ITERATIVE SOLUTION OF A CONSTRAINED TOTAL VARIATION REGULARIZED MODEL PROBLEM

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1 ITERATIVE SOLUTION OF A CONSTRAINED TOTAL VARIATION REGULARIZED MODEL PROBLEM Abstract. Te discretization of a bilaterally constrained total variation minimization problem wit conforming low order finite elements is analyzed and tree iterative scemes are proposed wic differ in te treatment of te non-differentiable terms. Unconditional stability and convergence of te algoritms is addressed, an application to piecewise constant image segmentation is presented and numerical experiments are sown. 1. Introduction In tis paper we investigate iterative scemes for te solution of te bilaterally constrained ROF problem wic consists in te minimization of te functional E(u) := Du () + α 2 u g 2 L 2 () + I K(u) in te set of functions wit bounded total variation Du (), were I K is te indicator functional of a convex set K L 2 () caracterized by two obstacle functions. Wile existence and uniqueness of a minimizer of E can be establised using te direct metod in te calculus of variations, te non-differentiability of te total variation and te indicator functional are callenging from a numerical point of view. Te minimization problem serves as a model problem for a wide class of functionals involving te total variation and te indicator functional wic comprises a bilateral constraint on te variable u. For example, in [5, 11] te above minimization problem is considered in te context of image denoising and te bilateral constraint is imposed in order for te restored image to ave pixel values lying in a certain range, e.g., in te interval [0, 255]. In [10] Can et al. sow te equivalence of a minimization problem occuring in image segmentation to te minimization of a functional tat resembles E wit te quadratic fidelity term replaced by a linear functional in u. Te application of a regularization term to a bilaterally constrained variable also occurs in te modelling of damage processes in continuum mecanics, were te bilaterally constrained Date: May 23, Matematics Subject Classification. 65K15, 49M27, 94A08. Key words and prases. total variation, bilateral constraint, finite elements, iterative scemes, image processing. 1

2 2 variable is te damage variable and te regularization term consists of a differentiable L p -norm (p 2) of te gradient of te damage variable (cf. [18]) or te total variation of te damage variable (cf. [20]) wile te constraint ensures tat te damage variable takes on values only in te interval [0, 1]. In te recent years many PDE-based algoritms ave been proposed for te classical ROF functional proposed by Rudin, Oser and Fatemi in [19], were te bilateral constraint is omitted, reacing from primal-dual metods, combinations of regularization and standard minimization tecniques and dual metods to te alternate direction metod of multipliers, augmented Lagrangian metods and split Brègman metods, see, e.g., [3, 22] for an overview of many of tose algoritms. Regarding te bilaterally constrained ROF problem Casas et al. proposed in [7] an active set strategy using te anisotropic BV -seminorm. Tis ansatz works wit te primal formulation of te problem and does not use regularization. However, a penalization approac is used to solve auxiliary subproblems. In [5], motivated by te dual approac of Cambolle in [8], Beck and Teboulle suggest a monotone fast iterative srinkage/tresolding algoritm (MFISTA) for te dual of te bilaterally constrained ROF problem for wic te convergence rate O(1/k 2 ) wit k being te iteration counter is proven. Yet, as in [8], te constant entering te convergence rate grows like O(1/ 2 ) wit being te mes size of te underlying grid. Can et al. proposed in [11] an augmented Lagrangian metod, wic as already been successfully applied to te ROF problem by Wu and Tai in [21]. Te aim of tis paper is to compare tree different approaces for te iterative minimization of E and to identify te most appropriate metod wit particular view to applications in materials science. To tis extent, we introduce a finite element discretization wit globally continuous piecewise affine finite elements. Te iterative algoritms differ in te treatment of te two callenging terms appearing in E. Te first approac uses splitting tecniques for bot te total variation and te bilateral constraint and is a variant of te augmented Lagrangian metod proposed in [11], were te involved norms are properly weigted in order to ensure unconditional stability of te metod, see also [3] were te autors proposed a properly weigted augmented Lagrangian metod for te classical ROF problem. Te next ansatz uses regularization to andle te non-differentiability of te total variation and te indicator functional and employs a semi-implicit subgradient flow wic turns out to be unconditionally stable and will be referred to as te Heron-penalty metod. For te classical ROF problem te Heron metod as already been introduced in [3]. Te tird algoritm uses a combination of te regularization and te splitting approac, i.e., using te observation made in [3] tat a regularization approac seems to be te most appropriate to deal wit te total variation, we employ a regularization for te total variation and a splitting tecnique for te bilateral constraint,

3 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 3 wic we call te Heron-split metod. Te outline of te paper is as follows. In Section 2 we give some basic notation and introduce te space of functions wit bounded variation as well as te globally continuous piecewise affine and te piecewise constant finite element space. We present te minimization problem in Section 3 and discuss existence and uniqueness of a minimizer. In Section 4 we consider te finite element discretization and prove convergence of discrete minimizers to te continuous solution before we present te tree algoritms and address teir stability in Section 5. In Section 6 we address te minimization of te functional E seg (u) := Du + α fu dx + I K (u), prove convergence of discrete energies to te minimal energy and recall a minimization problem arising in an image segmentation model discussed by Can et al. in [10]. We finally provide results of numerical experiments for te tree iterative scemes applied to te bilaterally constrained ROF problem and for a particular coice of g and two different values of α as well as for te image segmentation problem in Section Preliminaries 2.1. Notation. Let R d, d = 2, 3, be a bounded Lipscitz domain. We denote by te L 2 -norm on induced by te scalar product (v, w) := v w dx for scalar functions or vector fields v, w L 2 (; R l ), l {1, d}. Te Euclidean norm will be denoted by. We will work wit te space of functions of bounded variation BV () L 1 () wic contains all functions v L 1 () for wic te total variation given by Du () = Du := sup { u div q dx : q Cc 1 (; R d ), q 1 a.e. } is finite. Te space BV () equipped wit te norm u BV := u L 1 () + Du () is a Banac space. For more details concerning te space BV () we refer, e.g., to [1]. For a sequence (a j ) j N and a step size τ > 0 we let be te backward difference quotient. d t a j+1 := aj+1 a j τ

