Convergence of a Central Difference Discretization of ROF Model for Image Denoising

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1 Convergence of a Central Difference Discretization of ROF Model for Image Denoising Ming-Jun Lai and Jingyue Wang Department of Matematics Te University of Georgia Atens, GA October 8, 2009 Abstract We study a central difference discretization of Rudin-Oser-Fetami model for image de-noising. We sow tat te discrete solution u k converges to te continuous solution u in L 2 norm and a rate of convergence is given. 1 Introduction One of te most influential variational models for image denoising is te total variation based model proposed by Rudin, Oser and Fatemin in [16]. Tis model studies te following constrained minimization problem: arg min u BV (1) u wit u = g and u g 2 = σ 2, Ω Ω were g is te input data, σ is te standard deviation of te noise, Ω is te unit square [0, 1] 2, and u BV is te total variation (TV) of u defined in (2) below. Letting φ [C 1 0 (Ω)]2 be a compactly supported continuous mjlai@mat.uga.edu. Tis autor is partly supported by te National Science Foundation under grant DMS jywang@mat.uga.edu. Ω 1

2 function vector mapping Ω to R 2, te variation of a function u L 1 (Ω) is ten defined by u BV := Du := sup u φ. (2) Ω φ [C0 1(Ω)]2, φ 1 point-wise Ω For more details on functions of bounded variation, we refer te reader to [11]. Te existence and uniqueness of te minimizer of (1) ave been studied. See, e.g., [1]. Cambolle and Lions [4] proved tat te constrained problem (1) is equivalent to te following unconstrained problem: arg min u u BV + 1 2λ Ω u g 2. (3) Tey also proved more general results of existence and uniqueness of (1). For convenience, we denote by E(u) = u BV + 1 u g 2 (4) 2λ te ROF energy functional. It is a common practice to minimize te functional E(u) or minimize E(u) togeter wit oter constrained terms for image denoising, deblurring, dejittering, inpainting, segmentation, colorization, and etc. (cf., e.g., [4], [17], [15], [13], [12]). Wen implementing tese minimizations, te most commonly used discretization of te ROF model is based on te discrete energy functional E (u) = k 1 i,j=0 µ i,j ( u) i,j + 1 2λ k 1 i,j=0 µ i,j (u i,j g i,j ) 2, (5) were u is defined by a 2-dimensional matrix of size k k, µ i,j is related to te scale k. A simple coice of µ i,j is µ i,j = 1/k 2 and wit ( u) i,j = (( x u) i,j, ( y u) i,j ), ( x u) i,j = u i+1,j u i,j, ( y u) i,j = u i,j+1 u i,j, were = 1/k. On te boundary, u is assumed to satisfy te discrete Neumann boundary conditions: u 1,j = u 0,j, u k,j = u k 1,j, u i, 1 = u i,0, u i,k = u i,k 1. (6) 2

3 Te discrete function g i,j in (5) is te given input noised image. Many efficient algoritms ave been developed to find te numerical minimizer of (5). See te algoritms in [6], [2], [3] and [9]. Tere are oter coices for te discrete gradient operator u (cf. [3], [5], [19], and [18]). For example, in [19], te following approximation of te first term, tat is, te total variation (TV) term in E(u) is considered. u BV = k 1 i,j=0 2 { ( u i+1,j u i,j 4 ( u i+1,j u i,j ( u i,j u i 1,j ( u i,j u i 1,j ) 2 + ) 2 + ) 2 + ) 2 + ( u i,j+1 u i,j ( u i,j u i,j 1 ( u i,j+1 u i,j ( u i,j u i,j 1 ) 2 1/2 ) 2 ) 2 ) 2 + 1/2 1/ /2 } For anoter example, te autors in [5] studied te approximation of te first term in E(u) using te upwind variation sceme. Tese approximations lead to different forms of E. It is not ard to sow tat te solution of (5) Γ-converges to E (for te definition of Γ-convergence, we refer te reader to [7]). It terefore follows tat te sequence {u } of minimizers of E converges to u wic is te minimizer of E in L 1 (Ω) and E (u ) converges to E(u) as tends to zero (cf. [7]). It is interesting to know te rate of convergence and te convergence in oter norm, e.g., in L 2 norm. Te researcers in [19] and in [18] proved tat if te discrete energy E is equipped wit a discrete total variation as defined in (7) or in te upwind sceme discussed in [5], and if te discrete input data g is te projection of te continuous input data g by taking average of g on eac pixel, ten one can bound te error between te discrete minimizer u and te continuous minimizer u in L 2 norm by te Lipscitz norm of g provided tat g is in some Lipscitz space Lip(α, L 2 ), i.e., α derivatives of g in L 2 (Ω) for 0 < α 1. In tis paper we continue te studies in [18] and [19] and study anoter approximation of te TV term in E(u) by using central differences. Tat is, we sall consider te central-difference discrete ROF model wose energy (7) 3

