ecole Polytecnique, Centre de Mate matiques applique es, UMR CNRS 764, November 008. [7] A. Cambolle, D. Cremers, and T. Pock. A convex approac to minimal partitions. SIAM Journal on Imaging Sciences, 5(4):3 58, 0. 7

18 Figure 5: Relaxed solution u, (p ), (p ) for te input image (b) (top row) and (c) (bottom row) using te discretization (FE) (left, after 0 iterations of te adaptive algoritm) and (FD) (rigt). Figure 6: Te mes in te 5t and 0t iteration, te relaxed solution u, (p ), (p ) and χ for te input image (d) using te discretization (FE ) after 0 iterations of te algoritm. [8] A. Cambolle and G. Dal Maso. Discrete approximation of te Mumford-Sa functional in dimension two. Matematical Modelling and Numerical Analysis, 33 (4):65 67, 999. [9] A. Cambolle and J. Darbon. On total variation minimization and surface evolution using parametric maximum flows. International Journal of Computer Vision, 84(3):88 307, 009. [0] A. Cambolle and T. Pock. A first-order primal-dual algoritm for convex problems wit applications to imaging. Journal of Matematical Imaging and Vision, 40():0 45, 0. [] T. F. Can and L. A. Vese. A level set algoritm for minimizing te Mumford-Sa functional in image processing. In IEEE/Computer Society Proceedings of te st IEEE Worksop on Variational and Level Set Metods in Computer Vision, pages 6 68, 00. [] Y. Cen, T. A. Davis, W. W. Hager, and S. Rajamanickam. Algoritm 887: CHOLMOD, supernodal sparse Colesky factorization and update/downdate. ACM Transactions on Matematical Software, 35(3):: :4, 009. [3] D. C. Dobson and C. R. Vogel. Convergence of an iterative metod for total variation denoising. SIAM J. Numer. Anal., 34(5):779 79, Oct [4] I. Ekeland and R. Te mam. Convex analysis and variational problems. Society for Industrial and Applied Matematics, Piladelpia, PA, USA, 999. [5] L. Evans and R. Gariepy. Measure Teory and Fine Properties of Functions. CRC Press, 99. [6] X. Feng and A. Prol. Analysis of total variation flow and its finite element approximations. MAN, 37(3): , 00. [7] T. Goldstein and S. Oser. Te split Bregman metod for L -regularized problems. SIAM J. Imaging Sci., ():33 343, 009. [8] W. Han. A Posteriori Error Analysis Via Duality Teory, volume 8 of Advances in Mecanics and Matematics. Springer, 005. [9] M. Hintermu ller and K. Kunisc. Pat-following metods for a class of constrained minimization problems in function space. Tecnical report, Department of Computational an Applied Matematics, Rice University, Houston, Texas, July

19 Figure 7: Te original input image wit resolution and te adaptive mes after te 6 t iteration wit wite pixels in te center of te input image (first column). Second to fourt column: te relaxed solution u, te tresolded solution χ and te decile plots after te 0 t iteration for black pixels in te center (first row) and wite pixels in te center (second row) along wit te corresponding zoom of te center (wit zoom factor 8) using te discretization (FE). [30] M. Hintermüller and K. Kunisc. Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM J. Appl. Mat., 64(4):3 333, 004. [3] K. Jalalzai. Discontinuities of te minimizers of te weigted or anisotropic total variation for image reconstruction. arxiv:40.006, 04. [3] S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Teory, 8():9 37, 98. [33] L. Modica and S. Mortola. Un esempio di Γ -convergenza. Boll. Un. Mat. Ital. B (5), 4():85 99, 977. [34] J.-M. Morel and S. Solimini. Variational metods in image segmentation. Progress in Nonlinear Differential Equations and teir Applications, 4. Birkäuser Boston Inc., Boston, MA, 995. [35] D. Mumford and J. Sa. Optimal approximation by piecewise smoot functions and associated variational problems. Communications on Pure and Applied Matematics, 4(5): , 989. [36] M. Nikolova, S. Esedoḡlu, and T. F. Can. Algoritms for finding global minimizers of image segmentation and denoising models. SIAM Journal on Applied Matematics, 66(5):63 648, 006. [37] S. Özışık, B. Riviere, and T. Warburton. On te constants in inverse inequalities in L. Tecnical Report 0-9, Department of Computational & Applied Matematics, Rice University, 00. [38] T. Pock. Diagonal preconditioning for first order primal-dual algoritms in convex optimization. In IEEE International Conference on Computer Vision (ICCV), pages , 0. [39] T. Pock, D. Cremers, H. Biscof, and A. Cambolle. An algoritm for minimizing te Mumford-Sa functional. In IEEE t International Conference on Computer Vision, 009. [40] T. Pock, D. Cremers, H. Biscof, and A. Cambolle. Global solutions of variational models wit convex regularization. SIAM Journal on Imaging Sciences, 3(4): 45, 00. [4] T. Pock, T. Scoenemann, G. Graber, H. Biscof, and D. Cremers. A convex formulation of continuous multi-label problems. In ECCV08, pages III: , 008. [4] S. Repin. A posteriori error estimation for variational problems wit uniformly convex functionals. Matematics of Computation, 69:48 500, 000. [43] S. Repin. A Posteriori Estimates for Partial Differential Equations. Radon Series on Computational and Applied Matematics. Wal,

20 degrees of freedom Figure 8: First row: Input image u 0 composed by te superposition of two Gaussian kernels, numerical solutions u, (p ) and (p ) computed via te adaptive algoritm using discretization (FE ). Second row: χ, deciles and te error estimators in a log-log plot (err u in red, err χ in blue for te discretization (FE ) and err u in black, err χ in brown for te discretization (FE)) Figure 9: Te function plot ɛ err,ɛ u for te images (b) (red solid line) and (c) (blue dased line) using te discretization (FE) after te 0 t iteration. ε [44] R. T. Rockafellar. Convex Analysis. Princeton University Press, 997. [45] L. Rudin, S. Oser, and E. Fatemi. Nonlinear total variation based noise removal algoritms. Pysica D, 60:59 68, 99. [46] J. Sen. Γ-convergence approximation to piecewise constant Mumford-Sa segmentation. In Proceedings of te 7t International Conference on Advanced Concepts for Intelligent Vision Systems (ACIVS 005), volume 3708 of Lecture Notes in Computer Science, pages Springer, 005. [47] J. Wang and B. Lucier. Error bounds for finite-difference metods for Rudin-Oser-Fatemi image smooting. SIAM Journal on Numerical Analysis, 49(): , 0. 0