4 Finite element spaces. Trougout te paper we let c be a generic constant. We denote by (T ) >0 a family of uniform triangulations of wit mes sizes = max T T T were T is te diameter of te simplex T. For a given triangulation T te set N contains te corresponding nodes and we consider te finite element spaces of continuous, piecewise affine functions S 1 (T ) := { v C() : v T affine for all T T } and of elementwise constant functions (r = 1) or vector fields (r = d) L 0 (T ) r := { q L (; R r ) : q T constant for all T T }. We also denote by T l, l N, a triangulation of generated from an initial triangulation T 0 by l uniform refinements and te integer l being related to te mes size by = c2 l. Te set of nodes N l is defined correspondigly. We introduce on S 1 (T ) te discrete norm induced by te discrete scalar product (v, w) := z N β z v(z)w(z) for v, w S 1 (T ), were β z = ϕ z dx and ϕ z S 1 (T ) is te nodal basis function associated wit te node z N. Te norm and te L 2 -norm are related on S 1 (T ) as follows. Lemma 2.1. For v S 1 (T ) we ave v v (d + 2) 1/2 v. Te proof of Lemma 2.1 can be found, e.g., in [2, Lemma 3.9]. 3. Model problem For a function g L () and a parameter α > 0 we consider te constrained minimization problem defined by te functional E(u) := Du + α 2 u g 2 + I K (u) in te set of functions BV () L 2 (). Te convex set K is given by K := {v L 2 () : χ v ψ a.e.} wit χ, ψ BV () L () and χ < ψ almost everywere. We will refer to tis minimization problem as te bilaterally constrained ROF problem. Proposition 3.1 (Existence and Uniqueness). Tere exists a unique minimizer u BV () L 2 () of E. Proof. Te set of feasible functions is nonempty and te functional E is bounded from below. Hence, tere exists an infimizing sequence (u j ) j wic is bounded in BV () L 2 () due to te coercivity of E. Te compact embedding of BV () into L 1 () (cf. [1]) and te boundedness of te sequence imply te existence of a - not relabelled - weakly convergent subsequence

5 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 5 u j u in BV () and by passing to a furter subsequence u j u in BV () L 2 (). Since u j u in L 1 () tere exists a furter subsequence suc tat u j u almost everywere in. Tis implies tat χ u ψ almost everywere in since χ u j ψ for all j N. Te weak lower semicontinuity of E (cf. [1]) ten yields tat u is a minimizer of E. Uniqueness follows by te strict convexity of E. Remark 3.2. In fact, we ave u BV () L () since χ, ψ L (). We define te functionals G : L 2 () R { }, H : L 2 () R { } and F : L 2 () R { } by G(v) := α 2 v g 2, H(v) := I K (v) and Dv, if v BV () L 2 (), F (v) :=, if v L 2 () \ BV (). Lemma 3.3. Let u be te unique minimizer of E in BV () L 2 (). Ten for any v BV () L 2 () we ave α 2 u v 2 E(v) E(u). Proof. We note tat G is Frécet-differentiable, i.e., we ave G(v) = {δg(v)} wit δg(v)[w] = α(v g, w) for v, w L 2 (). Since u is a minimizer of E, we ave 0 E(u) = (F + G + H)(u). Standard arguments from convex analysis (cf. [1, Tm ]) yield δg(u) (F + H)(u). Te strong convexity of G and te subgradient inequality ten imply tat for arbitrary v BV () L 2 () α 2 u v 2 = δg(u)[v u] + G(v) G(u) F (v) + G(v) + H(v) F (u) G(u) H(u) wic proves te lemma. = E(v) E(u), 4. Finite element discretization For simplicity, we assume trougout te paper tat for a family of triangulations (T ) >0 te obstacle functions χ and ψ are suc tat tere exists 0 > 0 and χ, ψ S 1 (T ) for all 0 and we assume 0 in te sequel. Let us ten remark tat I K (v ) is finite for v S 1 (T ) if and only if χ(z) v (z) ψ(z) at all nodes z N. We ten seek a discrete minimizer u S 1 (T ) of E(u ) = u dx + α 2 u g 2 + I K (u )

6 6 in S 1 (T ), were we used tat Dv () = v L 1 () for v W 1,1 (). Existence and uniqueness can be proven following te arguments of te proof of Proposition 3.1. Proposition 4.1. Let u and u be te unique minimizers of E in BV () L 2 () and S 1 (T ), respectively. Ten we ave α 2 u u 2 c 1/2. Proof. Following te arguments in te proof of [2, Tm. 10.7] we note tat by Lemma 3.3 we ave for v = u and for arbitrary v S 1 (T ) tat (1) α 2 u u 2 E(u ) E(u) E(v ) E(u). Let u ε, S 1 (T ) be as in [2, Lemma 10.1], i.e., suc tat u ε, u wit respect to intermediate convergence. We define te truncation v v K for v L 1 () by v K := max{min{ψ, v}, χ} pointwise. One can ceck tat te mapping v v K is Lipscitz continuous wit respect to te L 1 -norm. Let J : L 1 () S 1 (T ) be te Cen-Nocetto quasi-interpolation operator introduced in [12] by Cen and Nocetto. We ten ave J v K if v K. Coosing v = J u K ε, in (1) and using J u K ε, L 1 () u K ε, L 1 () u ε, L 1 (), te binomial formula, te identity u K = u and te Lipscitz continuity of te truncation operator we obtain E(J u K ε, ) E(u) = J u K ε, dx + α 2 J u K ε, g 2 Du + α u g 2 2 u ε, dx Du + α (J u K ε, 2 u)(j u K ε, + u 2g) dx c(ε 1 + ε) Du () + α 2 J u K ε, uk L 1 () J u K ε, + u 2g L () c(ε 1 + ε) Du () + c α ( J u K ε, 2 uk ε, L 1 () + u K ε, ) uk L 1 () c(ε 1 + ε) Du () + c α ( ) uε, 2 L 1 () + u ε, u L 1 () c(ε 1 + ε) Du () + c( + 2 ε 1 + ε + ε) Du (), were we ave used te quasi-interpolation estimate v J v L 1 () c v L 1 () for all v W 1,1 () (cf. [12, 16]). Te coice ε = 1/2 concludes te proof.