4 functional is defined as follows E (u ) = J c (u ) + 1 2λ were te TV term J c is defined by k J c (u ) := u i+1,j u i 1,j 2 i,j=0 k µ i,j (u i,j gi,j) 2, (8) i,j=0 2 u i,j+1 + u i,j /2 µ i,j, (9) wit u i,j satisfying te discrete Neumann boundary conditions (6). Here te weigts µ i,j are given by 2 /4 (i, j) {(0, 0), (0, k), (k, 0), (k, k)} µ i,j = 2 /2 i = 0, k; 0 < j < k or j = 0, k; 0 < i < k (10) 2 0 < i, j < k. Te discrete input date g is defined by gi,j = 1 Ω i,j Ω i,j g wit Ω i,j = (Ω + (i 1/2, j 1/2)) Ω. We note tat Ω i,j = µ i,j. It is clear tat tis version (9) is simpler tan (7) or te upwind sceme. It turns out tat te computation is consequently simpler too. However, a problem of tis central-difference model is tat it does not deal well wit a special non-smoot data: a cessboard image. Neverteless, tis patological image never appen in real-life image processing. Even if we ave tis digital image for a fixed, we will not ave a cessboard image for all > 0. Tus tis central difference discrete ROF model makes sense for numerical calculation of te ROF model for image de-noising. We use tecniques similar to [19] to analyze tis discrete ROD model and obtained a convergence rate lower tan te rates in [19] under te same assumption on te input data g. Our results improve te result announced in a conference paper [14]. In fact, te results in [14] are a special case of te following teorems wit α = 1. Teorem 1 Suppose tat g Lip(α, L 2 (Ω)). Let u be te minimizer of E in (4) and u be te minimizer of E in (8) equipped wit te central difference TV operator. Ten E(u) E (u ) C(1 + 1 λ )( g Lip(α,L 2 (Ω)) + g 2 Lip(α,L 2 (Ω)) )α2 /(α+1). 4

5 Next we need more notation to state our next result. Let stand for te standard L 2 norm for te space L 2 (Ω) of all square integrable functions over Ω. For te discrete minimizer u, let I u be te piecewise constant injection of u into L 2 (Ω) space. A precise definition of I u will be given in (13) in te next section. Ten Teorem 2 Under te same assumptions in Teorem 1, we ave I u u 2 C(λ + 1)( g Lip(α,L 2 (Ω)) + g 2 Lip(α,L 2 (Ω)) )α2 /(α+1). Te convergence ere is more precise tan tat of te Γ convergence of u to u as discussed above. Te paper is organized as follows. We first explain some notations and definitions togeter wit some preliminary results wic are similar to tose in [19]. We need tese preliminary results in te next section. In Section 3, we present some basic lemmas wic are necessary for te proofs of our main results in Teorems 1 and 2. Ten we prove Teorems 1 and 2 in 4. We end tis paper wit some remarks in 5. 2 Preliminaries 2.1 Notations and Definitions A continuous image u is defined as a L 2 function on Ω = [0, 1] [0, 1]. We always assume tat te denoised image is in te space BV(Ω) of all functions of bounded variation. In te discrete setting, we consider te discrete domain Ω to be te set of all pairs (i, j) Z 2 wit 0 i, j k. A discrete image u is defined as a function on Ω. We define te discrete L p (Ω ) norms u L p (Ω ) := u (i,j) p µ i,j (i,j) Ω 1 p for 1 p, were µ i,j is te weigt at eac index (i, j) given in te introduction section. In particular, te L 2 term of g is defined by u g 2 L 2 (Ω ) = k u i,j gi,j 2 µ i,j. i,j=0 5

