ERROR BOUNDS FOR FINITE-DIFFERENCE METHODS FOR RUDIN OSHER FATEMI IMAGE SMOOTHING

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1 ERROR BOUNDS FOR FINITE-DIFFERENCE METHODS FOR RUDIN OSHER FATEMI IMAGE SMOOTHING JINGYUE WANG AND BRADLEY J. LUCIER Abstract. We bound te difference between te solution to te continuous Rudin Oser Fatemi image smooting model and te solutions to various finite-difference approximations to tis model. Tese bounds apply to typical images, i.e., images wit edges or wit fractal structure. Tese are te first bounds on te error in numerical metods for ROF smooting. Key words. Total variation, bounded variation, variational problems, finite-difference metods, image processing. AMS subject classifications. 65N06, 65N12, 94A08 1. Introduction. Image noise removal based on total variation smooting was introduced by Rudin, Oser, and Fatemi in [13]. Under tis ROF model, one supposes a true image f defined on Ω = [0, 1] 2 and a corrupted image g derived from f (by adding noise, etc.) wit f g 2 L 2 (Ω) = σ2. In an attempt to reconstruct f from g, one calculates a smooted image u tat minimizes v BV(Ω) = Dv subject to te constraint v g 2 L 2 (Ω) σ2. (1.1) Ω (Precise definitions are given later.) We deal wit te equivalent problem: If we calculate ḡ, te average of g on Ω, ten for any σ wit σ 2 < g ḡ 2 L 2 (Ω) tere exists a unique λ > 0 suc tat te minimizer of (1.1) is te minimizer u of te functional E(v) = 1 2λ v g 2 L 2 (Ω) + v BV(Ω). (1.2) Here λ is a positive parameter tat balances te relative importance of te smootness of te minimizer (important wen λ is large) and te L 2 (Ω) distance between te minimizer and te initial data (important wen λ is small). About te same time, Bouman and Sauer [1] proposed a discrete version of (1.2) in te context of tomograpy. Practically one discretizes E( ) to compute te minimizer of te discrete functional E ( ). We assume te discrete corrupted image g of resolution N N (N = 1/) is simply te piecewise constant projection of te continuous corrupted image g, and define te discrete functional E (v ) = 1 2λ vi gi J (v ), (1.3) i Tis work was partially supported by te Office of Naval Researc, Contract N J-1152, and te Institute for Matematics and its Applications, te University of Minnesota. Department of Matematics, University of Georgia, Atens, Georgia (jwang@mat.uga.edu). Department of Matematics, Purdue University, West Lafayette, Indiana (lucier@mat.purdue.edu). 1

2 were J is a discretized total variation. Te most commonly used J is te discrete variation J ++ used in [13] J ++ (v ) = i (v i+(1,0) v i ) 2 ( v i+(0,1) v ) 2 i + 2. (1.4) Efficient algoritms ave been developed to compute te discrete minimizer([2], [6], [3], [5]). In tis paper, we study te relationsip between te minimizer u of E( ) and te discrete minimizer u of E ( ). It is well known tat E Γ-converges to E in L 1. As a direct deduction u tends to u in L 1. Assuming te discrete variation J satisfies certain conditions tat we explain later, we give a bound of te L 2 norm of te difference between u and u in Teorem 4.2 in Section 4. Because te ROF model is often applied to images, an analysis of te error between solutions of discrete approximations and te solution of te continuous model itself sould apply to functions modeling images. Typical natural images ave little smootness, because of intensity discontinuities at te edges of objects and te fractal structure of many objects temselves (te leaves in a tree, air, etc.). Our results apply to functions in te Lipscitz spaces Lip(α, L 2 (Ω)), wic contain functions wit, rougly speaking, α derivatives in L 2 (Ω). Here 0 < α 1/2 for images wit edges : f BV(Ω) L (Ω) implies f Lip(1/2, L 2 (Ω)), wile functions wit fractal structure usually ave α < 1/2, see [7]. Our convergence results in Section 4 are proved for (1.3) wit J = J, a discrete variation obtained by symmetrizing J ++. Noneteless, our approac is quite general, and in Section 5 we obtain te same results for J U, an upwind discrete variation formulated in [12]. We remark tat an iterative metod for minimizing (1.3) wit J = J U was given in [5]. Wile te ROF model as proved to be tremendously influential, and as been te base of furter algoritms in image processing, we know of no oter results tat bound te difference between te solutions of te continuous problems and its finitedifference approximations. A finite element metod applied to te time-dependent gradient descent problem associated wit (1.2) was studied in [10]; we note tat teir Teorem 4 requires te initial data u 0 to ave two continuous derivatives on Ω so it does not apply to typical natural images wit edges. Te rest of tis section introduces notation and our main results. In Section 2 we compare discrete and continuous variational functionals. In Section 3 we note some properties of te minimizers of E( ) and E ( ) tat we use in Section 4 to first bound te discrete and continuous functionals at teir respective minimizers and ten to bound te L 2 difference between te discrete and continuous minimizers temselves. In Section 5, we prove a number of lemmas for te upwind discretization of te BV semi-norm tat allows us to prove similar error bounds for te discrete minimizer of te upwind sceme. Section 6 summarizes our results and points to variations tat appear elsewere. Finally, we note tat we present ere a sequence of lemmas, te proofs of wic are often omitted as a tedious calculation, or standard given te previous lemmas. We found wen developing tese results, owever, tat dividing te argument into tese smaller steps led to muc greater clarity, and we preserve tat structure in tis paper. 2