7 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 7 Remark 4.2. Te error estimate is suboptimal in te sense tat for u BV () L () we ave te approximation property inf u v c 1/2, v S 1 (T ) cf. [4, Tm. 6.5]. On igly symmetric triangulations and for an anisotropic version of te total variation tis quasioptimal convergence rate can be proved for related minimization problems, cf. [4]. 5. Iterative solution Te iterative minimization of E is difficult due to te non-differentiability of te seminorm and te occurrence of te constraints. In tis section we will discuss tree approaces to deal wit tese difficulties Splitting. We introduce for u S 1 (T ) te auxiliary variables p = u L 0 (T ) d and s = u S 1 (T ) and incorporate Lagrange multipliers λ L 0 (T ) d and η S 1 (T ) to enforce tese relations. Note tat since te sequence ( u ) >0 of discrete minimizers u of E may not be bounded in L 2 () but only in L 1 () we ave to measure tese quantities in a weaker norm tan te L 2 -norm. We terefore equip te space L 0 (T ) d wit te weigted L 2 -scalar product (p, q ) w := d p q dx and induced norm w. An inverse estimate sows tat ( u ) >0 is bounded wit respect to w, cf. [3]. We ten consider te consistently stabilized Lagrange functional L (u, p, s ; λ, η ) := p dx + α 2 u g 2 + I K (s ) + (λ, p u ) w + σ 1 2 p u 2 w + (η, s u ) + σ 2 2 s u 2. Te parameters σ 1 and σ 2 are assumed to be positive. We ten ave tat (u, p, s ; λ, η ) is a saddle point for L if and only if u minimizes E (cf. [21, Tm. 4.1] for te same result for te ROF functional witout bilateral constraint). Indeed, since L (u, p, s ; µ, ν ) L (u, p, s ; λ, η ) for all (µ, ν ) we obtain p = u and s = u. Ten we ave E(u ) = L (u, p, s ; λ, µ ) L (v, v, v ; λ, µ ) = v dx + α 2 v g 2 + I K (v ) = E(v ) for all v S 1 (T ), i.e., u minmizes E. Conversely, if u minimizes E we coose p = u and s = u. Since u is te unique minimizer of E we ave 0 E(u ) = F (u ) + G(u ) + H(u ),

8 8 cf. [17, Tm. 23.8]. In particular, tere exist λ F ( u ) and η H(u ) wit (2) α(u g) = div λ + η, cf. [17, Tm. 23.9], were te operator div : L 0 (T ) S 1 (T ) is defined by ( div µ, v ) = (µ, v ) for all v S 1 (T ), µ L 0 (T ), and F (q ) := q L 1 (). Wit our particular coice of p and s we ave L (u, p, s ; µ, ν ) = L (u, p, s ; λ, η ) for all (µ, ν ). One can sow tat te function (v, q, r ) L (v, q, r ; λ, η ) as a minimizer (v, q, r ). Te optimality conditions read σ 1 (q v, v ) w + σ 2 (r v, v ) +(λ, v ) w + (η, v ) = α(v g, v ), λ + σ 1 ( v q ) F (q ), η + σ 2 (v r ) H(r ). Using (2) and te particular coice of functions λ and η we observe tat te triple (u, p, s ) satisfies tese optimality conditions. Altogeter, we ave min u E(u ) = min (u,p,s ) max (λ,η ) L (u, p, s ; λ, η ). Due to te fact tat p is elementwise constant and tat mass lumping is used for te Lagrange multiplier η and te auxiliary variable s te minimization wit respect to p and s can be done explicitly on eac element and in eac node, respectively. We approximate a saddle-point wit te following iterative sceme (cf. [13]). Algoritm 5.1 (Split-split metod). Coose (p 0, s0 ) L0 (T ) d S 1 (T ) and (λ 0, η0 ) L0 (T ) d S 1 (T ) and set j = 0. (1) Compute te minimizer u j+1 of i.e., find u j+1 α(u j+1 S 1 (T ) suc tat u L (u, p j, sj ; λj, ηj ), g, v ) (λ j, v ) w σ 1 (p j uj+1, v ) w for all v S 1 (T ). (2) Compute te minimizing (p j+1, s j+1 ) of (η j, v ) σ 2 (s j uj+1, v ) = 0 (p, s ) L (u j+1, p, s ; λ j, ηj ),

9 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 9 i.e., define (p j+1, s j+1 ) L 0 (T ) d S 1 (T ) via p j+1 = 1 σ 1 ( σ1 u j+1 were (t) + = max{t, 0}, and for all z N. (3) Update { { s (z) = max χ(z), min λ j d) + u j+1 σ 1 u j+1 λ j σ 1 u j+1 λ j, (z) ηj (z) }}, ψ(z). σ 2 λ j+1 = λ j + σ 1(p j+1 u j+1 ), η j+1 = η j + σ 2(s j+1 u j+1 ), 1 and stop if σ 1 λ j+1 λ j w + p j+1 p j w ε stop and 1 σ 2 η j+1 s j ε stop. Increase j j + 1 and continue wit (1) oterwise. s j+1 η j + Te following stability estimate for te split-split metod is a consequence of general results for splitting metods, see, e.g., [13, 14], and arguments exploiting only te (strong) convexity of te involved functionals, wic ave been used by Wu and Tai in [21] for te classical ROF problem. It implies te convergence λ j+1 λ j w + p j+1 p j w 0 and η j+1 η j + s j+1 s j 0, i.e., Algoritm 5.1 terminates. Proposition 5.2. Let (u, p, s ; λ, η ) be a saddle-point of L wit p = u and s = u. For arbitrary initializations of Algoritm 5.1 we ave for any J 0 1( η η J λ λ J+1 2 w + σ1 p 2 p J J ( σ1 p j+1 2 p j 2 w + σ 2 s j+1 s j 2 j=0 2 w + σ2 s 2 s J σ 1 λ j+1 λ j 2 w + 1 σ 2 η j+1 η j 2 + α u u j+1 2) 1 2( η η λ λ 0 2 w + σ 2 1 u p 0 2 w + σ 2 2 u s 0 2 ) Regularization and penalization. Anoter way to approximate te discrete minimizer u of E is to regularize te BV -seminorm, introduce an auxiliary variable s = u and to penalize te difference s u wit a penalization parameter δ > 0 wile allowing u to penetrate te obstacles. We obtain te functional E,ε,δ (v, r ) := v ε dx + α 2 v g δ r v 2 + I K(r ), 2 )