6 In analogue of Sobolev norm, we define te discrete Sobolev norm as follows. Te first order forward finite differences of u at point i = (i 1, i 2 ) are + x u i,j = u i+1,j u i,j ; + y u i,j = u i,j+1 u i,j, were = 1/k is te step size. We can also define backward finite difference as x u i,j = u i,j u i 1,j ; y u i,j = u i,j u i,j 1. Tus, te second order finite difference along x direction is xx u i,j = + x u i,j x u i,j. Also yy u i,j can be similarly defined. We define u L 1 (Ω ), xx u L 1 (Ω ), yy u L 1 (Ω ) as xx u L 1 (Ω ) := u L 1 (Ω ) := k ( + x u i,j + + y u i,j )µ i,j ; (11) i,j=0 k xx u i,j µ i,j, yy u L 1 (Ω ) := k yy u i,j µ i,j. i,j=0 sup τ R 2, τ <t i,j=0 (12) Finally we need notation of (first-order) L p (Ω) modulus of smootness tat is defined by ( ) 1 ω(u, t) L p (Ω) = u(x + τ) u(x) p p dx. x,x+τ Ω Similarly we need a discrete modulus of smootness. Te discrete L p modulus of smootness is ω(u, m) L p (Ω ) := ( sup l Z 2, l m wit l = l 1 + l 2 for l Z 2. (i,j),(i,j)+l Ω u (i,j)+l u i,j p µ i,j ) 1 p. 6

7 2.2 Extension of functions To discuss te modulus of smootness of function v and discrete function v, we need to extend v L p (Ω) and v to all of R 2 and Z 2, respectively. Since we only consider te variation inside te Ω and Ω. Te extension sall not introduce extra variation on te boundary. For v L p (Ω), te extension is done by reflection along te boundary. First, Ext v(x, y) = v(x, y), (x, y) Ω. We ten reflect orizontally across te line x = 1, Ext v(x, y) = Ext v(2 x, y), 1 x 2, 0 y 1, and reflect again vertically across te line y = 1, Ext v(x, y) = Ext v(x, 2 y), 0 x 2, 1 y 2. Having defined Ext v on 2Ω, we ten extend Ext v periodically wit period (2, 2) on all of R 2. Te discrete extension is different from [19]. We use an odd extension ere. First we extend v to as follows: ten we reflect orizontally and ten vertically 2Ω := {(i, j) Z 2 0 i, j 2k} Ext v i,j = v i,j, (i, j) Ω ; Ext v(i,j) = Ext v(2k i,j), k + 1 i 2k, 0 j k, Ext v(i,j) = Ext v(i,2k j), 0 i 2k, k + 1 j 2k. Now tat Ext v is defined on 2Ω, we extend it periodically wit period (2k, 2k) to all of Z Injectors, Projectors and Smooting Operators We need to inject functions in te discrete space L 2 (Ω ) into L 2 (Ω). Te first one is te piecewise constant injector to inject discrete function u into L p (Ω): Let S i,j = (Ω + (i 1/2, j 1/2)), S i,j be a square wit center 7