3 1.1. Summary of main results. Te major difficulty to overcome in our analysis is to compare J (u ), te discrete variation of te minimizer of te discrete functional (1.3), wit u BV(Ω), te variation of te minimizer of te continuous functional (1.2). Indeed, if we consider P u i, te discrete function tat is computed as te average of u on subsquares Ω i = (i + Ω), ten for general u, no matter wic J we coose, we ave tat lim J (P u) u BV(Ω). 0 So E (P u) does not, in general, converge to E(u) (since P u P g L2 (Ω ) u g L2 (Ω) as 0), and te question of weter E (u ) does converge to E(u) requires a more subtle analysis. We note tat J ++ defined by (1.4) is a consistent approximation to u BV(Ω) for C 1 u: lim 0 J ++ (P u) = u BV(Ω) for smoot u. So, for general u BV(Ω), we first mollify u, computing S ǫ u = η ǫ u, were η ǫ is a mollifier and ǫ is a positive parameter tending to zero in a controlled way tat depends on, and compare E (P S ǫ u) to E(u). Mollifying u introduces an error in te L 2 (Ω) term of (1.2), but it reduces te te J term, making it closer to u BV(Ω). Tat s ow we compare te continuous u to te discrete u ; we also ave to go te oter way. To do tat we require J to ave a certain symmetry, so we consider first a symmetrized version J of J ++ and ten later an upwind discrete variation J U. In more-or-less complete analogy wit te continuous argument, we first compute S L u i, a discrete average of u i on (2L + 1) (2L + 1) squares (wit te point i in te middle of te square) and ten compute a piecewise linear interpolant Int S L u (1.19) of S L u i, comparing E(IntS Lu ) to E (u ). Now L is a positive parameter, depending on, tat tends to infinity as 0. Tus we prove te following two teorems: Teorem 1.1 (Functional difference). Let g Lip(α, L 2 (Ω)) and assume u is te minimizer of E(v) from (3.3) and u is te minimizer of E (v ) from (3.1). Ten tere is a constant C suc tat if ǫ = 1/(α+1) we ave E (P S ǫ u) E(u) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1). and if L is set to te integer part of α/(α+1) ten Finally, E(IntS L u ) E (u ) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1). E(u) E (u ) C λ g 2 Lip(α,L 2 (Ω)) α/(α+1). Teorem 1.2 (Minimizer difference). Let g Lip(α, L 2 (Ω)) and assume tat u is te minimizer of E(v) from (3.3) and u is te minimizer of E (v ) from (3.1). Ten tere is a constant C suc tat I u u 2 L 2 (Ω) C g 2 Lip(α,L 2 (Ω)) α/(α+1), were (I u )(x) = u i for x Ω i. 3

4 1.2. Basic notations. We consider te usual L p (Ω) spaces on Ω := [0, 1] 2 R 2, wit v Lp (Ω) := ( Ω v p) 1 p for 1 p <. (We assume te usual cange for p =.) We consider te discrete set Ω to be te set of all pairs i = (i 1, i 2 ) Z 2, Z te integers, wit 0 i 1, i 2 < N, = 1/N, and we refer to functions defined on Ω as discrete functions. So for discrete functions v = vi, we define te discrete Lp (Ω ) norms ( ) 1 v Lp (Ω ) := vi p 2 p for 1 p <. i Ω We define te translation operator for discrete functions by (T l (v )) i := v i+l for any l = (l 1, l 2 ) Z 2. To measure te size of a translation, we introduce l = max( l 1, l 2 ). Similarly, (T τ v)(x) = v(x + τ) for any τ = (τ 1, τ 2 ) R 2 and, for translations, we set τ = max( τ 1, τ 2 ). We often need to extend v L p (Ω) and v L p (Ω ) to all of R 2 and Z 2, respectively; we denote te extensions by Extv and Ext v. For v L p (Ω), we use te following procedure. First, Extv(x) = v(x), x Ω. We ten reflect across te line x 1 = 1, Extv(x 1, x 2 ) = Extv(2 x 1, x 2 ), 1 x 1 2, 0 x 2 1, and reflect again across te line x 2 = 1, Extv(x 1, x 2 ) = Extv(x 1, 2 x 2 ), 0 x 1 2, 1 x 2 2. Having defined Extv on 2Ω, we ten extend Extv periodically on all of R 2. We use te analogous construction of Ext v for discrete functions v. Note tat te value of Ext v at any point immediately outside Ω is te same as te value of v at te closest point inside Ω. For v L p (Ω) we define te (first-order) L p (Ω) modulus of smootness by ( ) 1 We also define ω(v, t) Lp (Ω) = sup τ R 2, τ <t ω(extv, t) Lp (2Ω) := v(x + τ) v(x) p dx x,x+τ Ω sup T τ Extv Extv Lp (2Ω). τ R 2, τ <t Te Lipscitz spaces Lip(α, L p (Ω)) consist of all functions v for wic v Lip(α,Lp (Ω)) := sup t α ω(v, t) Lp (Ω) < ; t>0 we set v Lip(α,Lp (Ω)) := v Lp (Ω) + v Lip(α,Lp (Ω)). We also need a discrete modulus of smootness. Te discrete L p (Ω ) modulus of smootness is ( ) 1 ω(v, m) L p (Ω ) := sup vi+l vi p 2 p. l Z 2, l m i,i+l Ω 4 p.

5 For Ext v we define similarly ω(ext v, m) Lp (2Ω ) = sup T l Ext v Ext v Lp (2Ω ). l Z 2, l m We ave te following relationsip between moduli of smootness and our extension operators; te lemma can be proved as in [8], page 182. Lemma 1.1 (Witney extension). For all 1 p tere exists a constant C suc tat for all v L p (Ω) and v L p (Ω ) and T τ Extv Extv Lp (2Ω) Cω(v, τ ) Lp (Ω) τ R 2 (1.5) T l Ext v Ext v Lp (2Ω ) Cω(v, l ) Lp (Ω ) l Z 2 (1.6) Moreover, for all positive t R, m Z we ave and ω(extv, t) Lp (2Ω) Cω(v, t) Lp (Ω) (1.7) ω(ext v, m) L p (2Ω ) Cω(v, m) L p (Ω ). (1.8) 1.3. Variation functionals. Te variation of a function v L 1 (Ω) is 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(Ω)] 1 2. Te variation of a function v L 1 (Ω) is ten defined to be v BV(Ω) := Dv := sup v φ. Ω φ [C0 1(Ω)]2, φ 1 pointwise Ω We note tat if v is in te Sobolev space W 1,1 (Ω), so tat its first distributional derivatives are in L 1 (Ω), ten v BV(Ω) = v. We need discrete analogues of te variation of a function. For and independently taking values in te set {+, } and any discrete function v we define J (v ) := (Ext vi (1,0) Ext vi i Ω Ω ) 2 ( Ext vi (0,1) + Ext vi ) 2 2. (1.9) We note tat te sum is over i Ω, and Ext v i = v i for all i Ω. Having defined J ++ (v ), J + (v ), J + (v ), and J (v ), we define for any nonnegative a, b, c, and d wit a + b + c + d = 1 J (v ) = a J ++ (v ) + b J + (v ) + c J + (v ) + dj (v ) (1.10) 5