10 10 were v ε = ( v 2 + ε 2 ) 1/2 and ε, δ > 0. Te introduction of s decouples te two nonlinearities. A similar approac as been used by Hintermüller et al. in [15] for te predual of te classical ROF problem and an alternate minimization tecnique as been employed for te numerical solution. Remark 5.3. Note tat in contrast to te oter two metods, te constraint u K is enforced only in te limit as δ 0. We can establis te existence of a discrete minimizer (u,ε,δ, s,ε,δ ) S 1 (T ) S 1 (T ) of E,ε,δ using te direct metod in te calculus of variations. Let Π K, be te projection operator from S 1 (T ) onto K wit respect to (, ), i.e., Π K, is defined by v (z), if χ(z) v (z) ψ(z), Π K, v (z) := ψ(z), if v (z) > ψ(z), χ(z), if v (z) < χ(z), for v S 1 (T ) and all z N. Te optimality conditions for a minimizing pair (u,ε,δ, s,ε,δ ) ten imply te relation s,ε,δ = Π K, u,ε,δ. Proposition 5.4. Let u S 1 (T ) and (u,ε,δ, s,ε,δ ) S 1 (T ) S 1 (T ) be te unique minimizer of E and a minimizer of E,ε,δ, respectively, and let (v, r ) S 1 (T ) S 1 (T ) be arbitrary. Ten we ave α (3) 2 v u,ε,δ 2 E,ε,δ (v, r ) E,ε,δ (u,ε,δ, s,ε,δ ), and in particular α 2 u u,ε,δ δ s,ε,δ u,ε,δ 2 c(ε + δ), α 2 u Π K, u,ε,δ 2 c(ε + δ). Proof. Te proof of (3) is analogous to te proof of Lemma 3.3. Since u K we can coose (v, r ) = (u, u ) in (3). Observing tat x x ε x +ε for x R d, using Lemma 2.1, te minimality property of u, te identity s,ε,δ = Π K, u,ε,δ, te estimate Π K, u,ε,δ L 1 () u,ε,δ L 1 (), a binomial formula and Young s inequality we deduce α 2 u u,ε,δ 2 E,ε,δ (u, u ) E,ε,δ (u,ε,δ, s,ε,δ ) = u ε dx + α 2 u g 2 u,ε,δ ε dx α 2 u,ε,δ g 2 1 2δ s,ε,δ u,ε,δ 2 ε + u dx + α 2 u g 2

11 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 11 u,ε,δ dx α 2 u,ε,δ g 2 1 2δ s,ε,δ u,ε,δ 2 ε + Π K, u,ε,δ dx + α 2 Π K,u,ε,δ g 2 u,ε,δ dx α 2 u,ε,δ g 2 1 2δ s,ε,δ u,ε,δ 2 ε 1 2δ s,ε,δ u,ε,δ 2 + α (s,ε,δ u,ε,δ )(s,ε,δ + u,ε,δ 2g) dx 2 ε 1 2δ s,ε,δ u,ε,δ δ s,ε,δ u,ε,δ 2 + α2 4 δ s,ε,δ + u,ε,δ 2g 2 = ε 1 4δ s,ε,δ u,ε,δ 2 + α2 4 δ s,ε,δ + u,ε,δ 2g 2, wic implies te second estimate. To prove te tird estimate we note tat similarly to Lemma 3.3 we ave α 2 v u 2 E(v ) E(u ) for arbitrary v S 1 (T ) K. Coosing v = s,ε,δ we deduce α 2 u s,ε,δ 2 s,ε,δ dx + α 2 s,ε,δ g 2 u dx α 2 u g 2 s,ε,δ ε dx + α 2 s,ε,δ g 2 u ε dx α 2 u g 2 + ε s,ε,δ ε dx + α 2 s,ε,δ g 2 u,ε,δ ε dx α 2 u,ε,δ g 2 + ε ε + α 2 s,ε,δ u,ε,δ s,ε,δ + u,ε,δ 2g, were te tird inequality is due to te fact tat u,ε,δ ε + α 2 u,ε,δ g 2 u,ε,δ ε + α 2 u,ε,δ g δ u,ε,δ s,ε,δ 2 u ε + α 2 u g 2. Te second estimate ten implies te tird estimate.

12 12 Remark 5.5. Wit a view to recovering te convergence rate from Proposition 4.1 we coose ε = 1/2 / and δ = 1/2 /α or ε = / and δ = /α to recover te optimal convergence rate O( 1/2 ), cf. [2, Rmk (ii)] and [4, Tm. 6.5]. In order to define a stable numerical metod for te approximation of a minimizer (u,ε,δ, s,ε,δ ) S 1 (T ) S 1 (T ) of E,ε,δ we consider for an auxiliary variable p = u,ε,δ 1/2 ε te functional v 2 + ε 2 Ẽ,ε,δ (v, q, r ) := + q2 2 dx + α 2 v g 2 2q 2 Note tat for fixed (v, r ) it follows tat wit optimal q given by + 1 2δ r v 2 + I K(r ). min Ẽ,ε,δ (v, q, r ) = E,ε,δ (v, r ) q L 0 (T ),q >0 q = ( v 2 + ε 2 ) 1/4. Due to te seperate convexity of te functional Ẽ,ε,δ wit respect to v, q and r we use a decoupled semi-implicit subgradient flow wose iterates can be determined explicitly. Algoritm 5.6 (Heron-penalty metod). Coose (u 0, p0, s0 ) S1 (T ) L 0 (T ) S 1 (T ) suc tat χ s 0 ψ and p0 ε. Set j = 0. (1) Compute te minimizer u j+1 of u Ẽ,ε,δ(u, p j, sj ) + 1 2τ u u j 2, i.e., u j+1 S 1 (T ) suc tat ( d t u j+1, v ) = u j+1 + α(u j+1 v (p j dx )2 for all v S 1 (T ). (2) Compute te minimizing (p j+1, s j+1 ) of g, v ) 1 δ (sj uj+1, v ) (p, s ) I,ε,δ (u j+1, p, s ) + 1 2τ p p j τ s s j 2, i.e., define (p j+1, s j+1 ) L 0 (T ) d S 1 (T ) via ( ( d t p j+1, q ) = p j+1 uj ε 2 (p j+1 ) 3 ) q dx