8 (i, j) and size. We define Ω i,j = S i,j Ω for 0 i, j k. Note tat Ω i,j is a alf square or quarter square for (i, j) wic are on te boundary or on four corners of Ω respectively. Ten (I u )(x) = u i,j for x Ω i,j, (13) Next let L be a linear injector mapping functions in te discrete space L 2 (Ω ) into a space of continuous, piecewise linear functions. L u = u i,jφ i,j, (14) (i,j) Ω were φ i,j(x, y) := φ i,j(x, y) := φ(x/ i, y/ j), (15) is a translation of dilated tent function φ on R 2 wic is supported on te exagon wit boundary vertices {(1, 0), ( 1, 1), (0, 1), ( 1, 0), (1, 1), (0, 1)} and is zero for all integers in R 2 except for (0, 0) were φ(0, 0) = 1. We also consider te piecewise constant projector of u L 1 (Ω) onto te space of discrete functions, defined by (P u) i,j = 1 Ω i,j Ω i,j u(x, y)dxdy, (i, j) Ω, were Ω i,j = µ i,j is te measure of Ω i,j. We will need continuous and discrete smooting operators. Assume tat η(x) is a a fixed non-negative, rotationally symmetric, mollifier wit support on te unit disk wic is C and as integral 1 wit x = (x, y). For ɛ > 0 we define te scaled function η ɛ (x) := 1 ( x ) ɛ 2 η, x = (x, y) R 2 ; ɛ Let S ɛ be a smooting operator defined by S ɛ u = (η ɛ Ext u) = η ɛ (x y) Ext u(x) dy, x 2Ω. R 2 Te discrete smooting operator S L is defined by (S L u ) i,j = 1 (2L + 1) 2 L i 1,j 1 = L Ext u (i,j)+(i 1,j 1 ) for (i, j) Ω 8

9 3 Basic Lemmas We now give basic lemmas tat will be used to prove te main results in te next section. We first give properties of injectors, projectors and smooting operators. Teir proofs can be found(wit minor modifications) in [19]. Te following lemmas bound te errors introduced by injectors and projectors wic can be verified easily. Lemma 3.1 Let v L 2 (Ω) and v being a discrete function. Ten tere exists a constant C suc tat te following properties old: P v L 2 (Ω ) v, (16) ω(p v, m) L 2 (Ω ) Cω(v, m), (17) v L 2 (Ω ) = I v, (18) ω(i v, m) Cω(v, m) L 2 (Ω ), (19) and v I P v Cω(v, ), (20) L v I v Cω(v, 1) L 2 (Ω ). (21) We next give some properties of te smooting operators wic will be used in te proofs for our main results. Tey can be verified straigtforwardly. Lemma 3.2 For any integer M > 0, ω(s L v, M) L 2 (Ω ) Cω(v, M) L 2 (Ω ), (22) S L v v L 2 (Ω ) Cω(v, L) L 2 (Ω ), (23) J c (S L v ) J c (v ), (24) 9

10 and xx S L v L 1 (Ω ) + yy S L v L 1 (Ω ) C L v L 1 (Ω ). (25) Furtermore, for all t > 0, ω(s ɛ v, t) L 2 (Ω) Cω(v, t) L 2 (Ω), (26) S ɛ v v Cω(v, ɛ) L 2 (Ω), (27) and S ɛ v BV v BV, (28) D 2 xs ɛ v L 1 (Ω) + D 2 ys ɛ v L 1 (Ω) C ɛ v BV. (29) Te first inequality in Lemma 3.2 sows tat te smooting operator does not decrease te smootness. Inequality (23) sows te error between u and smooted u can be bounded by its discrete modulus of continuity. Inequality (24) sows smooting does not increase te discrete total variation. Inequality (25) sows te second order difference of te smooted function can be bounded by its first order finite difference. Inequalities (26), (27), (28), and (29) are continuous analogues of (22), (23), (24), and (25), respectively. Te following lemma sows te consistency of te central-difference discrete variation. Tey can be proved by a standard inequality: x 2 + y 2 (x u) 2 + (y v) 2 + u 2 + v 2 and te fact tat te central difference approximates te first order derivatives wit error terms O() wen a function is not very smoot. Lemma 3.3 (TV Consistency) Tere exists a C > 0 suc for any discrete function v defined on Ω ( ) L v BV J c (Lv ) + C xx v L 1 (Ω ) + yy v L 1 (Ω ) (30) and for any v in L 1 (Ω), J c (P v) v BV + C( D xx v L 1 (Ω) + D yy v L 1 (Ω)). (31) 10

