An Approximation for the Mumford-Shah Functional

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 25, n pproximation for the Mumford-Shah Functional Luca Lussardi Dipartimento di Matematica Facoltà di Ingegneria, Università degli Studi di Brescia via Valotti 9, 2533 Brescia, Italy luca.lussardi@ing.unibs.it bstract We approximate, in the sense of Γ-convergence, the Mumford-Shah functional by means of a sequence of non-local integral functionals depending on the average of the absolute value of the gradient. Mathematics Subject Classification: 49Q20 Keywords: Γ-convergence, Mumford-Shah functional Introduction In the variational approach to many problems in computer vision (image segmentation, signal processing and so on an important rôle has been played by the Mumford-Shah functional, which is the most famous example of a free discontinuity functional (terminology introduced by DeGiorgi in []. The Mumford-Shah functional is given by MS(u = u 2 dx + ch n (S u where u SBV (, the space of special functions of bounded variation; S u is the approximate discontinuity set of u and H n is the (n -dimensional Hausdorff measure. Several approximation methods are known for the Mumford- Shah functional and, more in general, for free discontinuity functionals: the mbrosio & Tortorelli approximation (see [] and [3] via elliptic functionals, the Gobbino s approximation by finite difference methods (see [2] and many others (see [6], [7], [9], [0].

2 238 L. Lussardi In [5] Braides & Dal Maso approximate the Mumford-Shah functional by means of a sequence of non-local integral functionals given by F (u = ( f u 2 dy dx ( B (x with u H ( and, for instance, f(t =t /2. variant of this method is investigated in [4], [5] and [3] where the problem of the convergence of F (u = f ( u dy dx (2 B (x is considered; here f is a convex-concave function with f (t/ φ(t, as (, t (0, 0, where φ has linear growth at infinity and plays the rôle of the bulk energy density in the limit of F. Then, under the assumption on f in [3], the Mumford-Shah functional cannot be recovered by the Γ-convergence of F, since the bulk term in MS is given by u 2 dx and it has superlinear growth at infinity. question arise: is it possibile to recover the Mumford-Shah functional from (2 instead of (? The aim of this paper is to prove an approximation results for the Mumford-Shah functional, obtained adapting the results contained in [4], [5] and [3], by means of a sequence of functionals of type (2. The core of the proof is Theorem 4. where the lower bound for the Γ-limit is optimized by a sup of measures argument, while the upper bound descends from standard density results and general properties of Minkowsky content. 2 Preliminary Notes Functions of bounded variation. For a thorough treatment of BV functions we refer to [2]. Let be an open subset of R n ; the space BV ( of real functions of bounded variation is the space of the functions u L ( whose distributional derivative is representable by a measure R n -valued measure Du on. We denote by S u the approximate discontinuity set of u and by J u the set of approximate jump points of u. For a function u BV ( let Du = D a u + D s u be the (Lebesgue decomposition of Du into absolutely continuous and singular part. We denote by u the density of D a u; the measures D j u := D s u J u, D c u := D s u ( \ S u are called the jump part and the Cantor part of the derivative, respectively. We say that a function u BV ( is a special function of bounded variation (u SBV ( if D c u ( = 0; moreover we say that a function u L ( is

3 n approximation for the Mumford-Shah functional 239 a generalized special function of bounded variation (u GSBV ( if u T := ( T u T belongs to SBV ( for every T 0. If u GSBV (, the function u given by u = u T for L n -a.e. on { u T } turns out to be well-defined. Moreover, the set function T S u T is monotone increasing; therefore, we set S u = T>0 S u T. Supremum of measures. We recall the following useful result from measure theory, which can be found in [4]. Lemma 2. (supremum of measures Let be an open subset of R n and denote by ( the family of its open subsets. Let λ be a positive Borel measure on, and μ: ( [0, + a set function which is superadditive on open sets with disjoint compact closures (i.e. if, B and B =, then μ( B μ( +μ(b. Let (ψ i i I be a family of positive Borel functions. Suppose that μ( ψ i dλ for every ( and i I; then μ( 3 Main Results sup ψ i dλ i for every (. Let R n be a bounded open set with Lipschitz boundary, and consider the family (F >0 of non-local functionals L ( [0, + ] given by f ( u dy dx if u W, ( B F (u = (x (3 + otherwise, where f :[0, + [0, + is requested to satisfy the following conditions: ( for every >0, f is a non-decreasing continuous function with f (0 = 0; moreover, there exists a > 0 such that a 0as 0 and f is concave in (a, +. (2 f (t lim =. (,t (0,0 t 2 (3 f f uniformly on the compact subsets of (0, +, where f(t =f > 0 is a constant function. possibile choice for f is f (t =(t 2 / f.