13 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 13 for all q L 0 (T ), and { s j+1 (z) = max χ(z), min { δs j (z) + τuj+1 τ + δ (z) }}, ψ(z) for all z N. (3) Stop if d t u j+1 + d t p j+1 + d t s j+1 ε stop. Oterwise, increase j j + 1 and continue wit (1). Te following proposition guarantees stability and convergence of te algoritm. Proposition 5.7. Let τ > 0, u 0 S1 (T ), s 0 S1 (T ) wit χ s 0 ψ and p 0 L0 (T ) wit p 0 > 0 be arbitrary initializations of Algoritm 5.6 and let te sequences (u j ) j N, (s j ) j N and (p j ) j N be generated by Algoritm 5.6. Ten for arbitrary J N te stability estimate (4) τ J j=0 d t u j d t p j d t s j Ẽ,ε,δ(u J, pj, sj ) Ẽ,ε,δ(u 0, p0, s0 ), olds. Particularly, Algoritm 5.6 terminates and ( Ẽ,ε,δ (u j, pj, sj )) j N is monotonically decreasing. Furtermore, every convergent subsequence (u j l, sj l ) l N S 1 (T ) S 1 (T ) converges to a minimizer (u,ε,δ, s,ε,δ ) S 1 (T ) S 1 (T ) of E,ε,δ. Proof. Te proof of te stability estimate follows as in te proof of [3, Teorem III.4] by coosing d t u j+1, d t p j+1 and d t s j+1 in te optimality conditions defining te iterates in Algoritm 5.6 and using te seperate convexity of Ẽ,ε,δ. Te stability estimate (4) implies tat te sequence (u j ) j is bounded. We coose a subsequence (u j l ) l wic converges to some u,ε,δ S 1 (T ). Since d t u j 0 it follows tat uj l+1 u,ε,δ as well and by te equivalence of norms on finite-dimensional spaces we also ave u j l u,ε,δ and u j l+1 u,ε,δ. Furtermore, since d t p j l+1 0 te optimality condition for te iterates (p j l+1 ) l implies u j l ε 2 (p j p jl+1 l+1 ) 3 0, wic means tat p j l ( u,ε,δ 2 + ε 2 ) 1/4. Te optimality condition for te iterate u j l+1 and d t u j l+1 0 ten imply tat 1 δ (sj l uj l+1, v ) u,ε,δ v ( u,ε,δ 2 + ε 2 dx + α ) 1/2 (u,ε,δ g)v dx

14 14 as l for all v S 1 (T ), i.e., (s j l ) l converges to some function s,ε,δ S 1 (T ). Te optimality condition for te iterate s j l+1 reads ( d t s j l+1, r s j l+1 ) + 1 δ (uj l+1 s j l+1, r s j l+1 ) + I K (s j l+1 ) I K (r ) for all r K S 1 (T ). Tis means tat for l we ave 1 δ (u,ε,δ s,ε,δ, r s,ε,δ ) + I K (s,ε,δ ) I K (r ) for all r K S 1 (T ), i.e., (u,ε,δ, s,ε,δ ) satisfies te necessary and sufficient optimality conditions for a minimizer of E,ε,δ. Remark 5.8. Note tat due to Proposition 5.4 all minimizers (u,ε,δ, s,ε,δ ) of E,ε,δ are witin te ball in L 2 () around te unique minimizer u of E wit radius c(ε + δ). Terefore, by Proposition 5.7 te iterates (u j, sj ) generated by Algoritm 5.6 get arbitrarily close to tis ball around u Regularization and splitting. Alternatively, we may consider only a regularization of te BV -seminorm and seek a minimizer u,ε S 1 (T ) of te functional E ε (v ) := v ε dx + α 2 v g 2 + I K (v ). In order to enforce te constraint strictly we can use an augmented Lagrangian ansatz, i.e., work wit te regularized augmented Lagrangian functional L,ε (v, r ; ν ) := v ε dx + α 2 v g 2 + (ν, r v ) + σ 2 r v 2 + I K(r ) to approximate te unique minimizer u,ε S 1 (T ) of E ε. We can use te following iteration to approximate te unique minimizer u,ε. Algoritm 5.9 (Regularized splitting metod). Coose η 0 S1 (T ) and set j = 0. (1) Compute a minimizer (u j+1, s j+1 ) S 1 (T ) S 1 (T ) of (2) Update (u, s ) L,ε (u, s ; η j ). η j+1 and stop if 1 σ ηj+1 η j ε stop. = η j + σ(sj+1 u j+1 ). Remark Te stability (and convergence) of Algoritm 5.9 follows from te teory of augmented Lagrangian metods (cf. [13, 14]) using arguments from [21] wic avoid differentiability assumptions on te involved functionals but exploit teir (strong) convexity.

15 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 15 Since te minimizer in step (1) of Algoritm 5.9 is not directly accessible we consider for j N te augmented functional L,ε (u, p, s ; η j ) := u 2 + ε 2 + p2 2 dx + α 2 u g 2 2p 2 + (η j, s u ) + σ 2 s u 2 + I K(s ). Noting tat L,ε is separately convex in u, p and s we use as in te Heron-penalty metod a semi-implicit subgradient flow to approximate a minimizer (u j+1, s j+1 ) of L,ε (u, s ; η j ) in te (j + 1)-t iteration. Algoritm 5.11 (Heron-split metod). Coose (u 0, p0, s0 ) S1 (T ) L 0 (T ) S 1 (T ) and η 0 S1 (T ) suc tat χ s 0 ψ and p 0 ε. Set j = 0. (1) Set u j+1,0 = u j, pj+1,0 = p j and sj+1,0 = s j. (2) For l = 0,..., M 1: first compute te minimizer u j+1,l+1 of u L,ε (u, p j+1,l, s j+1,l ; η j ) + 1 2τ u u j+1,l 2, and ten compute te minimizing (p j+1,l+1, s j+1,l+1 ) of (p, s ) L,ε (u j+1,l+1, p, s ; η j ) + 1 2τ p p j+1,l τ s s j+1,l 2. (3) Set u j+1 (4) Update = u j+1,m, p j+1 = p j+1,m and s j+1 = s j+1,m. η j+1 = η j + σ(sj+1 u j+1 ), 1 and stop if σ ηj+1 η j ε stop. Oterwise, increase j j + 1 and continue wit (1). Remark (i) Instead of te fixed number M of inner iterations in step (2) of Algoritm 5.11 one can also stop te inner iteration using te stopping criterion d t u j+1,l+1 + d t p j+1,l+1 + d t s j+1,l+1 ε stop. (ii) In our experiments we will set M = 1 and use te stopping criterion d t u j+1 + d t p j+1 + d t s j σ ηj+1 η j ε stop in step (4) of Algoritm We will refer to tis setting as te Heron-split metod. 6. Application to piecewise constant segmentation In tis section we investigate te constrained minimization problem min u BV () Eseg (u) := min Du + α fu dx + I K (u) u BV ()