11 We remark tat for v / W 2,1 (Ω), te rigt and side of (31) is infinite, tus te inequality still olds. We also give properties of te minimizers of te continuous ROF model (3) and te discrete ROF model (8). Te proofs for tese lemmas can be easily adapted from te results in [19]. In te following lemma, we need to consider discrete variational functionals for discrete functions defined on 2Ω. For tese purposes we define J 2Ω c = (i,j) 2Ω u i+1,j u i 1,j 2 We also define te L 2 norm on L 2 (2Ω ) by 2 u i,j+1 + u i,j 1 2 v L 2 (2Ω ) = vi,j 2 2 (i,j) 2Ω 2 1/2 We note tat for discrete variations and L 2 norms defined on 2Ω, te weigts at eac point (i, j) is always 2 wic is sligtly different from J c defined in (9). Lemma 3.4 (Extension of Minimizers) If u is te minimizer of te functional 1/2. 2. E (v ) = 1 2λ v g 2 L 2 (Ω ) + J c(v ), (32) ten Ext u is te minimizer over all discrete functions v defined on 2Ω of te functional E 2Ω (v ) = 1 2λ v Ext g 2 L 2 (2Ω ) + J 2Ω c (v ) (33) wit periodic boundary conditions. Similarly, if u is te minimizer of ten Ext u is te minimizer of E(v) = 1 2λ v g 2 L 2 (Ω) + v BV(Ω) (34) E 2Ω (v) = 1 2λ v Ext g 2 L 2 (2Ω) + v BV(2Ω), (35) again wit periodic boundary conditions. 11

12 Recall tat if u and w are minimizers of (34) wit data g and, respectively, ten u w L 2 (Ω) g L 2 (Ω). (See [18] for a proof.) Similarly, for te two periodic problems (33) and (35) we ave and Ext u T l Ext u L 2 (2Ω ) Ext g T l Ext g L 2 (2Ω ) (36) Ext u T τ Ext u L 2 (2Ω) Ext g T τ Ext g L 2 (2Ω). (37) were T and T are translation operators and (T l (u )) i,j := u (i,j)+l for any l = (l 1, l 2 ) Z 2 (38) (T τ (u)) := u((x, y) + τ) for any τ = (τ 1, τ 2 ) R 2. (39) Wit (36) and (37), we can immediately obtain te following property of te smootness of te minimizers. Lemma 3.5 (Smootness bounds) Suppose tat u is te minimizer of E in problem (3) for input data g. Let Ext u be te extended u over R 2 as defined in Section 2.2. Ten, for ɛ > 0 and Ext u(x + ) u(x) L 2 (Ω) Cω(g, ɛ) L 2 (Ω), ɛ (40) ω(u, ɛ) L 2 (Ω) Cω(g, ɛ) L 2 (Ω). (41) Moreover, suppose tat u is te minimizer of E in (8). Let Ext u be te extended u over Z 2. Ten, for any integer L > 0, and T l (u ) u L 2 (Ω ) Cω(g, L) L 2 (Ω ), l L (42) ω(u, L) L 2 (Ω ) Cω(g, L) L 2 (Ω ) Cω(g, L) L 2 (Ω). (43) 12

13 In addition, te following properties about te minimizers u and u can be proved easily. u BV 1 2λ g 2, (44) u g g, (45) and J c (u ) 1 2λ g 2 L 2 (Ω ) 1 2λ g 2, (46) Finally we need te following u g L 2 (Ω ) g L 2 (Ω ) g. (47) Lemma 3.6 If u is te minimizer of E in (49), ten for any v BV, v u 2 2λ(E(v) E(u)). (48) Proof. Te result is classical. We give a proof for completeness. By te definition of E, E(v) E(u) = v BV u BV + 1 2λ ( v g 2 u g 2 ). Since u is te minimizer, (g u)/λ is in te sub-differential u BV, i.e., for any v, g u λ, v u v BV u BV. Ten E(v) E(u) g u λ, v u g u = λ, v u = 1 2λ v u λ ( v g 2 u g 2 ) + 1 2λ ( v u v u, u g ) 13