6 and define te special isotropic discrete variation J (v ) := 1 4( J++ (v ) + J + (v ) + J + (v ) + J (v ) ) ; J is invariant under rotations of Ω by 90 degrees, or under orizontal or vertical reflections. At times we consider discrete variational functionals for discrete functions defined on 2Ω ; for tese purposes we denote by J (v Ω ) te discrete variation defined in (1.9) and J 2Ω (Ext v ) te corresponding sum over 2Ω ; similarly we write J Ω (v ) = J (v ) and J Ω (v ) = J (v ) and we use te notation J 2Ω (Ext v ) for a J 2Ω ++ (Ext v ) + b J 2Ω + (Ext v ) + c J 2Ω + (Ext v ) + dj 2Ω (Ext v ) and J 2Ω (Ext v ) for te corresponding sum wit a = b = c = d = 1/4. We ave te following relationsips between continuous and discrete variations of functions and te continuous and discrete extension operators; te lemma is proved simply by considering te symmetries of J Ω any variation along te lines of reflection. Lemma 1.2 (TV symmetry). For any discrete function v, Tus, we ave J 2Ω and noticing tat Extv does not add J 2Ω (Ext v ) = 4J Ω (v ). (1.11) (Ext v ) = 4J Ω (v ) and for any l Z 2 J 2Ω (T l Ext v ) = 4J Ω (v ). Similarly, for any v BV(Ω), we ave Extv BV(2Ω) = 4 v BV(Ω). We also define a discrete anisotropic variation tat is analogous to te W 1,1 (Ω) Sobolev semi-norm: v W 1,1 (Ω ) = { Ext vi+(1,0) Ext vi + Ext vi+(0,1) Ext vi } 2. i Ω (1.12) Because a 2 + b 2 a + b 2 a 2 + b 2, tere exist positive constants C 1 and C 2 suc tat for any discrete function v and any discrete functional J C 1 v W 1,1 (Ω ) J (v ) C 2 v W 1,1 (Ω ). (1.13) For some intermediate estimates we need second-order continuous and discrete semi-norms, so we define for v in te Sobolev space W (2Ω) wit periodic boundary conditions (i.e., treating 2Ω as a torus) v W (2Ω) = D1v 2 + D2v, 2 2Ω and for periodic discrete functions v on 2Ω we define v W (2Ω ) as { vi+(1,0) 2v i + v i (1,0) + vi+(0,1) 2v i + } v i (0,1) 2. (1.14) i 2Ω 2 6 2

7 Note tat tese semi-norms do not include cross derivatives or differences, but we do not need tese in our estimates. Lemma 1.3 (TV difference). For any two discrete functionals J and J, and any discrete function v we ave J (v ) J (v ) Ext v W (2Ω ). (1.15) Proof. Te quantities summed in (1.9) are te norms of two-vectors of divided differences, wic we coose to write in te following way: J ++ (v ) J + (v ) ( ) = 1 Ext vi+(1,0) Ext v i Ext v i Ω i+(0,1) Ext vi 2 ( 1 Ext vi+(1,0) Ext ) v i ( 1)(Ext v i Ω i (0,1) Ext vi ) 2 ( ) 1 Ext vi+(1,0) Ext ( ) vi Ext v i Ω i+(0,1) Ext vi Ext vi+(1,0) Ext vi ( 1)(Ext vi (0,1) Ext vi ) ( 1 Ext vi+(1,0) Ext ) ( ) vi Ext v Ext v i Ω i+(0,1) Ext vi i+(1,0) Ext vi ( 1)(Ext vi (0,1) Ext vi ) 2 = 1 ( ) 0 Ext vi+(0,1) 2 Ext vi + Ext vi (0,1) 2 i Ω Ext v W (2Ω ). Analogous arguments apply to te oter differences Projectors, injectors, and smooting operators. We define te piecewise constant injector of discrete functions v into L p (Ω): (I v )(x) = vi for x Ω i, were Ω i := ( Ω+i ). Later we define an injector into a space of continuous, piecewise linear functions. We also consider te piecewise constant projector of v L 1 (Ω) onto te space of discrete functions, defined by (P v) i = 1 Ω i Ω i v, i Ω, were Ω i is te measure of Ω i. Lemma 1.4 (Injector and projector). Tere exists a constant C suc tat for all v L 2 (Ω) We also ave for any periodic v W (2Ω) 2 v I P v L2 (Ω) Cω(v, ) L2 (Ω). (1.16) P v W (2Ω ) v W (2Ω). (1.17) 7

8 Proof. Relationsip (1.16) is a special case of a general bound for te error in spline approximation; see [8], Teorem 7.3, page 225. To prove (1.17), we deal wit te differences in te orizontal direction. (P v) i+(1,0) 2(P v) i + (P )v i (1,0) i 2Ω = 1 i 2Ω = 1 2 i 2Ω = 1 2 i 2Ω = 2 i 2Ω 2Ω 2 2 (P v) i+(1,0) (P v) i (P v) i (P v) i (1,0) 2 [(v(x +, y) v(x, y)) (v(x, y) v(x, y))] dx dy Ω i [D 1 v(x + t, y) D 1 v(x + t, y)] dt dx dy 1 Ω i 0 0 Ω i 0 D 11 v(x + t + s, y) ds dt dx dy D 11 v dx dy (excange te order of integration and sum over i) Arguing similarly in te vertical direction, we see tat (1.17) olds. We need anoter map taking v L 2 (Ω ) to L 2 (Ω), in te form of a piecewise linear interpolant of te discrete values of vi. To tis end, let φ be te box spline function wose support is te exagon D in Figure 1.1 wit φ being linear on eac triangle in Figure 1.1 and { 1, i = (0, 0), φ(i) = 0, i (0, 0), i Z 2. x 3 1 x 2 x 4 x 1 1 x 5 x 6 Fig D, te support of φ We dilate and translate φ to obtain te function ( x ( ( 1 φ i (x) := φ i + 2, 1. (1.18) 2))) We see tat supp φ i is D dilated by and translated by ( i + ( 1 2, 1 2)). 8