16 16 wit f L () and K := {v L 2 () : 0 v 1 a.e.}. Tis constrained minimization problem occurs, e.g., in te piecewise constant segmentation of images discussed in [10] were one aims at minimizing for a given image g L () te nonconvex functional MS(Σ, c 1, c 2 ) := Dχ Σ + α (c 1 g) 2 dx + α (c 2 g) 2 dx \Σ Σ wit χ Σ being te caracteristic function of te unknown set Σ. Te autors propose an alternating minimization wit respect to te unknown variables Σ, c 1 and c 2. For a fixed set Σ te optimal c 1 and c 2 are given by c 1 = 1 Σ Σ g dx and c 2 = 1 \ Σ \Σ g dx. However, te major difficulty amounts to solving te minimization problem defined by te functional MS(, c 1, c 2 ) for fixed c 1 and c 2. Berkels et al. considered a relaxed functional of MS(, c 1, c 2 ) in [6], employed a finite element discretization and used te primal-dual metod proposed by Cambolle and Pock in [9] using adaptive refinement tecniques as well. Te crucial observation is tat one may solve tis global optimization problem defined by MS(, c 1, c 2 ) by computing a minimizer u of E seg wit χ 0, ψ 1 and f := (c 1 g) 2 (c 2 g) 2. Ten, for almost every γ [0, 1], te set Σ := {x R d : u(x) γ} is a minimizer of MS(, c 1, c 2 ), cf. [10, Tm. 2]. One can establis te existence of a minimizer u BV () K of E seg, owever, te minimizer may not be unique. Restricting te minimization of E seg to S 1 (T ) ten also provides a minimizer u S 1 (T ). Analogously to Lemma 4.1 we obtain te following estimate. Lemma 6.1. Let u and u be minimizers of E seg on BV () and on S 1 (T ), respectively. We ten ave E seg (u ) E seg (u) c 1/2. Proof. Te proof is analogous to tat of Lemma 4.1 and we terefore omit it. Te autors in [10] furtermore observed tat te functional Ẽ seg (u) := Du + βθ(u) + αfu dx, were θ(ξ) := max{0, 2 ξ 1 2 1}, as te same minimizers as Eseg provided tat β > α 2 f L (), cf. [10, Claim 1]. Tey employ a regularization of te BV -seminorm and of te function θ and use an explicit gradient descent sceme to approximate a minimizer of Ẽseg and E seg, respectively.

17 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 17 We remark tat tis procedure may be practical in tis specific situation, owever, te construction of te functional Ẽ seg only enables one to deal wit tis particular constrained minimization problem defined by E seg. Since we can adapt te iterative scemes presented in Section 5 to te minimization of E seg after making minor adjustments we will also present numerical experiments for te piecewise constant segmentation problem in te next section. We conclude tis section wit a result wic is analogous to Proposition 5.4. Proposition 6.2. Let u S 1 (T ) and (u,ε,δ, s,ε,δ ) S 1 (T ) S 1 (T ) be minimizers of E seg and of te functional E seg,ε,δ (v, r ) := v ε dx + α fu dx + 1 2δ v r 2 + I K(r ), respectively. Ten we ave E seg (u ) E seg,ε,δ (u,ε,δ, s,ε,δ ) + 1 4δ u,ε,δ s,ε,δ 2 c(ε + δ). Proof. Observing tat for a R d we ave a a ε a + ε and tat Π K u,ε,δ = s,ε,δ, and using te arguments in te proof of Proposition 5.4 we obtain E seg (u ) E,ε,δ (u,ε,δ, s,ε,δ ) E,ε,δ (u, u ) E,ε,δ (u,ε,δ, s,ε,δ ) = u ε dx + α fu dx u,ε,δ ε dx α fu,ε,δ dx 1 2δ s,ε,δ u,ε,δ 2 ε + u dx + α fu dx u,ε,δ dx α fu,ε,δ dx 1 2δ s,ε,δ u,ε,δ 2 ε + Π K u,ε,δ dx + α fπ K u,ε,δ dx u,ε,δ dx α fu,ε,δ dx 1 2δ s,ε,δ u,ε,δ 2 ε 1 2δ s,ε,δ u,ε,δ 2 + α f u,ε,δ s,ε,δ ε 1 2δ s,ε,δ u,ε,δ δ s,ε,δ u,ε,δ 2 + α 2 δ f 2. Tis proves te assertion. Remark 6.3. In order to recover te convergence rate from Lemma 6.1 we coose δ = 1/2 α f.

18 18 7. Numerical Experiments 7.1. Constrained ROF. We investigate two examples wic differ only in te coice of te parameter α > 0. In bot cases te algoritms are initialized as follows: we set p 0 = λ0 = 0, η0 = 0, s0 = (χ + ψ)/2 and σ 1 = 3/2 (see [3]) in Algoritm 5.1. We start Algoritm 5.6 wit ε =, τ = 1, u 0 = (χ + ψ)/2, p0 = ε1/2 and s 0 = u0, and Algoritm 5.11 wit ε =, τ = 1, u 0 = (χ + ψ)/2, p0 = ε1/2, s 0 = u0 and η0 = 0. To compare te accuracy of te algoritms we use precomputed approximate minimizers ũ of E generated by Algoritm 5.1 wit σ 2 = α, (ũ,ε,δ, s,ε,δ ) of E,ε,δ generated by Algoritm 5.6 wit bot δ = /α and δ =, and ũ,ε of E ε generated by Algoritm 5.11 wit σ = α after 10 4 iterations. Te stopping criteria ave been cosen as follows: Split-split metod: (5) ũ u j 10 3 Heron-penalty metod: (6) max{ ũ,ε,δ u j, s,ε,δ s j } 10 3 Heron-split metod: (7) ũ,ε u j 10 3 Example 7.1. We let d = 2, = (0, 1) 2, χ 1/5, ψ 1/2 and α = 50. Moreover, given a uniform triangulation T 5 of wit mes size = 22 5, we let g S 1 (T 5 ) to be te continuous, piecewise affine approximation of te caracteristic function χ B1/5 (x ) of te closed ball wit radius 1/5 around te center x of, i.e., g is defined by { 1, if z B g(z) := 1/5 (x ) 0, else for z N 5. We ten set g := g + ξ 5 were ξ 5 S 1 (T 5 ) wose coefficient vector is a sample of a random variable uniformly distributed in te interval [ 1/2, 1/2]. In Table 1 we displayed te iteration numbers needed by te algoritms to satisfy te stopping criteria (5), (6) and (7). Since te optimal coice of te step sizes σ 2 and σ for te split-split metod and te Heron-split metod, respectively, are not clear, we ran te algoritms for tree different coices of step sizes. As one can observe te coice σ 2 = α and σ = α, respectively, lead to te smallest iteration numbers wic is due to te balance of fitto-data and fit-to-constraint, i.e., wen σ 2 = σ = α is cosen te splitsplit algoritm and te Heron-split metod try to recover te datum g and to satisfy te bilateral constraint simultaneously. Regarding te Heronpenalty metod, we ran te algoritm for te coices δ = /α and δ =. Obviously, due to a worse conditioning, te iteration numbers for te coice