14 4 Proof of te Main Results Recall te ROF continuous and discrete energy functionals are defined by E(v) = v BV + 1 2λ v g 2 ; (49) E (v ) = J c (v ) + 1 2λ v g 2 L 2 (Ω ) (50) wit input image g = P g. We first give te bound on te difference of te continuous energy and te discrete energy. 4.1 Bound On te Difference of te Energy Functionals We begin wit te following property of modulus of smootness. Te proof is trivial and ence is omitted ere. Lemma 4.1 If v is in L 2 (Ω), for > 0, ω(v, ) L 1 (Ω) Cω(v, ) L 2 (Ω). (51) If v is a discrete function on Ω, for interger m > 0, ω(v, m) L 1 (Ω ) Cω(v, m) L 2 (Ω ), (52) were in bot inequalities, C depends on Ω. Lemma 4.2 Suppose tat g Lip(α, L 2 (Ω)). If u, u are te minimizers of E and E in (50) and(49), respectively. If L = 1/(α+1), ten E(u) E(L S L u ) E (u ) + C(1 + 1 λ )( g Lip(α,L 2 (Ω)) + g 2 Lip(α,L 2 (Ω)) )α2 /(α+1). Proof. We first ave E(u) E(L S L u ) by te definition of u. Now we sall bound te energy E(L S L u ). By (30) and te properties of discrete smooting operator (24) and (25) in Lemma 3.2, ( ) L S L u BV J c (S L u ) + C xx S L u L 1 (Ω ) + yy S L u L 1 (Ω ) J c (u ) + C L u L 1 (Ω ). 14

15 Note u L 1 (Ω ) C ω(u, 1) L 1 (Ω ) C ω(u, 1) L 2 (Ω ) by (52) C ω(g, 1) L 2 (Ω ) by (43) C g Lip(α,L 2 (Ω)) α 1. Hence, we ave L S L u BV J c (u ) + C L g Lip(α,L 2 (Ω)) α 1. On te oter and, te L 2 term of E ( L S L u ) can be written as L S L u g = (L S L u I S L u ) + (I S L u I u ) We apply (21), (18) and (20) to obtain + (I u I g ) + (I g g). L S L u g Cω(S L u, 1) L 2 (Ω ) + S L u u L 2 (Ω )+ Ten by (22), (23) and (43), we ave u g L 2 (Ω ) + Cω(g, ). L S L u g u g L 2 (Ω ) + C g Lip(α,L 2 (Ω))(L) α + C g Lip(α,L 2 (Ω)) α. Note by (47), We ave u g L 2 (Ω ) g. L S L u g 2 u g 2 L 2 (Ω ) + C g 2 Lip(α,L 2 (Ω)) ((L)2α + 2α + (L) α α + (L) α + α ), were C is a positive constant. It follows tat E(L S L u ) = L S L u BV + 1 2λ L S L u g 2 J c (u ) + C L g Lip(α,L 2 (Ω)) α λ u g 2 L 2 (Ω ) + C λ g 2 Lip(α,L 2 (Ω)) ((L)2α + 2α + (L) α α + (L) α + α ) 15

16 Seting te largest error terms (L) α and α 1 /L equal, i.e., setting L = 1/(α+1), we obtain te result. Using similar metod we can prove te following Lemma 4.3 Suppose tat g W 1,2. Let u, u be te minimizers of E, E in (49) and (50), respectively. If ɛ = 1/(α+1), ten E (u ) E (P (S ɛ u)) E(u) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1). Proof. First of all, we ave E (u ) E (P S ɛ u) by te definition of u. By (31) J c (P S ɛ u) S ɛ u BV + C( D xx S ɛ u L 1 + D yy S ɛ u L 1). By (31), (28), (29) and (44) J c (P S ɛ u) u BV + C ɛ u BV u BV + C λɛ g 2. Te L 2 (Ω ) term of E (P S ɛ u) can be estimated as follows, P S ɛ u g L 2 (Ω ) = I P (S ɛ u) I g We apply (20), (27) and (43) I P (S ɛ u) S ɛ u + S ɛ u u + u g + g I g. P S ɛ u g L 2 (Ω ) Cω(S ɛ u, ) + ω(u, ɛ) + u g L 2 + Cω(g, ) Note tat u g L 2 g L 2 u g L 2 + C g Lip(α,L 2 (Ω))(ɛ α + α ). by (45), we ave P S ɛ u g 2 L 2 (Ω ) u g 2 L 2 +C g 2 Lip(α,L 2 (Ω)) (ɛ2α + 2α +ɛ α α +ɛ α + α ). We now summarize above to get E (u ) E (P S ɛ u) E(u) + C λɛ g 2 L 2 + C λ g 2 Lip(α,L 2 (Ω)) (ɛ2α + 2α + ɛ α α + ɛ α + α ). Again, setting te largest error term ɛ α equal /ɛ, we ave ɛ = 1/(α+1). Tis completes te proof. 16