9 We define te interpolant Int v by Int v = i Z 2 Ext v i φ i. (1.19) We ten ave te following lemma. Lemma 1.5 (Piecewise linear injector). For any v be in L 2 (Ω ) we ave Int v BV(Ω) = 1 2 (J ++(v ) + J (v )). (1.20) Additionally, tere exists a constant C suc tat for all discrete functions v I v Int v L2 (Ω) Cω(v, 1) L 2 (Ω ). (1.21) Proof. Te proofs of (1.20) and (1.21) are just calculations, wic can be found in [14]. For (1.20), te J ++ terms come from triangles wit te orientation of te triangle in te upper-rigt quadrant of Figure 1.1, and te J terms come from triangles wit te orientation of te triangle in te lower-left quadrant. We need bot continuous and discrete smooting operators, wic we define as follows. Assume tat η(x) is a a fixed nonnegative, rotationally symmetric, function wit support in te unit disk; furter, suppose tat η is C and as integral 1. For ǫ > 0 we define te scaled function η ǫ (x) := ǫ 2 η(x/ǫ), x R 2 ; we smoot a function v L p (Ω), 1 p, by computing (S ǫ v)(x) := (η ǫ Extv)(x) = η ǫ (x y) Extv(y) dy, x R 2. R 2 Our discrete smooting operator is defined simply as S L v := It s clear from tese definitions tat 1 (2L + 1) 2 l L T l Ext v. T l S L Ext v = S L T l Ext v and T τ S ǫ Extv = S ǫ T τ Extv (1.22) and tat for any 1 p and S L v Lp (Ω ) S L v Lp (2Ω ) Ext v Lp (2Ω ) 4 v Lp (Ω ) (1.23) S ǫ v Lp (Ω) S ǫ v Lp (2Ω) Extv Lp (2Ω) 4 v Lp (Ω). (1.24) For tese continuous and discrete smooting operators we ave te following results. Lemma 1.6 (Smooting operators). For all v L 2 (Ω) and all discrete functions v, we ave J (S L v ) J (v ) and S ǫ v BV(Ω) v BV(Ω). (1.25) 9

10 Tere exists a constant C > 0 suc for all M, t > 0, ω(s L v, M) L 2 (Ω ) Cω(v, M) L 2 (Ω ) and ω(s ǫ v, t) L2 (Ω) Cω(v, t) L2 (Ω). (1.26) Furtermore, S L v v L2 (Ω ) Cω(v, L) L2 (Ω ) and S ǫ v v L2 (Ω) Cω(v, ǫ) L2 (Ω). (1.27) We also ave S ǫ v W (2Ω) C ǫ v BV(Ω) and S L v W (2Ω ) C L v W 1,1 (Ω ). (1.28) Proof. Te first two inequalities follow simply because te BV semi-norm and J are convex and symmetric on 2Ω ; see Lemma 1.2. Te two inequalities (1.26) follow from te definitions of S L and S ǫ and Lemmas 1.1 and 1.2. Te next two inequalities follow from te definitions of S L, S ǫ, and te properties of te discrete and continuous moduli of smootness; see also (1.5) of Lemma 1.1. Te bound on te discrete W (2Ω ) semi-norm is a typical inverse inequality; to deal wit te differences in te orizontal direction, S Lvi+(1,0) 2S Lvi + S Lv i (1,0) 2 2 i 2Ω = 1 Ext vi+(1,0)+l 2 Ext vi+l + Ext v i+l (1,0) (2L + 1) i 2Ω l L = 1 Ext vi+(l+1,l 2 ) (2L + 1) 2 Ext vi+(l,l 2 ) 2 i 2Ω l 2 L Ext vi (L,l 2 ) Ext vi (L+1,l 2 ) 2 2 (sum over l 1 ) { 1 Ext vi+(l+1,l 2 ) Ext v i+(l,l 2 ) (2L + 1) 2 2 i 2Ω l 2 L Ext + vi (L,l 2 ) Ext v } i (L+1,l 2 ) 2 C (2L + 1) Ext vi+(1,0) Ext v i i 2Ω C L v W 1,1 (Ω ) 2 2 For te bound on te W (2Ω) semi-norm, again we deal wit derivatives in only one direction. We prove 2Ω D2 1S ǫ v C ǫ Ω Dv. 10

11 In fact 2Ω D 2 1S ǫ v = sup φ C 1 0 = sup φ C 1 0 = sup φ C 1 0 (2Ω), φ 1 (2Ω), φ 1 (2Ω), φ 1 = sup φ C0 1 (2Ω), φ 1 (D1S 2 ǫ v) φ R 2 (D 1 S ǫ v) D 1 ( φ) R 2 D 1 (η ǫ Extv) D 1 ( φ) R 2 (Extv) D 1 (D 1 η ǫ φ); R 2 note tat all but te first of tese integrals are over R 2. Notice D 1 η ǫ φ D 1 η ǫ L 1 φ C ǫ, and D 1 η ǫ φ C0 (2Ω ǫ ), were 2Ω ǫ := {x dist(x, 2Ω) ǫ}; terefore D1S 2 ǫ v C D Extv C D Extv C Dv. ǫ 2Ω ǫ ǫ Ω ǫ Ω 2Ω were Ω = {(x, y) x, y 3}. 2. Relationsips between discrete and continuous variation and functionals. We need to compare continuous and discrete variation functionals, so we ave te following tecnical lemma, wic is proved in te appendix. Lemma 2.1 (TV bound). Tere exists a C > 0 suc for any J and any v L 1 (Ω) and for any v defined on Ω J (P v) v BV(Ω) + C Extv W (2Ω) (2.1) Int v BV(Ω) J (v ) + C Ext v W (2Ω ). (2.2) Our goal is to bound te difference between various continuous and discrete convex functionals defined on L 2 (Ω) and L 2 (Ω ). We fix λ > 0. Given g L 2 (Ω), we consider te (unique) minimizer u of te functional E(v) = 1 2λ v g 2 L 2 (Ω) + v BV(Ω) and te (unique) minimizer u of te functional E (v ) = 1 2λ v P g 2 L 2 (Ω ) + J (v ), were J is any of te discrete variational functionals defined above. Most of our analysis concerns itself wit te special case J = J. It is difficult to compare u and u directly, because J (u ) and u BV(Ω) could be far apart, in general, even if u u as 0. However, tere are smooted versions 11

