The Convergence of a Central-Difference Discretization of Rudin-Osher-Fatemi Model for Image Denoising

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1 Te Convergence of a Central-Difference Discretization of Rudin-Oser-Fatemi Model for Image Denoising Ming-Jun Lai 1, Bradley Lucier 2, and Jingyue Wang 3 1 University of Georgia, Atens GA 30602, USA mjlai@mat.uga.edu 2 Purdue University, West Lafayette IN 47907, USA lucier@mat.purdue.edu 3 University of Georgia, Atens GA 30602, USA jwang@mat.uga.edu Abstract. We study te connection between minimizers of te discrete and te continuous Rudin-Oser-Fatemi models. We use a centraldifference total variation term in te discrete ROF model and treat te discrete input data as a projection of te continuous input data into te discrete space. We employ a metod developed in [13] wit sligt adaption to te setting of te central-difference total variation ROF model. We obtain an error bound between te discrete and te continuous minimizer in L 2 norm under te assumption tat te continuous input data are in W 1,2. 1 Introduction One of te most influential variational models for image denoising is te total variation based model proposed by Rudin, Oser and Fatemi(ROF) [10]. 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 as follows. We consider functions φ in te space of C 1 functions from Ω to R 2 wit compact support, i.e., [C0 1 (Ω)] 2. Te variation of a function u L 1 (Ω) is ten defined to be u BV := Du := sup u φ. Ω φ [C0 1(Ω)]2, φ 1 point-wise Ω For more details on functions of bounded variation, we refer te reader to [9]. Ω X.-C. Tai et al. (Eds.): SSVM 2009, LNCS 5567, pp , c Springer-Verlag Berlin Heidelberg 2009

2 Te Convergence of a Central-Difference Discretization of ROF Model 515 Te existence and uniqueness of te minimizer of (1) ave been studied by Lions, Oser and Rudin [11] and more completely by Acar and Vogel [1]. Cambolle and Lions [4] proved tat te constrained problem (1) is equivalent to te following unconstrained problem: arg min u BV 1 u g 2. (2) u 2λ Tey also proved more general results of existence and uniqueness of (1). We later call E(u) = u BV 1 2λ u g 2 (3) te ROF energy functional. On te computing side, te most commonly used discrete variational model is based on te discrete energy E (u) = 1 i,j=0 μ i,j ( u) i,j 1 2λ Ω 1 i,j=0 μ i,j (u i,j g i,j ) 2, (4) were u is defined by a 2-dimensional matrix of size, μ i,j is related to te scale. A simple coice of μ i,j is μ i,j =1/ 2. Tere are several possible coices for te discrete gradient operator u [3], [5], and [13]. A common coice is wit ( u) i,j =(( x u) i,j, ( y u) i,j ), ( x u) i,j = u i1,j u i,j, ( y u) i,j = u i,j1 u i,j, were =1/. On te boundary, u is assumed to satisfy te discrete Neumann boundary conditions: u 1,j = u 0,j, u,j = u 1,j, (5) u i, 1 = u i,0, u i, = u i, 1. (6) Te discrete function g i,j is te input image. Many efficient algoritms ave been developed to find te numerical minimizer of (4) [6], [2], [3]. It is not ard to sow tat E Γ -converges to E (for te definition of Γ - convergence, we refer te reader to [7]), terefore, te sequence {u }, minimizers of E,convergestou, te minimizer of E, inl 1 (Ω) and E (u ) converges to E(u) as tends to (cf. [7]). It is interesting to now te rate of convergence and te convergence in oter norm, e.g., in L 2 norm. It is also interesting see te difference between te continuous minimizer and te discrete minimizer. Te autors in [13] proved tat if te discrete energy E is equipped wit a symmetrical discrete total variation as defined in (7) and te discrete input data g is te projection of te

3 516 M.-J. Lai, B. Lucier, and J. Wang continuous input data g by taing average of g on eac pixel, one can bound te error between te discrete minimizer u and te continuous u in L 2 norm by te Lipscitz norm of g provided tat g is in some Lipscitz space. u TV = 1 i,j=0 2 { ( ) 2 ( ) 2 u i1,j u i,j u i,j1 u 1/2 i,j 4 ) 2 ( ) 2 u ( i1,j u i,j u i,j u i,j 1 ) u ( i,j u 2 ( ) i 1,j u i,j1 u 2 i,j ) u ( i,j u 2 ( ) i 1,j u i,j u 2 i,j 1 1/2 1/2 1/2 } (7) In tis paper, we extend te study in [13], [12] to te discrete ROF model equipped wit a central-difference TV term wic is muc simpler tan te symmetrical discrete TV term. Te ideas for te study in tis paper is exactly te same to te ones in [13]. However, a problem of te central-difference model is tat it does not deal well wit some non-smoot data, for example, a cessboard image. Tus we ave to adapt te study in [13] sligtly to tis situation and put a stronger assumption on te input data g in order to establis te convergence. We can still get a similar error bound if te input data g W 1,2. More precisely, our main results are Teorem 1. If g W 1,2, u is te minimizer of E in (3) and u is te minimizer of E in (4) equipped wit te central-difference TV operator, we will give te definition in (10), ten E(u) E (u ) C(1 1 λ )( g W 1,2 g 2 W 1,2)1/2. and Teorem 2. If g W 1,2, u is te minimizer of te functional E in (3) and u is te minimizer of te functional E in (10), ten I u u 2 C(λ 1)( g W 1,2 g 2 W 1,2)1/2. were I u is te piecewise constant injection of u into L 2 space. Te definition of I u will be given in (14) in te next secion. 2 Preliminaries A continuous image u is defined as a L 2 function on Ω R 2. In practice, we always assume Ω to be te unit square [0, 1] [0, 1].

4 Te Convergence of a Central-Difference Discretization of ROF Model 517 We assume te output of denoised image to be in te space of bounded variation. In te discrete settings, we consider te discrete set Ω to be te set of all pairs i =(i 1,i 2 ) Z 2 wit 0 i 1,i 2. A discrete image u is defined as a function on Ω. We always use superscripts to indicate a function is a discrete function troug tis paper. For discrete functions, we define te discrete l p (Ω ) norms 1 u lp (Ω ) := p u i p μ i for 1 p i Ω were μ i is te measure of te discrete space at eac index i. Te simplest coice of μ i is μ i =1 for i Ω. 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 Δ x u i = u i 11,i 2 u i 1,i 2 ; Δ y u i = u i 1,i 21 u i 1,i 2, were =1/ is te step size. We can also define bacward finite difference as Δ x u i = u i 1,i 2 u i 1 1,i 2 ; Δ y u i = u i 1,i 2 u i 1,i 2 1. One can define te second order finite difference as Δ xx u i = Δ x u i Δ x u i. Also Δ yy u i can be similarly defined. We define u l 1, Δ xx u l 1, Δ yy u l 1 as u l 1 := i ( Δ x u i Δ y u i )μ i; (8) Δ xx u l 1 := i Δ xx u i μ i, Δ yy u l 1 := i Δ yy u i μ i. (9) In tis paper, we sall study te error bound for te central-difference discrete ROF model of wic te energy functional is defined as follows E c (u )=J c (u ) 1 2λ u g 2 c. (10) were te BV term J c is defined by J c (u ):= Δ c x u i 2 Δ c y u i 2 μ i, (11) i Ω

5 518 M.-J. Lai, B. Lucier, and J. Wang and Δ c x u i and Δc y u i at i := (i 1,i 2 ) are defined by Δ c x u i = u i 11,i 2 u i 1 1,i 2 2, Δ c y u i = u i 1,i 21 u i 1,i Here u satisfies te discrete Neumann boundary condition: u 1,j = u 1,j, u i, 1 = u i,1, u 1,j = u 1,j, u i,1 = u i, 1. Te discrete space measure μ i = Ω i were Ω i is te intersection of Ω and te square wit center i and size. Ω i := Ω [i 1 2,i 1 2 ] [i 2 2,i 2 ]. (12) 2 It is straigtforward to calculate 2 /4 (i 1,i 2 ) {(0, 0), (0,),(, 0), (, )} μ i = 2 /2 i 1 =0,;0<i 2 <or i 2 =0,;0<i 1 < 2 0 <i 1,i 2 < (13) Te l 2 term is defined by u g 2 c = u i,j gi,j 2 μ i,j. i,j=0 We often need to extend u L p (Ω) and u l p (Ω ) to all of R 2 and Z 2, respectively; we denote te extensions by Ext u and Ext u.foru L p (Ω), we use te following procedure. First, Ext u(x) =u(x), x Ω. We ten reflect orizontally across te line x 1 =1, Ext u(x 1,x 2 )=Extu(2 x 1,x 2 ), 1 x 1 2, 0 x 2 1, and reflect again vertically across te line x 2 =1, Ext u(x 1,x 2 )=Extu(x 1, 2 x 2 ), 0 x 1 2, 1 x 2 2. Having defined Ext u on 2Ω, wetenextendext u periodically wit period (2, 2) on all of R 2. We use a similar construction for discrete functions u.firstweextendu to 2Ω := {i =(i 1,i 2 ) Z 2 0 i 1,i 2 2} as follows: Ext u i = u i, i Ω ;

6 Te Convergence of a Central-Difference Discretization of ROF Model 519 ten we reflect orizontally Ext u (i 1,i 2) =Ext u (2 i 1,i 2), 1 i 1 2, 0 i 2, and ten vertically Ext u (i 1,i 2) =Ext u (i 1,2 i 2), 0 i 1 2, 1 i 2 2. Now tat Ext u is defined on 2Ω, we extend it periodically wit period (2, 2) to all of Z 2. Note tat wit tis definition, te value of Ext u at any point immediately outside Ω is te same as te value of u at te closest point inside Ω. We sometimes need to inject or project functions into L 2 (Ω) or discrete space l 2 (Ω ) respectively. We use te piecewise constant injector to inject discrete function u into L p (Ω): (I u )(x) =u i for x Ω i. (14) We also define an injector L into a space of continuous, piecewise linear functions. In fact, L is te linear interpolation of discrete points {u i } on a triangulation of vertices Z 2. L u = u i φ i. (15) i Ω Here φ i is a dilated and translated tent function, φ i (x) :=φ i 1,i 2 (x 1,x 2 ):=φ(x 1 / i 1,x 2 / i 2 ), (16) were φ is te tent function wic is continuous on R 2, supported in te exagon sown in Fig. 1, linear on eac triangle as sown in Fig. 1, and satisfies te following { 0 (i1,i φ(i 1,i 2 )= 2 ) Z 2 \(0, 0) 1 (i 1,i 2 )=(0, 0) We also consider te piecewise constant projector of u L 1 (Ω) onto te space of discrete functions, defined by (P u) i = 1 u, i Ω, Ω i Ω i were Ω i = μ i is te measure of Ω i defined in (12). We need bot continuous and discrete smooting operators, wic we define as follows. Assume tat η(x) is a a fixed non-negative, rotationally symmetric, mollifier wit support in te unit dis tat is C and as integral 1. For ɛ>0 we define te scaled function η ɛ (x) := 1 ( x ) ɛ 2 η, x R 2 ; ɛ

7 520 M.-J. Lai, B. Lucier, and J. Wang ( 1, 1) (0, 1) ( 1, 0) (0, 0) (1, 0) (0, 1) (1, 1) Fig. 1. Te Support of φ we smoot a function u L p (Ω), 1 p, by computing (η ɛ Ext u)(x) = η ɛ (x y) Extu(y) dy, x 2Ω. R 2 Te discrete smooting operator S L is defined by (S L u ) i = 1 (2L 1) 2 L j 1,j 2= L u i(j 1,j 2) for i Ω For u L p (Ω) we define te (first-order) L p (Ω) modulus of smootness by ( ) 1 ω(u, t) L p (Ω) = sup u(x τ) u(x) p p dx. τ R 2, τ <t x,xτ Ω We also define ω(ext u, t) L p (2Ω) := sup τ R 2, τ <t Ext u( τ) Ext u L p (2Ω). We also ave need of a discrete modulus of smootness. To begin, we define te translation operator (T l (u )) i := u il for any l =(l 1,l 2 ) Z 2. (17) We define te norm l = l 1 l 2 on Z 2, and ten te discrete l p modulus of smootness is ( ) 1 ω(u,m) l p := sup u il p u i p μ i. l Z 2, l m i,il Ω For Ext u we define similarly ω(u,m) l p (2Ω ) = sup T l u u l p (2Ω ). l Z 2, l m

8 Te Convergence of a Central-Difference Discretization of ROF Model Basic Properties We begin wit te following properties. Lemma 1. (Contraction) Let u, v be te minimizers for input data f and g in problem (2) respectively, u v L 2 f g L 2. See a proof in [13] or [12]. Wit te above property, one can ave te following Lemma 2. (Continuity of translation) Assume u is te minimizer of E in problem (2) for input data g. Extendu to Ext u over R 2 by symmetric extension as defined before. Ten Ext u(x ) Ext u(x) L2 (Ω) ω(g, ) L2 (Ω). Remar 1. One can conclude from Lemma 2 tat ω(u, ) L2 (Ω) ω(g, ) L2 (Ω). (18) Remar 2. Similar tecniques allow one to sow tat tis result also olds for tediscretecaseofu and g were u is te minimizer of te discrete energy E wit te symmetric discrete TV operator J c,andu is extended on Z 2 as before. In fact, te corresponding discrete version is. T l (u ) u l 2 (A) Cω(g, l ) l 2 (A), (19) were A is te index set {i := (i 1,i 2 ):0 i 1,i 2 }. For any discrete image v, te discrete modulus of continuity is ω 1 (v,m) l 2 (A) := sup T l (v ) v l 2 (A n1,n 2 ) (20) 0< l m wit T l being te translation operator defined in (17) and A n1,n 2 := {(i, j) :(i, j) A, (i n 1,j n 2 ) A}. Lemma 3. (Maximum principle) Suppose u is te minimizer of E.Ifg L.Ten u g. Te following lemmas bound te errors introduced by injectors and projectors defined before respectively. Lemma 4. Let u L 2 (Ω) and u l 2 (Ω ). Ten tere exists a constant C suc tat te following properties old: a) P u l 2 u L 2;

9 522 M.-J. Lai, B. Lucier, and J. Wang b) c) d) e) ω(p u, m) l 2 Cω(u, m) L 2. u l 2 = I u L 2; ω(i u,m) l 2 Cω(u,m) L 2. u I P u L 2 Cω(u, ) L 2. Te following lemma bounds te difference between te two injectors we defined in (14) and (15). Lemma 5 L u I u L 2 Cω(u, 1) l 2 Te following lemmas sow te properties of te smooting operators Lemma 6 S L u u l 2 ω(u,l) l 2, (21) and J c (S L u ) J c (u ), (22) Δ xx S L u l 1 Δ yy S L u l 1 C L u l 1. (23) Te first inequality in Lemma 6 sows te error between u and smooted u can be bounded by its discrete modulus of continuity. Te second inequality sows smooting does not increase te discrete total variation. Te last inequality sows te te second order difference of te smooted function can be bounded by its first order finite difference. Lemma 7 is te continuous case of Lemma 6. Lemma 7 η ɛ u u L 2 ω(u, ɛ) L 2, (24) and η ɛ u BV u BV, (25) D xx u ɛ L 1 D yy u ɛ L 1 C ɛ u BV. (26)

10 Te Convergence of a Central-Difference Discretization of ROF Model Proof of te Main Result 4.1 Main Idea Recall te ROF continuous and discrete energy functionals are defined by E(v) = v BV 1 2λ v g 2 ; (27) E (v )=J c (v ) 1 2λ v g 2 c (28) wit input image g = P g. To study te difference between E (u ) and E(u), it sould first be noticed tat E and E are two different functionals defined on different spaces. E is defined on te continuous BV(Ω) space wile E is a discrete operator defined on discrete function space. Terefore, some connection between tese two operators sould be built. We use two energy bounds to bridge tem. First, given a discrete minimizer u of functional E, we inject u into L 2 space by function L S L u wit E(L S L u ) less tan E (u ) plus some error. Te construction of L S L u is done by first smooting u as S L u, ten linearinterpolating S L u. Assuming u is te minimizer of E, weave E(u) E(L S L u ) E (u )e g,, (29) were e g, is te error to be bounded in te next section, wic depends on initial g and mes size, and tends to zero as tends to zero. Te second energy bound is similar but taen in te opposite direction. Based on u, we construct a smooted" discrete function P η ɛ u by first smooting" it, ten projecting it into discrete function space, wit E (P η ɛ u) less tan E(u) plus an error term e g, similar to e g,. By te definition of u,weave From (29) we see from (30) ten we conclude tat Tis will complete our error bound. E (u ) E (P η ɛ u) E(u)e g,. (30) E(u) E (u ) e g, ; E (u ) E(u) e g, ; E (u ) E(u) max{e g,,e g,}. 4.2 Setc of te Proof Proposition 1. If g W 1,2,andu, u are te minimizers of E, E in (28), (27) respectively, ten E(u) E (u )C(1 1 λ )( g W 1,2 g 2 W 1,2)1/2.

11 524 M.-J. Lai, B. Lucier, and J. Wang Proof. We sall bound te energy E(L S L u ). It is straigtforward to calculate its TV term (albeit, te computation is tedious) tat L S L u BV J c (S L u )C ( Δ xx S L u l 1 Δ yy S L u l 1). By te property of discrete smooting operator (22) and (23) in Lemma 6, L S L u BV J c (u ) C L u l 1. By Holder s inequality and Lemma 2, u l 1 is bounded by u l 1 = ( Δ x u i Δ y u i ) μ i i { } 1/2 { } 1/2 C Δ x u i 2 μ i Δ y u i 2 μ i C i ( T(1,0) u u T (0,1) u u ) C ω(g, 1) l 2 by (19) C g W 1,2 i We ave L S L u BV J c (u ) C L g W 1,2. Te L 2 term of E ( L S L u ) can be written as L S L u g L 2 = (L S L u I S L u )(I S L u I u ) (I u I g )(I g g) L 2 u g c C(L) g W 1,2 Applying properties of injectors and projectors, Lemma 4 and Lemma 5 and noting te assumption L 1 and te fact tat we obtain Tus u g c g c g, L S L u g 2 L 2 u g 2 c C(L) g 2 W 1,2. E(L S L u )= L S L u BV 1 2λ L S L u g 2 L 2 J c (u ) C L g W 1,2 1 2λ u g 2 c C λ (L) g 2 W 1,2 = E (u ) C L g W 1,2 C λ (L) g 2 W 1,2.

12 Te Convergence of a Central-Difference Discretization of ROF Model 525 Setting L = 1/2, we obtain te result of tis proposition. Using similar metod we prove te following Proposition 2. If g W 1,2,andu, u are te minimizers of E, E in (27), (28) respectively, ten E (u ) E(u)C(1 1 λ )( g W 1,2 g 2 W 1,2)1/2. Combining Propositions 1 and 2 immediately yields te following Teorem 1. If g W 1,2,andu, u are te minimizers of E, E in (27), (28) respectively, ten E(u) E (u ) C(1 1 λ )( g W 1,2 g 2 W 1,2)1/2. Next we need te following lemma Lemma 8. If u is te minimizer of E in (27), ten for any v BV, v u 2 2λ(E(v) E(u)). (31) A proof of tis Lemma can be found in [13] or [12]. It ten follows Teorem 2. If g W 1,2,andu, u are te minimizers of E, E in (27), (28) respectively, ten I u u 2 C(λ 1)( g W 1,2 g 2 W 1,2)1/2. Remar 3. In tis paper, we ave proved te error bound for te discrete ROF model equipped wit a central-difference TV term using te metod suggested in [13]. Tis model is simpler in form tan te model studied in [13], were a symmetrical TV term is used. Tis model is also sligtly easier to be computed by Cambolle s metod (cf. [3]). However we notice tat te central-difference model fails to deal wit a class of data, for example a cessboard image. Tus we ave to put some stronger assumption on te initial data(in W 1,2 )) to obtain te error bound wic may not be satisfied by all real images. However tis result still sows te metod in [13] can be extended to oter symmetric discrete TV operators. It is also interesting to study furter if a similar error bound for tis model can be obtained witout tis assumption imposed. References 1. Acar, R., 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, U.C.L.A (2001)

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