17 4.2 Proofs of Main Teorems We are now ready to prove our main results in tis paper. Te Proof of Teorem 1. Combining Lemma 4.2 and Lemma 4.3 immediately yields te following E(u) E k (u ) C(1 + 1 ) ( g λ ) 2 Lip(α,L 2 (Ω)) + g Lip(α,L 2 (Ω)) α2 /(α+1) wic is te result of Teorem 1. Te Proof of Teorem 2. We sall prove te following ( ) I u u 2 C(1 + λ) g 2 Lip(α,L 2 (Ω)) + g Lip(α,L 2 (Ω)) α2 /(α+1). To do so, we first use Lemma 3.6 to get L S L u u 2 2λ(E(L S L u ) E(u)) were L = 1/(α+1). By Lemma 4.2 and Lemma 4.3, E(L S L u ) E (u ) + C(1 + 1 λ ) ( g 2 Lip(α,L 2 (Ω)) + g Lip(α,L 2 (Ω)) Tus, E(u) E (u ) C(1 + 1 λ ) ( g 2 Lip(α,L 2 (Ω)) + g Lip(α,L 2 (Ω)) L S L u u 2 C(1 + λ) ) α2 /(α+1) ) α2 /(α+1). ( ) g 2 Lip(α,L 2 (Ω)) + g Lip(α,L 2 (Ω)) α2 /(α+1). Next we use inequality (53), Lemma 3.5, Lemma 3.2 to obtain I u u 2 = I u I S L u + I S L u L S L u + L S L u u 2 (53) 3( I u I S L u 2 + I S L u L S L u 2 + L S L u u 2 ) 3( u S L u 2 L 2 (Ω ) + (Cω(g, 1) L 2 (Ω )) 2 + L S L u u 2 ) C(ω(u, L) L 2 (Ω )) 2 + C(ω(g, 1) L 2 (Ω )) 2 + L S L u u 2 ) C g 2 Lip(α,L 2 (Ω)) (L)2α + C g 2 Lip(α,L 2 (Ω)) ( ) 2α + C(1 + λ) g 2 Lip(α,L 2 (Ω)) + g Lip(α,L 2 (Ω)) α2 /(α+1). wic can be rewritten in te form of te statement of Teorem 2. Here we ave used C for a generic constant wic may be different in te different lines above. 17

18 5 Conclusion and Remarks In tis paper, we proved te error bound for te discrete ROF model equipped wit a central-difference TV term using te same ideas in [19]. Tis model is simpler in form tan te models studied in [19] and [18]. Tis model is also sligtly easier to be computed by Cambolle s metod (cf. [3]). On te oter and, te results in tis paper sow tat te ideas in [19] are powerful wic can be extended to deal wit oter symmetric discrete approximation of TV term. It is also interesting to study furter weter te convergent result can be establised for oter discrete variations, for example ( max{ (u i+1,j u i,j )(u i,j u i 1,j ), 0} 2 + max{ (u i,j+1 u i,j )(u i,j u i,j 1 ), 0} 2 to replace te first order difference ( (ui+1,j u i,j ) (u i,j+1 u i,j ) 2 ) 1/2 2 ) 1/2 te discrete TV term. We leave te study to te interested reader. Acknowledgement: Te first autor would like to tank Professors Antonin Cambolle, Micael Felsberg, Francois Lauze, Jan Modersitzki, Xueceng Tai, and Joacim Weickert for interesting and useful discussions during te poster presentation at te second international conference on te Scale Space and Varational Metods in Computer Vision in Voss, Norway during June 1 5, References [1] Acar, R. and Vogel, C. R.: Analysis of bounded variation penalty metods for ill-posed problems, Inverse Problems 10 (1994), [2] Carter, J. L.: Dual Metods for Total Variation-Based Image Restoration, P.D. tesis, 2001, U.C.L.A. [3] Cambolle, A.; An algoritm for total variation minimization and applications, Journal of Matematical Imaging and Vision 20 (2004),

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