12 of u and u, close to u and u, wose continuous and discrete variations are close, as te following Lemma sows. Lemma 2.2 (TV consistency). Tere exists a constant C suc tat for any discrete function v L 2 (Ω ) and any positive integer L we ave Int S L v BV(Ω) J (v ) + C L J (v ). (2.3) Furtermore, tere is a constant C suc tat for any v BV(Ω) and any positive ǫ and any discrete functional J, we ave J (P S ǫ v) v BV(Ω) + C ǫ v BV(Ω). (2.4) Proof. For te first inequality, we ave from (2.2) in Lemma 2.1 (wit J = J ) IntS L v BV(Ω) J (S L v ) + C S L v W (2Ω ), wile from (1.25) in Lemma 1.6 J (S L v ) J (v ) and (1.28) in te same lemma S L v W (2Ω ) C L v W 1,1 (Ω ) C L J (v ). Te second inequality follows from (1.13). Combining te previous inequalities gives (2.3). For (2.4), we ave from (2.1) in Lemma 2.1 J (P S ǫ v) S ǫ v BV(Ω) + C S ǫ v W (2Ω), wile (1.25) yields S ǫ v BV(Ω) v BV(Ω) and (1.28) gives S ǫ v W (2Ω) C ǫ v BV(Ω). Combining tese tree inequalities yields (2.4). Now we compare discrete and continuous energy functionals. Lemma 2.3 (Comparing discrete and continuous energies). Tere exists a constant C > 0 suc tat for all J and for all v BV(Ω) E (P S ǫ v) E(v) + C ǫ v BV(Ω) + C λ v g ) L (Ω)( 2 ω(v, )L2 (Ω) + ω(v, ǫ) L2 (Ω) + ω(g, ) L2 (Ω) (2.5) + C ( ω(v, ) 2 λ L2 (Ω) + ω(v, ǫ)2 L 2 (Ω) + ω(g, ) )2 L 2 (Ω). Furtermore, if J = J, ten for all discrete functions v E(IntS L v ) E (v ) + C L J (v ) + C λ v P g L2 (Ω )( ω(v ), L) L2 (Ω ) + ω(g, ) L2 (Ω) (2.6) + C ( ω(v, L) 2 L λ 2 (Ω) + ω(g, ) )2 L 2 (Ω). 12

13 Proof. We ave E (P S ǫ v) = J (P S ǫ v) + 1 2λ P S ǫ v P g 2 L 2 (Ω ). (2.7) From (2.4), we see tat te first term on te rigt is bounded by Now v BV(Ω) + C ǫ v BV(Ω). (2.8) P S ǫ v P g 2 L 2 (Ω ) = I P S ǫ v I P g 2 L 2 (Ω), and te quantity on te rigt can be written as (I P S ǫ v S ǫ v) + (S ǫ v v) + (v g) + (g I P g) 2 L 2 (Ω) v g 2 L 2 (Ω) + 2 v g L 2 (Ω) (I P S ǫ v S ǫ v) + (S ǫ v v) + (g I P g) L2 (Ω) + (I P S ǫ v S ǫ v) + (S ǫ v v) + (g I P g) 2 L 2 (Ω). v g 2 L 2 (Ω) + 2 v g L 2 (Ω) ( ) I P S ǫ v S ǫ v L2 (Ω) + S ǫ v v L2 (Ω) + g I P g L2 (Ω) + C ( I P S ǫ v S ǫ v 2 L 2 (Ω) + S ǫv v 2 L 2 (Ω) + g I P g 2 L (Ω)). 2 From (1.27) we can bound S ǫ v v L2 (Ω) Cω(v, ǫ) L2 (Ω) and from (1.16) we know tat I P g g L2 (Ω) Cω(g, ) L2 (Ω). We also ave from (1.16) and (1.26) I P S ǫ v S ǫ v L2 (Ω) Cω(S ǫ v, ) L2 (Ω) Cω(v, ) L2 (Ω). Tus, P S ǫ v P g 2 L 2 (Ω ) v g 2 L 2 (Ω) + C v g L 2 (Ω) ( ) ω(v, ) L 2 (Ω) + ω(v, ǫ) L 2 (Ω) + ω(g, ) L 2 (Ω) + C ( ω(v, ) 2 L 2 (Ω) + ω(v, ǫ)2 L 2 (Ω) + ω(g, )2 L 2 (Ω)). Using tis inequality as well as (2.8) in (2.7) yields (2.5). Now let v be any discrete function. Ten E(IntS L v ) = Int S L v BV(Ω) + 1 2λ IntS Lv g 2 L 2 (Ω). (2.9) By (2.3), te first term on te rigt is bounded by J (v ) + C L J (v ). (2.10) 13

14 Now IntS L v g 2 L 2 (Ω) = (Int S L v I S L v ) + (I S L v I v ) + (I v I P g) + (I P g g) 2 L 2 (Ω) I v I P g 2 L 2 (Ω) + 2 I v I P g L2 (Ω) (IntS L v I S L v ) + (I S L v I v ) + (I P g g) L2 (Ω) + (IntS L v I S L v ) + (I S L v I v ) + (I P g g) 2 L 2 (Ω) I v I P g 2 L 2 (Ω) + 2 I v I P g L2 (Ω) ( IntS L v I S L v L 2 (Ω) + I S L v I v ) L 2 (Ω) + I P g g L 2 (Ω) + C( IntS L v I S L v 2 L 2 (Ω) + I S L v I v 2 L 2 (Ω) + I P g g 2 L 2 (Ω) ) Since, for all discrete v, I v L 2 (Ω) = v L2 (Ω ), te quantity above is bounded by v P g 2 L 2 (Ω ) + 2 v P g L2 (Ω ) ( Int S L v I S L v L2 (Ω) + S L v v ) L2 (Ω ) + I P g g L2 (Ω) + C( IntS L v I S L v 2 L 2 (Ω) + S Lv v 2 L 2 (Ω ) + I P g g 2 L 2 (Ω) ) From (1.27) we ave By (1.21) and (1.26) we ave S L v v L2 (Ω ) Cω(v, L) L2 (Ω ). Int S L v I S L v L2 (Ω) Cω(S L v, 1) L2 (Ω ) Combining tese inequalities, we ave Cω(v, 1) L2 (Ω ) Cω(v, L) L2 (Ω ). Int S L v g 2 L 2 (Ω) v P g 2 L 2 (Ω ) + C v P g L2 (Ω )( ω(v ), L) L2 (Ω ) + ω(g, ) L2 (Ω) + C ( ω(v, L) 2 L 2 (Ω) + ω(g, )2 L 2 (Ω)). Combining tis inequality wit (2.9) and (2.10) yields (2.6). 3. Properties of te continuous and discrete minimizers. We need to discuss some properties of minimizers of te discrete and continuous functionals. We begin by comparing functionals on Ω and Ω and te corresponding functionals on 2Ω and 2Ω. We remind te reader of te notations used in Lemma 1.2. Lemma 3.1 (Extending minimizers). If u is te minimizer of te functional E Ω (v ) = E (v ) = 1 2λ v g 2 L 2 (Ω ) + JΩ (v ), (3.1) 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 ) + J2Ω (v ) (3.2) 14

15 wit periodic boundary conditions. Similarly, if u is te minimizer of ten Extu is te minimizer of E Ω (v) = E(v) = 1 2λ v g 2 L 2 (Ω) + v BV(Ω) (3.3) E 2Ω (v) = 1 2λ v Extg 2 L 2 (2Ω) + v BV(2Ω), (3.4) again wit periodic boundary conditions. Furtermore, if u and w are minimizers of (3.3) wit data g and, respectively, ten u w L2 (Ω) g L2 (Ω); similarly for te discrete and continuous minimizers of (3.1) (3.4). Tus, for te two periodic problems (3.2) and (3.4) we ave and Ext u T l Ext u L2 (2Ω ) Ext g T l Ext g L2 (2Ω ) (3.5) Extu T τ Extu L2 (2Ω) Extg T τ Extg L2 (2Ω). (3.6) Proof. Beginning wit Lemma 1.2, te discrete extension result can be proved wit a tedious calculation, wic can be found in [14]. Te rest of te teorem is standard. Te results of te next lemma follow quickly from te previous one and Lemma 1.1 and we present tem witout proof. Lemma 3.2 (Smootness bounds). Assume u is te minimizer of E(v) from (3.3) and u is te minimizer of E (v ) from (3.1). Ten and ω(u, ǫ) L2 (Ω) Cω(g, ǫ) L2 (Ω) (3.7) ω(u, L) L2 (Ω ) Cω(P g, L) L2 (Ω ) Cω(g, L) L 2 (Ω). (3.8) 4. Proof of te main teorems. We now bound te difference between discrete and continuous functionals at teir respective minimizers. Teorem 4.1 (Functional difference). Assume u is te minimizer of E(v) from (3.3) and u is te minimizer of E (v ) from (3.1). Ten if ǫ = 1/(α+1) we ave E (P S ǫ u) E(u) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1). (4.1) and if L is set to te integer part of α/(α+1) ten Finally, E(IntS L u ) E (u ) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1). (4.2) E(u) E (u ) C λ g 2 Lip(α,L 2 (Ω)) α/(α+1). (4.3) 15

16 Proof. We mainly use Lemma 2.3 and Lemma 3.2. By (2.5) of Lemma 2.3 E (P S ǫ u) E(u) + C ǫ u BV(Ω) + C λ u g ) L (Ω)( 2 ω(u, )L2 (Ω) + ω(u, ǫ) L2 (Ω) + ω(g, ) L2 (Ω) + C λ ( ω(u, ) 2 L2 (Ω) + ω(u, ǫ)2 L 2 (Ω) + ω(g, )2 L 2 (Ω)). We ten note u g L2 (Ω) g L2 (Ω) and u BV(Ω) 1 2λ g 2 L 2 (Ω) and apply (3.7) to obtain E (P S ǫ u) E(u) + C ǫλ g 2 L 2 (Ω) + C λ g ) L (Ω)( 2 ω(g, ǫ)l2 (Ω) + ω(g, ) L2 (Ω) + C λ ( ω(g, ) 2 L 2 (Ω) + ω(g, ǫ)2 L 2 (Ω)). Now, ω(g, t) L2 (Ω) g Lip(α,L2 (Ω))t α, t > 0. Tus E (P S ǫ u) E(u) + C ǫλ g 2 L 2 (Ω) + C λ g L 2 (Ω) g Lip(α,L 2 (Ω))(ǫ α + α ) + C λ g 2 Lip(α,L 2 (Ω)) (ǫ2α + 2α ) E(u) + C λ g 2 Lip(α,L 2 (Ω)) ( ǫ + ǫα + α + ǫ 2α + 2α ) We know at a minimum 1 > ǫ >, so setting te largest error terms /ǫ and ǫ α equal, i.e., setting ǫ = 1/(α+1), we ave E (P S ǫ u) E(u) + C λ g 2 Lip(α,L 2 (Ω)) (α/(α+1) + α + 2α + 2α/(α+1) ). Tus we obtain (4.1). We point out tat (4.1) olds for any discrete variation J defined in (1.10). More generally it olds for any discrete variation satisfying Lemma 2.2. Similarly, if one begins wit (2.6), notes tat u P g L2 (Ω ) P g L2 (Ω ) g L 2 (Ω) and J (u ) 1 2λ P g 2 L 2 (Ω ) 1 2λ g 2 L 2 (Ω) and applies (3.8), one finds on setting L to te integer part of α/(α+1) tat E(IntS L u ) E (u ) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1), wic is (4.2). Because u and u are minimizers of teir respective functionals, we ave E (u ) E (P S ǫ u) E(u) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1) (4.4) 16

17 and E(u) E(IntS L u ) E (u ) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1). (4.5) Ten (4.3) is proved. To sow te error bound for minimizers, we need te following result, wic can be proved easily using classical arguments. Lemma 4.1. Assume u is te minimizer of E(v) from (3.3) and u is te minimizer of E (v ) from (3.1). Ten for any v BV(Ω), Also, for any v defined on Ω, v u 2 L 2 (Ω) 2λ(E(v) E(u)). (4.6) v u 2 L 2 (Ω ) 2λ(E (v ) E (u )). (4.7) Teorem 4.2 (Minimizer difference). Let g Lip(α, L 2 (Ω)). Assume tat u is te minimizer of E(v) from (3.3) and u is te minimizer of E (v ) from (3.1). Ten I u u 2 L 2 (Ω) C g 2 Lip(α,L 2 (Ω)) α/(α+1). Proof. We apply (4.7) wit v = P S ǫ u and ǫ = 1/(α+1) : P S ǫ u u 2 L 2 (Ω ) 2λ(E (P S ǫ u ) E (u )) 2λ [ (E(u) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1) ) + ( E(u) + C λ g 2 Lip(α,L 2 (Ω)) α/(α+1) ) ]. Te first substitution is by (4.1); te second is by (4.5). Tus we ave P S ǫ u u 2 L 2 (Ω ) C g 2 Lip(α,L 2 (Ω)) α/(α+1). (4.8) Ten I u u 2 L 2 (Ω) = I u I P S ǫ u + I P S ǫ u S ǫ u + S ǫ u u 2 L 2 (Ω) 3 ( I u I P S ǫ u 2 L 2 (Ω) + I P S ǫ u S ǫ u 2 L 2 (Ω) + S ǫ u u 2 ) L 2 (Ω) Because I v L2 (Ω) = v L 2 (Ω ) for any v it follows from (4.8) I u I P S ǫ u 2 L 2 (Ω) = P S ǫ u u 2 L 2 (Ω ) C g 2 Lip(α,L 2 (Ω)) α/(α+1). To bound I P S ǫ u S ǫ u L 2 (Ω), by (1.16), (1.26), and (3.7), we ave I P S ǫ u S ǫ u L2 (Ω) Cω(S ǫ u, ) L2 (Ω) (4.9) Cω(u, ) L2 (Ω) Cω(g, ) L2 (Ω). 17

18 Finally by (1.27) and (3.7) S ǫ u u L2 (Ω) Cω(u, ǫ) Cω(g, ǫ). (4.10) Tus combining (4.8), (4.9) and (4.10), we ave ) I u u 2 L 2 (Ω) ( g C 2 Lip(α,L 2 (Ω)) α/(α+1) + ω(g, ) 2 L 2 (Ω) + ω(g, ǫ)2 L 2 (Ω) C ( g 2 Lip(α,L 2 (Ω)) α/(α+1) + g 2 L 2 (Ω) 2α + g 2 L 2 (Ω) 2α/(α+1)) Because te first term dominates te oters, we ave I u u 2 L 2 (Ω) C g 2 Lip(α,L 2 (Ω)) α/(α+1). 5. Error bound for te upwind sceme. In tis section, we prove te error bound for te upwind sceme. Te upwind discrete gradient operator is defined by ( )vi = 1 Ext v i Ext v i+(1,0) Ext v i Ext v i (1,0) Ext v i Ext v i+(0,1) Ext v i Ext v i (0,1). (5.1) Te upwind discrete variation is ten defined by J U (v ) = i Ω ( )v i 0 2 (5.2) were 0 is te vector (0, 0, 0, 0), and p q and p q are componentwise maximum and minimum, respectively, of te vectors p, q R 4. In oter words, we include a difference in te vector norm of te it term in (5.2) only if v is increasing into vi. Noting canges in te following proofs (and one sees little cange in te images temselves) if we cange componentwise maximum ( ) to componentwise minimum ( ) in (5.2). In teir paper, Oser and Setian [12] were solving Hamilton-Jacobi equations were tis substitution could not be made: teir problem, unlike ours, as a true notion of wind. To prove te result for te upwind sceme, we need to adapt to J U te previous lemmas involving J. First we sall prove te convexity of J U. Lemma 5.1. J U is convex Proof. First note tat for two vectors p, q R n, it is easy to verify 0 (p + q) 0 p 0 + q 0, were inequality p q means p i q i for eac index i. Tus, (p + q) 0 p 0 + q 0. (5.3) 18

19 i ( We apply (5.3) to eac term in (5.2) of J U λf + (1 λ)g ), were 1 > λ > 0 and f and g are discrete functions, to find tat ( J U λf + (1 λ)g ) = ( )(λf + (1 λ)g ) i 0 2 i i { λ( )f i 0 + (1 λ)( )g i 0 } 2 = λj U (f ) + (1 λ)j U (g ). In te following we use te notation x and y defined for {+, } by x v i = Ext v i (1,0) Ext v i, y v i = Ext v i (0,1) Ext v i. (5.4) Note tat te divided differences are applied to te extended discrete function and tat te difference is zero if i Ω and te oter index is outside Ω. Using tese operators, we can write J U (v ) = i Ω + x v i 0 x v i 0 + y v i 0 y v i 0 2. (5.5) Te following lemma corresponds to (1.13). Lemma 5.2. J U is equivalent to W 1,1 (Ω ), were W 1,1 (Ω ) is te discrete semi-norm defined in (1.12). Proof. Trivially, 1 2 { + x vi 0 + x vi y vi 0 + y vi 0 } 2 i i { + x v i x v i y v i y v i 0 2} 2 { + x v i 0 + x v i y v i 0 + y v i 0 } 2 Te middle sum is J U (v ), so we need to prove tat te last sum equals v W 1,1 (Ω ). Note tat + x vi 0 + x vi+(1,0) 0 = + x vi x vi 0 = + x vi, so te absolute value of eac orizontal and vertical difference in v is included precisely once in te last sum, so it equals v W 1,1 (Ω ). Te following lemma corresponds to Lemma 1.3. Lemma 5.3. JU (v ) J (v ) Ext v W (5.6) (2Ω ) were J is any discrete variation defined in (1.9). Proof. We only prove te case for J = J ++. Te oter cases are te same. 19

20 Note tat + x v i 2 = + x v i ( + x )v i 0 2, so we can write J ++ (v ) in a similar way to J U (v ) as J ++ (v ) = + x v i 0 + x vi 0 i Ω + y vi y vi 0 Tus, J U (v ) J ++ (v ) + x vi 0 = x vi 0 + x vi 0 i Ω + y vi 0 + x vi 0 y vi 0 + y vi y vi 0 0 x vi 0 + x vi 0 i Ω 0 2 y vi 0 + y vi 0 ( x vi 0 + x vi 0 + y v 0 + y v 0 ) 2. i Ω Because a 0 b 0 a b, we ave J U (v ) J ++ (v ) ( x vi + x vi + y vi + y vi ) 2 i Ω Ext v W (2Ω ). We use Lemma 5.2 and Lemma 5.3 to prove te following lemma tat corresponds to Lemma 2.1. Lemma 5.4. Tere exists a C > 0 suc for any v L 1 (Ω) and for any v defined on Ω J U (P v) v BV(Ω) + C Extv W (2Ω) (5.7) Intv BV(Ω) J U (v ) + C Ext v W (2Ω ). (5.8) Proof. Te second inequality can be proved by simply combining (2.2) and (5.6). To prove te first inequality, again we assume tat Extv W (2Ω), oterwise it is trivial. We apply Lemma 5.3 wit v = P v, ten Ten by (2.1) in Lemma 2.1 J U (P v) J (P v) + C Ext P v W (2Ω ). J U (P v) v BV(Ω) + C Extv W (2Ω) + C Ext P v W (2Ω ) = v BV(Ω) + C Extv W (2Ω) + C P Extv W (2Ω ) v BV(Ω) + C Extv W (2Ω). 20

21 Table 6.1 L 2 (Ω) errors on grids of size 128, 256, and 512, and differences σ = g u L 2 (Ω) of 16, 32, and 64, wit initial data a multiple of te caracteristic function of a disk (6.1); columns 1 3 are te results wit te anisotropic approximation J = J ++ to BV(Ω) ; columns 4 6 are te results wit te upwind approximation J = J U ; α is te estimated order of convergence, u u L2 C α. L 2 (Ω) difference between continuous and discrete solutions anisotropic J upwind J resolution σ = 16 σ = 32 σ = 64 σ = 16 σ = 32 σ = α Te last line follows from (1.17). Lemma 5.5 is te counterpart of te first inequality (1.25) in Lemma 1.6. Lemma 5.5. J U (S L v ) J U (v ). Proof. Te result comes from te symmetry and convexity of J U. Te proof is exactly te same as te proof for J in Lemma 1.6. We note tat te proofs of Lemmas 3.1 and 4.1 carry over directly to J U, and we obtain te following teorem for te upwind discrete variation. Teorem 5.1 (Error bounds for upwind sceme). Assume u is te minimizer of E(v) from (3.3) for g Lip(α, L 2 (Ω)) and u is te minimizer of te discrete functional Ten E (v ) = 1 2λ v P g 2 L 2 (Ω ) + J U(v ). I u u 2 L 2 (Ω) C g 2 Lip(α,L 2 (Ω)) α/(α+1). Te proof is te same as te proof for te symmetric discrete variation J. 6. Discussion and extensions. We remark tat our error bounds are not optimal in an approximation-teory sense. In general, wit suitably smoot piecewise polynomials, one can approximate a function in Lip(α, L p (Ω)) to order α in L p (Ω) for 0 < p <. So one can approximate Lip(α, L 2 (Ω)) functions in L 2 (Ω) to order α ; in contrast, we acieve an error bound of α/(2α+2). Te caracteristic function of a disk is in Lip(α, L p (Ω)) for α = 1/p; if g is te caracteristic function of a disk, ten te minimizer u of E(v) is again te caracteristic function of a disk (for λ small enoug). Tus one as α = 1/2 and one can expect at most a convergence rate of 1/2 in L 2 (Ω). Our results bound te L 2 (Ω) error by C α/(2(α+1)) = C 1/6. In [5] some numerical experiments were conducted; wit permission we reprint a table sowing te results of tese computations wit initial data g = 255χ x ( 1 2, 1 2 ) 1 4, (6.1) Wile te computations were iterative, te iterated approximation to te true discrete solution was provably witin a distance in L 2 (Ω) of 1/4 to te true discrete solution. We note tat te upwind sceme as sligtly smaller errors (because of less smooting in te discrete solution at te edges of te disk) and a sligtly iger 21

22 estimated rate of convergence. In bot cases, te estimated rate of convergence is strictly between our bound of 1/6 and te optimal rate of 1/2 ; we don t know weter tis difference is real. Estimating real rates of convergence for data in Lip(α, L 2 (Ω)) is difficult for many reasons. Even te optimal asymptotic rate of convergence, O( α ), is quite slow, so one needs very small to be convinced tat te parameter is in te asymptotic regime. Furtermore, a function in Lip(α, L 2 (Ω)) for α 1/2 is, in general, not even bounded; if it s te caracteristic function of a set, ten tis set need not ave bounded perimeter. For bot tese reasons, computing wit generic Lip(α, L 2 (Ω)) data is quite difficult. We do not ave an opinion on wat te true rate of convergence migt be. Somewat weaker results were proved by te first autor in [14] for te functional J (v ) = 1 2( J++ (v ) + J (v ) ). Te arguments tere exploit te fact tat for tis particular J Intv BV(Ω) = J (v ); tey also require tat g Lip(β, L 1 (Ω)) L (Ω), wic implies tat g Lip(α, L 2 (Ω)) for α = β/2, and tey acieve te same convergence rate of α/(2α+2). Finally, similar tecniques ave been applied to analyze a central difference approximation to v BV(Ω) in [11]; tere te same convergence rate of approximation O( 1/4 )(α = 1) was acieved, but for quite smoot functions: g is required to be in te Sobolev space W 1,2 (Ω), a space tat does not contain images wit edges. 7. Appendix. We include ere te proof of a tecnical lemma. Proof of Lemma 2.1. One proves te second inequality simply by combining (1.20) and (1.15). As for te first inequality, te left and side is finite for v L 1 (Ω), so if Extv / W (2Ω), we re done. So we assume tat Extv W (2Ω) and we prove (2.1) for J = J ++, te oter cases being te same. We denote P v by v and use te divided differences + x vi and + y vi from (5.4). In te argument tat follows, we write v for Extv. Ten + x vi 1 2 D 1 v = v i+(1,0) v i 1 Ω i Ω 2 D 1 v i = 1 1 [v(x +, y) v(x, y)] dx dy 1 Ω i Ω i 2 D 1 v. Ω i Te integrand of te first integral can be rewritten as an integral of D 1 v, ten combining tese two integrals and once again rewriting te integrand as an integral of te second derivative of v, we ave + x v i 1 2 Ω i D 1 v = 1 3 = 1 3 Ω i 0 t Ω i 0 (D 1 v(x + t, y) D 1 v(x, y)) dt dx dy 0 22 D 2 1v(x + s, y) ds dt dx dy.

23 Terefore + x v i = 1 2 Ω i D 1 v Ω i t 0 0 D 2 1v(x + s, y) ds dt dx dy. ( ) Similarly for + y vi ; so we can bound te norm of + vi = + x vi + y vi by + vi 1 ( ) ( Ω i D 1 v t Ω Ω i D 2 v 3 i 0 0 D2 1v(x + s, y) ds dt dx dy t 0 D2 2v(x, y + s) ds dt dx dy Ω i 0 ) 1 2 Ω i Dv t Ω i 0 0 t Ω i 0 0 D 2 1v(x + s, y) ds dt dx dy D 2 2v(x, y + s) ds dt dx dy. (7.1) ( ) Te last line follows from te fact tat f f 2 + g 2 (by Jensen s inequal- g ity) and a 2 + b 2 a + b. To bound te discrete total variation J ++ (v ), we sum (7.1) over all indices i Ω wit weigt 2 at eac index. We obtain J ++ (v ) Dv + e x + e y, were e x = i 1 C 0 C t 0 0 t Similarly for e y ; terefore J ++ (v ) Ω 0 Ω i Ω t 0 0 D 2 1v(x + s, y) ds dt dx dy { } D1v(x 2 + s, y) dx dy ds dt Ω { } D1v 2 dx dy ds dt Ω D 2 1v. Ω Dv + C ( D1v 2 + D2v ) 2. Ω REFERENCES [1] C. Bouman and K. Sauer, Bayesian estimation of transmission tomograms using segmentation based optimization, IEEE Trans. Nuclear Science, 39 (1992), pp [2] J. L. Carter, Dual Metods for Total Variation-Based Image Restoration, P.D. tesis, 2001, U.C.L.A. [3] A. Cambolle, An algoritm for total variation minimization and applications, J. Mat. Imaging Vision, 20 (2004), pp

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