19 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 19 Split-split Heron-penalty Heron-split σ 2 δ σ 1 α 1/ /α 1 α 1/ 2/ / / / Table 1. Iteration numbers wit (5), (6), (7) for Example 7.1 and σ 1 = 3/2 for te split-split metod and τ = 1 for te Heron-penalty and te Heron-split metod ũ u j, δ = α ũ s j, δ = α ũ u j, δ = ũ s j, δ = 0.1 Error Number of iterations Figure 1. L 2 -error between approximate minimizer ũ of E (generated by te split-split metod) and iterates u j, sj of te Heron-penalty metod, = 22 6, ε =, for Example 7.1. Note tat te iterates u j serve as good approximations of ũ wen δ = /α. δ = /α are muc larger tan for δ =. However, for δ = te iterates u j generated by te Heron-penalty metod stay far from te minimizer u of E and te distance corresponds to te magnitude of α as predicted by Proposition 5.4. Tis effect is also illustrated in Figure 1 were te L 2 -error

20 20 between te approximate minimizer ũ of E generated by te split-split metod and te iterates u j and sj, respectively, generated by te Heronpenalty metod for bot δ = /α and δ = is plotted against te number of iterations. For δ = /α we see tat te final iterates u N and sn serve as good approximations of ũ and te approximation error is of order. Yet, for δ =, te iterates u j do not ave practical approximation properties and one would ave to decrease te mes size significantly to obtain a similar accuracy as for δ = /α. Still, one can also observe tat te different coices of δ do not considerably affect te approximation properties of te iterates s j, i.e., one may set δ = and coose te final iterate sn as an approximation of ũ. Example 7.2. Te setting is as in Example 7.1 except for te parameter α, wic is set to α = 500 ere. In Table 2 te iteration numbers for Example 7.2 are displayed. We see tat te effect of iger iteration numbers for te split-split metod and te Heron-split metod for te coice σ 2 = σ = 1 and for te coice δ = /α in te Heron-penalty metod are now even more pronounced due to te coice of a large α. Moreover, Figure 2 sows once again tat te iterates u j generated by te Heron-penalty metod for δ = may not be used as a good approximation of ũ since te constant α affects te L 2 -distance between u j and ũ wile te iterates s j seem to define accurate approximations of ũ almost independently of te coice of δ. Te overall conclusion of te experiments is tat te split-split metod wit σ 2 = α, Heron-penalty metod wit δ = and te Heron-split metod wit σ = α lead to te smallest iteration numbers and ave a similar performance. An advantage of te Heron-penalty metod is tat te coice of parameters is clear wile te coice of te step sizes σ 2 and σ in te split-split metod and te Heron-split metod, respectively, remain unclear in general. However, te penalization parameter δ in te Heron-penalty metod as to scale as /α for te iterates u j to be close to u. Split-split Heron-penalty Heron-split σ 2 δ σ 1 α 1/ /α 1 α 1/ 2/ / / / Table 2. Iteration numbers wit (5), (6), (7) for Example 7.2 and σ 1 = 3/2 for te split-split metod and τ = 1 for te Heron-penalty and te Heron-split metod.

21 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION ũ u j, δ = α ũ s j, δ = α ũ u j, δ = ũ s j, δ = Error Number of iterations Figure 2. L 2 -error between approximate minimizer ũ of E (generated by te split-split metod) and iterates u j, sj of te Heron-penalty metod, = 22 6, ε =, for Example 7.2. Again, te iterates u j approximate ũ properly wen δ = /α Piecewise constant segmentation. In tis section we report numerical experiments for te binary image segmentation model presented in Section 6. We consider te two-pase image segmentation of te image cameraman. To tis extent, we let g be te continuous piecewise affine function wit coefficient vector wose entries consist of te (scaled) grayvalues of te image ranging from 0 to 1. We furter let α = 1500, χ 0, ψ 1, and consider te following algoritm of [10] wic involves te solution of a constrained T V -minimization problem. Algoritm 7.3. Coose c 0 1, c0 2 R and γ [0, 1]. Set k = 0. (1) Compute for f = (c k 1 g)2 (c k 2 g)2 an approximate minimizer ũ k+1 S 1 (T ) of te functional E seg (v ) = v + α fv dx + I K (v ) using eiter te split-split metod or te Heron-penalty metod or te Heronsplit metod.

22 22 (2) Define u k+1 for z N. Set (3) Set (4) Stop if by c k+1 1 = 1 Σ k+1 u k+1 (z) = { 1, if ũ k+1 (z) γ, 0, if ũ k+1 (z) < γ, Σ k+1 = {T T : u k+1 T 1}. Σ k+1 g dx and c k+1 2 = MS(Σ k+1, c k+1 1, c k+1 2 ) MS(Σ k, c k 1, ck 2 ) MS(Σ k+1, c k+1 1, c k+1 2 ) Oterwise, increase k k + 1 and continue wit (1). 1 \ Σ k+1 g dx. \Σ k+1 ε stop. Using a uniform triangulation wit mes size 2 8 we cose c 0 1 = 1 and c 0 2 = 0 and set te tresolding parameter to γ = 1/2. We set ε stop = 10 3 in step (4). In step (2), we initialize te algoritms as in te preceding subsection and used ε = 1/2 for te Heron-penalty and te Heron-split metod. We set δ = 1/2 in te Heron-penalty metod and used σ 2 = σ = α f in te split-split and te Heron-split metod and τ = 1 in te Heronpenalty and te Heron-split metod. Te stopping criteria in step (2) were cosen as follows: (8) (9) (10) Split-split metod: ( λ j+1 λ j 2 w + 3 p j+1 ( 1 σ2 2 η j+1 η j 2 + sj+1 s j 2 p j ) 1/2 2 w 10 3 ) 1/ Heron-penalty metod: ( dt u j d t p j d t s j+1 2 ) 1/ Heron-split metod: ( dt u j+1 ( 1 σ 2 ηj d t p j+1 η j 2 + sj+1 s j 2 2) 1/ ) 1/ Figure 3 sows te original image and te outputs of Algoritm 7.3 using te split-split metod, te Heron-penalty metod and te Heron-split metod in step (2), respectively, were we used te iterates s j as approximations of a minimizer in step (2) in te Heron-penalty metod. Te tree wite orizontal lines in te outputs are due to image conversion. One can observe tat all outputs are visually almost te same. Te output of te split-split metod and te Heron-split metod visually do not differ at all, wereas and and