4 240 L. Lussardi Remark 3. Let δ (0, ; by(2 there exists t δ > 0 and δ > 0 such that f (t ( + δt 2 / for any 0 t t δ and 0 δ. Since φ(t =t 2 is convex and f is concave in (a, +, with a 0, we get f (t ( + δt 2 / for any t 0 and sufficiently small. Then f (t/ ( + δt 2 for any t 0. The main result is the following convergence result. Theorem 3.2 Let (F >0 be as in (3, with f satisfying conditions (- (2-(3. Then (F Γ-converges, w.r.t. the strong L -topology, as 0, to F : L ( [0, + ] given by u 2 dx +2f H n (S u if u GSBV ( F(u = + otherwise. Moreover we have a compactness property: Theorem 3.3 (compactness Let ( j be a positive infinitesimal sequence and let (u j be a sequence in L ( such that u j M, and F j (u j M for a suitable constant M independent of j; then there exists a subsequence (u jk converging in L ( to a function u SBV (. For the sequel we will need a localization of F : for every open subset of, we set f ( u dy dx if u W, ( B F (u, = (x + otherwise. Clearly, F (, coincides with the functional F defined in (3. The lower and upper Γ-limits of ( F (, will be denoted by F (, and F (,, respectively. 4 Lower bound and compactness Theorem 4. For any u GSBV ( and for any open subset of F (u, u 2 dx +2f H n (S u. Proof. Step. First we show that F (u, u 2 dx +2f H n (S u, u SBV (.

5 n approximation for the Mumford-Shah functional 24 Fix δ (0,, T>0and η>0 small; consider the family (g >0 given by ( t g (t =( δφ T if 0 t< and g (t = { ( δ [ ( ( φ T +(φ T (t ]} (f η if t, with φ T (t =t 2 if 0 t<t and φ T (t =2Tt T 2 if t T. The function g depends on, δ, T and η, but, for simplicity, we drop the dependence by δ, T and η. By(2 there exists t δ > 0 such that, for sufficiently small, f (t ( δφ T (t/ whenever 0 t t δ ; from convexity of φ T and from uniform convergence of f on compact subsets of (0, + we get f g, for sufficiently small. Thus: ( for every >0, g is a non-decreasing continuous function with g (0 = 0; moreover, there exists a > 0(a = such that a 0as 0 and g is concave in (a, +. g (t (2 lim (,t (0,0 ( δφ T (t/ =. Moreover it turns out that, denoting by g(t = 2( δtt (f η, (3 g g uniformly on the compact subsets of [0, +. (4 There exists L>0 such that g (s g (t L s t, s, t > 0. Then, since F (u, g ( B (x u dy dx, u W, (, (4 we get, by Theorem 3. in [3], F (u, ( δ φ T ( u dx +2 for all u BV (, where ϑ(x, t =g ( ωn ω n S u + 2( δt D c u ( 0 ϑ(x, t dt dh n (x u + (x u (x ( t 2 n.

6 242 L. Lussardi By arbitrariness of δ (0, we have F (u, φ T ( u dx +2 ϑ(x, t dt dh n (x (S u [0,] +2T D c u (. (5 s sup φ T (t =t 2 and sup[tt(f η] = f, for t>0, by Lemma 2. we T T,η obtain F (u, u 2 dx +2 f dt dh n (x (S u [0,] = u 2 dx +2f H n (S u. Step 2. Let u GSBV (, and T > 0. By definition, u T SBV ( and u u T. Thus for every sequence u j u in L ( we get F (u, lim inf j + F j (u T j. By Step, asut j ut in L (, we obtain F (u, u T 2 dx +2f H n (S u T. By taking the limit as T + and recalling the definition of u and S u we conclude. Proof of Theorem 3.3. Let ( j be a positive infinitesimal sequence and let (u j be a sequence in L ( such that u j M, and F j (u j M for a suitable constant M independent of j. Then by (4 and by compactness Theorem 3.2 in [3], there exists a subsequence (u jk converging to u BV (. Suppose D c u ( 0; then, by taking the limit as T + in (5, F (u would be +, which contradicts F j (u j M. Thus D c u ( = 0 and then u SBV (. 5 Upper bound In this last section we conclude the proof of Theorem 3.2. s usual, first we will take into account a suitable dense subset of SBV (: let W( be the space of all functions w SBV ( satisfying the following properties: i H n (S w \ S w =0; ii S w is the intersection of with the union of a finite member of (n - dimensional simplexes; iii w W k, ( \ S w for every k N

7 n approximation for the Mumford-Shah functional 243 where SBV 2 ( = {u SBV ( : u L 2 (, H n (S u < + }. In [8] the density property of W( in SBV ( is proved. More precisely: Theorem 5. ssume that is Lipschitz. Let u SBV 2 ( L (. Then there exists a sequence (w j in W( such that w j u strongly in L (, w j u strongly in L 2 (, R n, lim sup h w j u and lim sup φ(w j +,w j,ν w j dh n φ(u +,u,ν u dh n j + S wj S u for every upper semicontinuous function φ such that φ(a, b, ν =φ(b, a, ν whenever a, b R and ν S n. Theorem 5.2 Let u GSBV (; then F (u u 2 dx +2f H n (S u. Proof. Since the upper Γ-limit of F coincides with the upper Γ-limit of the relaxed functional F, we get F (u lim sup 0 F (u. It can be easily seen (see [5], Proposition 3.6 that, for >0 fixed, we have F (u = ( f B (x Du (B (x dx. (6 Step. First we consider the case u W(. Let S = {x :d(x, S u <}; then we can split F as follows: F (u = ( f \S B (x Du (B (x dx + ( B (x Du (B (x dx. S f Since u W, ( \ S, the first integral becomes ( u dy dx f ( \S f Moreover since B (x B (x u dy u(x B (x u dy dx. a.e. x, by (2 and from the dominated convergence Theorem (see Remark 3. we get f ( u dy dx u 2 dx. B (x

8 244 L. Lussardi We estimate now the second integral ( B (x Du (B (x S f Let η>0 small; by uniform convergence of f on compact subsets of (0, + and by monotonicity property of f, for sufficiently small ti holds f (t f + η, for any t 0. Thus S f ( B (x Du (B (x dx. dx S (f + η. Since S u is the union of (n -dimensional simplexes, by standard results on Minkowsky content we have S /2 H n (S u, and then S (f + η 2(f + ηh n (S u. We conclude by arbitrariness of η. Step 2. In the case u SBV 2 ( L ( the thesis descends from Theorem 5. and from lower semicontinuity of F. Finally it is easy to conclude by truncation arguments and again by lower semicontinuity of F. References [] L. mbrosio and V.M. Tortorelli, pproximation of functionals depending on jumps by elliptic functionals via Γ-convergence, Comm. Pure ppl. Math., XLIII (990, [2] L. mbrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, [3] L. mbrosio and V.M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7, VI( (992, [4]. Braides, pproximation of free-discontinuity problems, volume 694 of Lecture Notes in Mathematics, Springer Verlag, Berlin, 998. [5]. Braides and G. Dal Maso, Non-local approximation of the Mumford- Shah functional, Calc. Var., 5( (997, [6]. Braides and. Garroni, On the non-local approximation of freediscontinuity problems, Comm. Partial Differential Equations, 23(5-6 (998,

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