23 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 23 (a) Original image (b) Split-split (c) Heron-penalize (d) Heron-split Figure 3. Original image and outputs of Algoritm 7.3 using te split-split, Heron-penalty and Heron-split metod in step (2), respectively (orizontal wite lines are due to image conversion). te image generated by te Heron-penalty metod as two black dots wic do not appear in te oter two images. Tese dots are not contained in te output of te Heron-penalty metod wen using δ = 1/2 α f. Te final values of c 1 and c 2 were c 1 = and c 2 = for bot te splitsplit metod and te Heron-split metod and c 1 = , c 2 = for te Heron-penalty metod. Te final values of te energy MS(Σ, c 1, c 2 ) were , and for te split-split metod, te Heronpenalty metod and te Heron-split metod, respectively. Te split-split metod needed 5000 iterations in total for termination (6 outer iterations), wile te Heron-penalty metod needed 1382 iterations in total (7 outer iterations) and te Heron-split metod needed 857 iterations in total (6 outer iterations). Te Heron-penalty metod wit δ = 1/2 α f needed 1675

24 24 iterations in total (6 outer iterations). Using u j generated by te Heronpenalty metod as approximations for te minimizer in step (2) led to te same results as for te case using s j. Acknowledgements. Te autors acknowledge financial support for te project Finite Element approximation of functions of bounded variation and application to model of damage, fracture and plasticity (BA 2268/2-1) by te DFG via te priority program Reliable Simulation Tecniques in Solid Mecanics, Development of Non-standard Discretization Metods, Mecanical and Matematical Analysis (SPP 1748). References [1] H. Attouc, G. Buttazzo, G. Micaille, Variational Analysis in Sobolev and BV Spaces, MPS/SIAM Series on Optimization, Vol. 6, Piladelpia, [2] S. Bartels, Numerical Metods for Nonlinear Partial Differential Equations, Springer, Heidelberg, [3] S. Bartels, M. Milicevic, Stability and Experimental Comparison of Prototypical Iterative Scemes for Total Variation Regularized Problems, Computational Metods in Applied Matematics, 2016, doi: /cmam [4] S. Bartels, R. H. Nocetto, A. J. Salgado, A Total Variation Diminising Interpolation Operator and Applications, Matematics of Computation, Vol. 84, 2015, pp [5] A. Beck, M. Teboulle, Fast Gradient-Based Algoritms for Constrained Total Variation Image Denoising and Deblurring Problems, IEEE Transactions on Image Processing, Vol. 18, No. 11, 2009, pp [6] B. Berkels, A. Effland, M. Rumpf, A Posteriori Error Control for te Binary Mumford-Sa Model, Mat. Comp. (to appear). [7] E. Casas, K. Kunisc, C. Pola, Regularization by Functions of Bounded Variation and Application to Image Enancement, Appl. Mat. Optim., Vol. 40, 1999, pp [8] A. Cambolle, An algoritm for total variation minmization and applications, J. Mat. Imaging Vision, Vol. 20, 2004, pp [9] A. Cambolle, T. Pock, A First-Order Primal-Dual Algoritm for Convex Problems wit Applications to Imaging, J Mat Imaging Vis, Vol. 40, 2011, pp [10] T. F. Can, S. Esedoglu, M. Nikolova, Algoritms for Finding Global Minimizers of Image Segmentation and Denoising Models, SIAM J. Appl. Mat., Vol. 66, No. 5, pp [11] R. H. Can, M. Tao, X. Yuan, Constrained Total Variation Deblurring Models and Fast Algoritms Based on Alternating Direction Metod of Multipliers, SIAM J. Imaging Sci., Vol. 6, 2013, pp [12] Z. Cen, R. H. Nocetto, Residual Type A Posteriori Error Estimates for Elliptic Obstacle Problems, Numerisce Matematik, Vol. 84, 2000, pp [13] M. Fortin, R. Glowinski, Augmented Lagrangian Metods, Studies in Matematics and its Applications, Vol. 15, Nort-Holland Publising Co., Amsterdam, [14] R. Glowinski, Numerical Metods for Nonlinear Variational Problems, Springer, New York, [15] M. Hintermüller, C. N. Rautenberg, J. Han, Functional-Analytic and Numerical Issues in Splitting Metods for Total Variation-Based Image Reconstruction, Inverse Problems, Vol. 30, No. 5, 2014, , doi: / /30/5/ [16] R. H. Nocetto, L. B. Walbin, Positivity Preserving Finite Element Approximation, Matematics of Computation, Vol. 71, No. 240, 2001, pp [17] R. T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, 1970.

25 NUMERICAL METHODS FOR CONSTRAINED TV MINIMIZATION 25 [18] T. Roubiček, J. Valdman, Perfect Plasticity wit Damage and Healing at Small Strains, its Modelling, Analysis, and Computer Implementation, SIAM J. Appl. Mat., Vol. 76, 2016, pp [19] L. I. Rudin, S. Oser, E. Fatemi, Nonlinear Total Variation Based Noise Removal Algoritms, Pysica D, Vol. 60, 1992, pp [20] M. Tomas, Quasistatic Damage Evolution wit Spatial BV -Regularization, Discrete Contin. Dyn. Syst. Ser. S, Vol. 6, 2013, pp [21] C. Wu, X.-C. Tai, Augmented Lagrangian Metod, Dual Metods, and Split Bregman Iteration for ROF, Vectorial TV, and Higer Order Models, SIAM J. Imaging Sci., Vol. 3, 2010, pp [22] M. Zu, Fast Numerical Algoritms for Total Variation Based Image Restoration, P.D. tesis, University of California, Los Angeles, Department of Applied Matematics, Matematical Institute, University of Freiburg, Hermann-Herder-Str. 9, Freiburg i. Br., Germany address: Department of Applied Matematics, Matematical Institute, University of Freiburg, Hermann-Herder-Str. 9, Freiburg i. Br., Germany address: