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1 COMMUNICATIONS IN INFORMATION AN SYSTEMS c 2004 Inernaional Press Vol. 4, No. 1, pp , Sepember THE NEW FAMILY OF CRACKE SETS AN THE IMAGE SEGMENTATION PROBLEM REVISITE MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO edicaed o Sanjoy Mier on he occasion of his 70h birhday. Absrac. The objec of his paper is o inroduce he new family of cracked ses which yields a compacness resul in he W 1,p -opology associaed wih he oriened disance funcion and o give an original applicaion o he celebraed image segmenaion problem formulaed by Mumford and Shah [21]. The originaliy of he approach is ha i does no require a penalizaion erm on he lengh of he segmenaion and ha, wihin he se of soluions, here exiss one wih minimum densiy perimeer as defined by Bucur and Zolésio in [3]. This heory can also handle N-dimensional images. The paper is compleed wih several variaions of he problem wih or wihou a penalizaion erm on he lengh of he segmenaion. In paricular, i revisis and recass he earlier exisence heorem of Bucur and Zolésio [3] for ses wih a uniform bound or a penalizaion erm on he densiy perimeer in he W 1,p -framework. 1. Inroducion. The objec of his paper is o inroduce he new family of cracked ses which yields a compacness heorem in he W 1,p -opology associaed wih he oriened disance funcion 1 and o give an original applicaion o he celebraed image segmenaion problem formulaed by Mumford and Shah [21]. The originaliy of our approach is ha i does no require a penalizaion erm on he lengh of he segmenaion and ha, wihin he se of soluions, here exiss one wih minimum densiy perimeer as defined by Bucur and Zolésio in [3]. The paper is compleed wih several variaions of he problem wih or wihou a penalizaion erm on he lengh of he segmenaion. In paricular we revisi and recas he earlier exisence heorem of [3] in he W 1,p -framework. The heory is no limied o 2 problems and can handle N-dimensional images. Cracked ses form a very rich family of ses wih a huge poenial ha is no fully exploied in he image segmenaion problem. Indeed hey can no only be used o pariion he frame of an image, bu also o deec isolaed cracks and poins provided an objecive funcion sharper han he one of Mumford and Shah be used. For insance, in view of he connecion beween image segmenaion and fracure heory [1], he heory may have poenial applicaions in problems relaed o he deecion of fracures or cracks or fracure branching and Acceped for publicaion on November 3, This research has been suppored by Naional Sciences and Engineering Research Council of Canada research gran A 8730 and by a FCAR gran from he Minisère de l Éducaion du Québec. Cenre de recherches mahémaiques e éparemen de mahémaiques e de saisique, Universié de Monréal, C. P. 6128, succ. Cenre-ville, Monréal (Qc), Canada H3C 3J7, delfour@crm.umonreal.ca CNRS and INRIA, INRIA, 2004 roue des Lucioles, BP 93, Sophia Anipolis Cedex, France, Jean-Paul.Zolesio@sophia.inria.fr 1 Also referred o as he algebraic or signed disance funcion. 29

2 30 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO segmenaion in geomaerials [2], bu his is way beyond he scope of his paper. Some iniial consideraions abou he numerical approximaion of cracked ses can be found in [13]. In problems where he shape or he geomery is a design, conrol, or idenificaion variable, merics are used o measure he disance beween objecs, o specify opologies o make sense of coninuiy and compacness, and o obain meaningful opimaliy condiions. From he purely heoreical viewpoin i is now becoming clear ha he meric consruced from he W 1,p norm 2 on he oriened disance funcion (signed or algebraic disance funcion) is playing a cenral and naural role in he analysis of such problems. For insance, convergence and compacness in ha opology imply he same properies in all oher opologies consruced from disance funcions o or characerisic funcions of a se, is complemen, or is boundary. Earlier compacness resuls using he uniform cone propery or he densiy perimeer or more recen ones using he uniform cusp propery also hold in he finer W 1,p -opology (cf. [6, 7, 8] and he recen book [11] for an exensive analysis of merics on subses of he Euclidean space). In addiion, i also plays a key role in oher geomeric idenificaion problems and he characerizaion of he space of soluions of he evoluion equaion of he oriened disance funcion for iniial ses wih hin boundary evolving in a velociy field [12]. In 2 we review he definiions, properies, and he merics associaed wih he oriened disance funcion. In 3 we review he ses wih finie h-densiy perimeer and inroduce he new families of cracked ses. We give he associaed compacness heorems in he srong W 1,p -meric opology on he oriened disance funcion. In 4 we discuss he formulaion ( 4.1) of he N-dimensional image segmenaion problem and give he main exisence heorem for cracked ses wihou penalizaion erm or bound on he perimeer ( 4.2). 4.3 makes use of he h-densiy perimeer which is a relaxaion of he (N 1)-dimensional upper Minkowski conen in wo ways: an example of a segmenaion whose soluion has wo open conneced pars bu an inerface of inie lengh; and a complemen o he exisence heorem of 4.2 by proving ha among all he soluions here is one wih minimum h-densiy perimeer. In 4.4 we go back o he formulaion of Bucur and Zolésio wih respec o he family of ses wih a bounded h-densiy perimeer. We give an exisence heorem in he case of a uniform bound and no penalizaion erm and in he case of a penalizaion erm and no uniform bound on heir perimeer. Finally, as a corollary, we give he exisence for hose wo cases wihin he family of cracked ses. In his paper he words se, image, andobjec will be used equivalenly. Noaion 1.1. Given an ineger N 1, m N and H N 1 will denoe he N- dimensional Lebesgue and (N 1)-dimensional Hausdorff measures. The inner produc 2 This opology was inroduced in [9, 10] and furher invesigaed in [11].

3 THE NEW FAMILY OF CRACKE SETS 31 and he norm in R N will be wrien x y and x. Thecomplemen { x R N : x } and he boundary of a subse of R N will be respecively denoed by or R N \ and by or Γ. The disance funcion d A (x) from a poin x o a subse A of R N is defined as { y x : y A}. 2. Oriened disance funcion and is properies efiniions and properies. Given a subse of R N wih boundary Γ, heoriened disance funcion is defined as (2.1) b (x) def = d (x) d (x). There is a one-o-one correspondence beween b and he equivalence class (2.2) [] b def = { R N :Γ =Γand = } = { R N : b = b }. In general d = d d in and d in = d = d d, bu we only have b b b in. where in and are he inerior and closure of. For convex ses we have b = b ; for ses verifying he uniform segmen propery 3 we have b = b = b in.whenis a closed submanifold of R N of codimension greaer or equal o one, hen =Γand b = d = d Γ. The erminology and he noaion emphasize he fac ha b coincides wih he exerior normal o he boundary (when i exiss). The funcion b offers definie concepual and echnical advanages over he funcion d and makes i possible o simulaneously deal wih open N-dimensional subses and embedded submanifolds of R N in he same framework. In he lieraure, i usually appears as he disance o he boundary d Γ wih a change of sign across he boundary and is referred o as he algebraic or signed disance funcion. efiniion (2.1) and he associaed equivalence classes seem o have been firs inroduced in 1994 in [9]. The funcion b capures many of he geomeric properies of he se. For insance, in 1994 i was showed in [9, 11] ha he propery ha is convex if and only if d is convex remains rue wih b in place of d. The funcion b is Lipschiz coninuous of consan 1, and b exiss and b 1almoseverywhereinR N. Thus b W 1,p loc (RN ) for all p, 1 p. The poins 3 issaidosaisfyheuniform segmen propery if r >0, λ >0 such ha x Γ, d R N, d =1, for which for all y B r(x), (y, y + λd) in.

4 32 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO of R N where he gradien of b does no exis can be divided ino wo caegories: he ones on he boundary Γ and he ones ouside of Γ. efiniion 2.1. The se of projecions of a poin x R N ono he boundary Γ of a se, Γ, Π Γ (x) def = { p R N : b (x) = p x } since b (x) = d Γ (x); heskeleon of (2.3) Sk() def = { x R N :Π Γ (x) is no a singleon } (by definiion Sk() R N \Γ); he se of cracks of C() def = { x R N : b 2 (x) exiss bu b (x) does no exis }. The erminology crack is used here in a very broad sense. C() can conain subses of arbirary co-dimension. In dimension N = 2hecorners along a piecewise smooh boundary belong o C(). We recall basic properies. Theorem 2.1. Le be a subse of R N wih Γ. (i) For all x Γ, b 2 (x) exiss and b2 (x) =0; for all x/ Γ b 2 (x) exiss b (x) exiss. Hence b (x) exiss if and only if x/ Sk() C(). Moreover Sk() = { x R N : b 2 (x) does no exis } and Sk() R N \Γ and C() Γ have zero m N -measure. (ii) The projecion p Γ (x) of a poin x/ Sk() ono he boundary Γ of is given in erms of b (2.4) p Γ (x) =x 1 2 b2 (x) =x b (x) b (x). (iii) The Hadamard semi-derivaive 4 of b 2 always exiss (2.5) v R N, d H b 2 (x; v) =2 min (x p) v. p Π Γ(x) (iv) For all poins x/ Γ, he Hadamard semiderivaive of b exiss and (2.6) v R N, d H b (x; v) = 1 min (x p) v. b (x) p Π Γ(x) For all poins x Γ, d H b (x; v) exiss if and only if v R N b (x + v) (2.7), lim exiss. Proof. (i) and (ii) Cf. [11], Chaper 5, Theorem 4.4 and Chaper 8 5, 2, 3, and p. 369). (iii) Cf. [11], Theorem 3.1 (iii), p (iv) Obvious. 4 A funcion f : R N R has a Hadamard semi-derivaive in x in he direcion v if d H f(x; v) def f(x + w) f(x) = lim exiss w v (cf [11], Chaper 8, efiniion 2.1 (ii)).

5 THE NEW FAMILY OF CRACKE SETS Srong, weak, and uniform meric opologies efiniions and ses wih hin boundary. efiniion 2.2. (i) The boundary Γ of a subse of R N is said o be hin 5 if m N (Γ) = 0; oherwise i is said o be hick. (ii) Given a nonempy subse of R N, define he families (2.8) C b () def = { b : and Γ } (2.9) Cb 0 def () = {b C b () :m N (Γ) = 0}. The space Cb 0() corresponds o he subfamily of subses of RN wih a hin boundary ha is a more naural family han he family C b () in applicaions Srong meric opologies. In his paper we specialize o he following complee merics 6 associaed wih b over he subses of a bounded open hold-all (2.10) (2.11) ρ C() ([ ], []) def = b b C() =max b (x) b (x) x ρ W 1,p ()([ ], []) def = b b W 1,p () { 1/p = b b p + b b dx} p. The space C b () is a complee meric space for he merics (2.10) and (2.11), bu he space Cb 0() is complee only wih respec o he meric (2.11) (e.g. [11], Chaper 5)7. The meric (2.10) is he analogue wih b of he Hausdorff meric defined from d.for he purpose of he paper we shall use he convenien erminology Hausdorff meric, bu i should be remembered ha his meric wih b is differen from he classical Hausdorff meric wih d.thew 1,p -opologies are all equivalen for 1 p< (cf. [11] Theorem 5.1, Chaper 5, p. 226) Weak W 1,p and Hausdorff meric opologies. C b () isalsocomplee for he weak W 1,p -opologies which are also all equivalen for 1 p<. The weak W 1,p -convergence of sequences of oriened disance funcions is equivalen o he srong convergence in he C()-opology of uniform convergence (cf. [11] Theorem 5.2 (i)-(ii), Chaper 5, p. 228). For ses wih hin boundaries he srong and weak W 1,p ()-convergences of elemens of Cb 0() oanelemenofc0 b () areequivalen. Lemma 2.1. Given a bounded open subse of R N,le{ n } be a sequence of subses of such ha Γ n and m(γ n )=0. Furher assume ha here exiss 5 This erminology is no o be confused wih he one of hin se in Capaciy Theory. 6 Oher complee merics can be defined wih d, d, d Γ in place of b. 7 The compleeness of he meric (2.10) is no a rivial consequence of he classical proof in [14] of he compleeness of he Hausdorff meric associaed wih d. To our bes knowledge he merics (2.10) and (2.11) were firs inroduced by [9] in 1994.

6 34 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO such ha Γ and m(γ) = 0. Then b n b in W 1,2 ()-weak b n b in W 1,2 ()-srong, and hence in W 1,p ()-srong for all p, 1 p<. Proof. Same proof as in par (ii) of he proof of Theorem 10.1 in [11]. Since, for all n 1, m(γ n )=0=m(Γ), b =1= b n almos everywhere in (cf. [11], Theorem 3.2, p. 215). As a resul b n b 2 dx = b n 2 + b 2 2 b n b dx =2 (1 b n b ) dx 2 (1 b 2 ) dx =2 χ Γ dx =0. Therefore b n b in L 2 () N -srong and b n b in W 1,2 ()-srong, since he convergence b n b in L 2 ()-srong follows from he weak convergence in W 1,2 (). The convergence in W 1,p ()-srong follows from he equivalence of he opologies on C b () (cf. [11]. Chaper 5, Theorem 5.1 (i)) Oher meric opologies. The following heorem is cenral. I shows ha convergence and compacness in he meric ρ W 1,p () will imply he same properies in all oher opologies (cf. [11], Theorem 5.1, Chaper 5, p. 226). Recall ha b + = d, b = d, and b = d Γ,andhaχ in = d, χ in = d,and χ Γ =1 d Γ a.e. in R N. Theorem 2.2. Le be a bounded open subse of R N.Themap (2.12) b (b +,b, b ) =(d,d,d ):C b () W 1.p () W 1.p () 3 and for all p, 1 p<, hemap b (χ,χ in,χ in ): W 1,p () L p () 3 are coninuous. 3. Some families of ses. In his secion we review families of ses and heir properies ha will be used in he paper: he ses wih a finie densiy perimeer and he new cracked ses. We give he main associaed compacness heorems o deal wih he exisence of minimizing soluions in 4. Noaion 3.1. Given h>0 he open and closed ubular neighborhoods of a se A are defined as (3.1) U h (A) def = { x R N : d A (x) <h } A h def = { x R N : d A (x) h }. Recalling ha d Γ (x) = b (x) we also have U h (Γ) = { x R N : b (x) <h }.

7 THE NEW FAMILY OF CRACKE SETS Ses wih finie densiy perimeer. This family of ses inroduced in 1996 by Bucur and Zolésio [3] is based on a relaxaion of he (N 1)-dimensional upper Minkowski conen which leads o he compacness Theorem 3.1. We recall he definiion and give he proof of he compacness for he W 1,p -opology under a uniform bound on he h-densiy perimeer. efiniion 3.1. Le h>0 be a fixed real and asubseofr N wih nonempy boundary Γ. Consider he quoien (3.2) P h (Γ) def = sup 0<k<h m N (U k (Γ)). 2k Whenever P h (Γ) is finie, we say ha has a finie h-densiy perimeer. I was shown in [3] ha, whenever P h (Γ) is finie, for all 0 < k < h, Γ U k (Γ) and m(γ) m(u k (Γ)) kc. By leing k go o zero we ge m(γ) = 0. The compacness resul of [3] can now be sharpened and recas in he W 1,p -opology from which convergence in all oher opologies of Theorem 2.2 follows. We also recover he lower semiconinuiy of he h-densiy perimeer. Theorem 3.1. Le be a bounded open subse of R N and { n }, Γ n, be a sequence of subses of. Assume ha (3.3) h >0 and c>0 such ha n, P h (Γ n ) c. Then here exis a subsequence { nk } andasubse, Γ, of such ha (3.4) (3.5) P h (Γ) lim P h(γ n ) c n p, 1 p <, b nk b in W 1, p (U h ())-srong. Proof. The proof essenially ress on Lemmas 2.1 and he fac ha P h (Γ) c implies m N (Γ) = 0. Since is bounded, he family of oriened disance funcions C b () iscompacinc() andw 1,p ()-weak for all p, 1 p< (cf. [11], Theorem 2.2 (ii), p. 210, and Theorem 5.2 (iii), p. 228). So here exis b C b () and a subsequence, sill indexed by n, such ha b n b in he above opologies. Moreover, for all k, 0<k<h,andallε, 0<ε<h k, N(ε) > 0 such ha n N(ε), U k ε (Γ n ) U k (Γ) U k+ε (Γ n ) (cf. proof of par (i) of Theorem 9.2 in [11], p. 251). As a resul for all n N(ε), (3.6) m N (U k ε (Γ n )) k ε m N (U k (Γ)) 2(k ε) k 2k m N (U k (Γ)) 2k m N (U k+ε (Γ n )) 2(k + ε) m N (U k (Γ)) 2k m N (U k+ε (Γ n )) 2(k + ε) k + ε k lim n P h(γ n ) k + ε k. k + ε k P h (Γ n ) k + ε k c k + ε k

8 36 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO Going o he limi as ε goes o zero in he second and fourh erms k, 0 <k<h, m N (U k (Γ)) 2k lim n P h(γ n ) c P h (Γ) lim n P h(γ n ) c m N (Γ) = 0. The heorem now follows from he fac ha m N (Γ) = 0 and Lemma 2.1. Corollary 3.1. Le be a bounded open subse of R N and, Γ, be asubseof such ha (3.7) h >0 and c>0 such ha P h (Γ) c. Then he mapping b P h (Γ ):C b () R {+ } is lower semiconinuous in for he W 1,p ()-opology. Proof. Since we have a meric opology, i is sufficien o prove he propery for W 1,p ()-converging sequences {b n } o b. From ha poin on he argumen is he same as he one used o ge (3.6) in he proof of Theorem 3.1 afer he exracion of he subsequence. Remark 3.1. I is imporan o noice ha even if { n } is a W 1,p -convergen sequence of bounded open subses of R N wih a uniformly bounded perimeer, he limi se need no be an open se or have a nonempy inerior in such ha b = b in. I would be emping o say ha b n b implies d n d and use he open se in = for which d = d in o conclude ha b = b in. This is incorrec as can be seen on he following example shown in Figure 1. Consider a family { n } of 1 n n 1 1 Fig. 1. W 1,p -convergence of a sequence of open subses { n : n 1} of R 2 wih uniformly bounded densiy perimeer o a se wih empy inerior open recangles in R 2 of widh equal o 1 and heigh 1/n, n 1, an ineger. Their densiy perimeer is bounded by 4 and he b n s converge o b for equal o he line of lengh 1 which has no inerior The new families of cracked ses. In his secion we inroduce new families of hin ses which are well-suied for image segmenaion. They are more general han ses which are locally he epigraph of a coninuous funcion in he sense ha hey include domains wih cracks, and ses ha can be made up of componens of differen co-dimensions. The Hausdorff (N 1) measure of heir boundary is no

9 THE NEW FAMILY OF CRACKE SETS 37 necessarily finie. Ye compac families (in he W 1,p -opology) of such ses can be consruced. Firs recall he following definiions of he lim and limsup for he following differenial quoien of a funcion f : V (x) R N R defined in a neighborhood V (x) ofapoinx R N in he direcion d R N lim lim sup lim lim sup f(x + d) f(x) f(x + d) f(x) f(x + w) f(x) f(x + w) f(x) def = lim δ 0 0<<δ def = lim def = lim δ 0 def = lim δ 0 sup δ 0 0<<δ 0<<δ w d R N<δ (,w) (0,v) sup 0<<δ w d R N <δ (,w) (0,v) f(x + d) f(x) f(x + d) f(x) f(x + w) f(x) f(x + w) f(x). They are lower and upper semiderivaives 8 of he ini ype. However we shall no inroduce a new noaion since he lim and limsup are more explici. efiniion 3.2. (i) A se in R N, Γ, issaidobeweakly cracked if (3.8) x Γ, d R N, d =1, such ha lim sup (ii) A se in R N, Γ, issaidobecracked if (3.9) x Γ, d R N, d =1, such ha lim (iii) A se in R N, Γ, issaidobesrongly cracked 9 if (3.10) x Γ, d R N, d =1, such ha lim 0 d Γ (x + w) d Γ (x + d) d Γ (x + d) > 0. > 0. > 0. Srongly cracked implies cracked, and cracked implies weakly cracked since lim d Γ (x + w) lim lim sup d Γ (x + d) d Γ (x + d) lim sup d Γ (x + w). 8 Noe ha he (, d), 0 <<δ, is allowed in he and he sup of he las wo definiions and he consrain (, w) (0,v)canberemoved. 9 Here he definiion is f(x + d) f(x) def f(x + d) f(x) lim = lim. 0 δ 0 0< <δ

10 38 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO The special erminology of efiniion 3.2 is moivaed by he fac ha he boundary of such a se has zero N-dimensional Lebesgue measure (cf. Lemma 3.1) and can be made up of cusps, poins, cracks or hairs as shown in Figure 2. This erminology is Fig. 2. Example of a wo-dimensional srongly cracked se inroduced here o provide an inuiive descripion of he ses. The weakly cracked propery is verified in any poin of he boundary where he gradien of d Γ does no exis; in boundary poins where he gradien exiss i is no idenically 0. This is a very large family of ses ha includes domains which are locally he epigraph of a coninuous funcion. There are obvious variaions of he above definiions and he forhcoming compacness Theorem 3.2 by replacing d Γ by d, d or b. Lemma 3.1. Le, Γ, be a subse of R N, x Γ, andd, d =1,bea direcion in R N. If he semiderivaive dd Γ (x; d) does no exis hen lim sup d Γ (x + w) > 0. Proof. Since he funcion d Γ is Lipschizian, he limi of he quoien dd Γ (x; d) def d Γ (x + d) d Γ (x) = lim exiss if and only if he limi of he quoien d Γ (x + w) d Γ (x) lim exiss (cf [11], Chaper 8, Theorem 2.1(i)). Moreover, by he Lipschiz coninuiy, d Γ (x + w) d Γ (x) w = w d and hence he lim and he limsup of he quoien exis and are finie. For x Γ d Γ (x) = 0. Therefore dd Γ (x; d) does no exiss if and only if lim sup d Γ (x + w) > lim d Γ (x + w) 0

11 THE NEW FAMILY OF CRACKE SETS 39 since he las erm is nonnegaive. This complees he proof. Theorem 3.2. (i) A weakly cracked se is hin, ha is m(γ) = 0, and Γ in in. Moreover in any poin x Γ eiher b (x) exiss and is differen from zero or b (x) does no exiss (se of cracks). (ii) Given a cracked se, foreachx Γ here exiss a direcion d R N, d =1, such ha 1 l(x) def d Γ (x + d) = lim > 0 and for all ε, 0 <ε<l(x), hereexissδ>0 such ha, 0 <<δ, d Γ (x + d) (l(x) ε) x + C(δ cos ω, ω, d) R N \Γ, sin ω = l(x) ε, 0 <ω π/2 (C(λ, ω, d) isheopenconein0 of direcion d, heigh λ and aperure ω). (iii) Given a srongly cracked se, foreachx Γ here exiss a direcion d R N, d =1, such ha 1 l(x) def d Γ (x + d) = lim > 0 0 and for all ε, 0 <ε<l(x), hereexissδ>0 such ha, 0 < <δ, d Γ (x + d) (l(x) ε) x ± C(δ cos ω, ω, d) R N \Γ, sin ω = l(x) ε, 0 <ω π/2. Proof. (i) We already know ha d Γ exiss almos everywhere in R N and ha, whenever i exiss, 0, if x Γ d Γ (x) = 1, if x/ Γ (cf. [11], Chaper 4, Theorem 3.2 (i)). Therefore if d Γ (x) exissinapoinx Γ, d Γ (x) =0andforalld, d =1, lim sup d Γ (x + w) = lim sup d Γ (x + w) d Γ (x) = d Γ (x) d =0 which conradics he weakly cracked propery. Hence he poins of Γ are poins where d Γ (x) does no exis which is iself a se of zero measure. Moreover, if b (x) exiss in a poin x, hen b (x) =0andforalld, d =1, b (x + w) lim b (x + w) b (x) = lim = b (x) d.

12 40 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO If b (x) =0,henforalld b (x + w) lim lim b (x + w) b (x) = lim d Γ (x + w) = lim b (x + w) = b (x) d =0 =0 and d (x) = 0 which conradics our assumpion. Therefore if b (x) exissina poin x, b (x) 0. For any x Γ inroduce he noaion l(x) def = lim sup By assumpion l(x) > 0andforallδ>0 l(x) sup 0<<δ w d <δ d Γ (x + w). d Γ (x + w). Hence here exis sequences { n }, n 0, and {w n }, w n d, such ha 0 < l(x)/2 d Γ(x + n w n ) R N \Γ x n def = x + n w n x and necessarily Γ in in in in. (ii) Given a cracked se, for each x Γ here exiss a direcion d R N, d =1, such ha l(x) def d Γ (x + d) = lim > 0 and for all ε, 0<ε<l(x), here exiss δ>0 such ha, 0 <<δ, d Γ (x + d) (l(x) ε). Recall ha since d Γ is Lipschizian of consan one, we necessarily have 1 l(x) for d = 1. Therefore x + C(δ cos ω, ω, d) R N \Γ, sin ω = l(x) ε, 0 <ω π/2. (iii) Similar o he proof of par (ii). Theorem 3.3. Le be a bounded open subse of R N and α>0 and h>0 be

13 real numbers 10. Consider he families THE NEW FAMILY OF CRACKE SETS 41 (3.11) (3.12) F(, h, α) def = C h,α b () def F s (, h, α) def = : Γ and x Γ, d, d =1, such ha = {b : F(, h, α)}. : d Γ (x + d) 0<<h α Γ and x Γ, d, d =1, such ha (C h,α b ) s () def = {b : F s (, h, α)}. d Γ (x + d) 0< <h α Then C h,α b () and (C h,α b ) s () are compac in W 1,p (), 1 p<. Proof. (i) The family C h,α b () is conained in C b () which is compac in he uniform opology of C() and in he weak opology of W 1,p (), 1 p< (cf. [11], Chaper 4, Theorem 2.2 (ii), p. 210 and Theorem 5.2 (iii). p. 228). Therefore, given a sequence {b n } in C h,α b (), here exis a subsequence, sill denoed {b n }, and b C b () such ha b n b in W 1,p ()-weak and C(). In addiion, by definiion of he elemens of C h,α b (), condiion (3.9) is verified and each n has hin boundary. We wan o show ha b C h,α b (). Once his is proven, from Theorem 3.2 (i), has a hin boundary. Hence he weak W 1,p () convergence implies he srong W 1,p () convergence by Lemma 2.1. In view of he coninuiy of he map b d Γ = b : C() C(), d Γn d Γ in C(). From Lemma 10.1 in Chaper 5 of [11], given x Γ, here exiss a subsequence of {b n }, sill denoed {b n },andfor each n 1poinsx n Γ n B(x, 1/n). Hence x n x. By assumpion n 1, d n R N, d n =1, such ha d Γn (x n + d n ) α. 0<<h Since he d n s have norm one, here exis a subsequence, sill denoed {d n },andd, d = 1, such ha d n d. Fix, 0<<δ.Givenε>0, here exiss N such ha for all n N x n x <ε, d n d <δ n <ε, d Γn d Γ C() <ε. 10 In view of he fac ha he disance funcion d Γ is Lipschizian wih consan 1, we necessarily have 0 <α 1.

14 42 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO Fix n = N and consider he following esimaes d Γ (x + d) d Γ n (x n + d n ) d Γ n (x + d n ) d Γn (x n + d) d Γ(x + d) d Γ (x n + d) α ε d d n x x n ε>0, 0<<h d Γ (x + d) d Γ(x n + d) d Γn (x n + d) d Γ d Γn C) α 4ε 0<<h α 4ε d Γ (x + d) α. Therefore b C h,α b () and his complees he proof of he compacness. (ii) The proof for (C h,α b ) s () is idenical wih obvious changes. 4. The segmenaion of N-dimensional images Problem formulaion. Typically he image segmenaion funcional of Mumford and Shah [21] aims a idenifying wo dimensional objecs in a wo dimensional frame as shown in Figure 3. frame grey level image I open se Fig. 3. Image I of objecs and is segmenaion in an open 2- frame. In his secion we specialize o he segmenaion of N-dimensional images where he segmenaion could poenially be composed of objecs of codimension greaer or equal o one. To represen he se of Figure 2 as he boundary of an open se one can use he unbounded plane R 2 minus all he curves and he poins in Figure 2. If i is imporan ha he open se be bounded, a fixed open frame is inroduced. The open se in Figure 4 is hen defined as he inerior of he bounded open frame minus all he curves and poins used o draw he picure. efiniion 4.1. Le be a bounded open subse of R N wih Lipschizian boundary. (i) An image in he frame is specified by a funcion f L 2 (). (ii) We say ha { i } i I is an open pariion of if { i } i I is a family of

15 THE NEW FAMILY OF CRACKE SETS 43 open frame Fig. 4. The 2- srongly cracked se of Figure 2 in an open frame disjoin conneced open subses of such ha m N ( i I i )=m N () and m N ( i I i )=0. enoe by P() he family of all such open pariions of. Givenanopenpariion{ i } i I of, associae wih each i I, a funcion ϕ i H 1 ( i ). In is inuiive form he problem formulaed by Mumford and Shah [21] aims a finding an open pariion P = { i } i I in P() soluion of he following minimizaion problem (4.1) ε ϕ i 2 + ϕ i f 2 dx P P() ϕ i H 1 ( i) i i I for some fixed consan ε>0. Observe ha wihou he condiion m N ( i I i )= m N () he empy se would be a soluion of he problem or a phenomenon of he ype discussed in Remark 3.1 and he phenomenon of Figure 1 could occur. The quesion of exisence requires a more specific family of open pariions or a penalizaion erm which preserves he lengh of he inerfaces in some appropriae sense: (4.2) ε ϕ i 2 + ϕ i f 2 dx + ch N 1 ( i I i ) P P() ϕ i H 1 ( i) i i I for some c>0. The choice of a relaxaion of he (N 1)-Hausdorff measure H N 1 is criical. Here he finie perimeer of Caccioppoli reduces o he perimeer of since he characerisic funcion of i I i is almos everywhere equal o he characerisic funcion of. In ha conex, he relaxaion of he (N 1)-dimensional upper Minkowski conen by Bucur and Zolésio [3] is much more ineresing in view of he associaed compacness Theorem 3.1. Anoher way of looking a he problem would be o minimize he number I of open subses of he open pariion, bu his seems more difficul o formalize. As a final remark, i is clear ha he image of Figure 4 is no an L 2 ()-funcion and ha is idenificaion would require a sharper deecion funcional han he one of (4.2).

16 44 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO 4.2. Cracked ses wihou he perimeer. In his secion we specialize he compac family of Theorem 3.3 o ge he exisence of a soluion o he following minimizaion problem (4.3) F(,h,α) open, m N ()=m N () ε ϕ 2 + ϕ f 2 dx. ϕ H 1 () This allows for open ses wih H N 1 (Γ) =. The pair (h, α) are he conrol parameers of he segmenaion. Recall he characerizaion of Theorem 3.2 (ii) which says ha in each poin of he boundary here exiss a small open cone of uniform heigh and aperure which does no inersec he boundary. Theorem 4.1. Given a bounded open frame R N wih a Lipschizian boundary and real numbers h, α, andε>0, hereexisanopensubse of in F(, h, α) such ha m N ( )=m N () and y H 1 ( ) soluions of problem (4.3) Technical lemmas. The proof of he exisence heorems will require he following echnical resuls. Lemma 4.1. Given a subse A of R N wih nonempy boundary A, an open subse R N such ha b A = b if and only if d A = d in A or equivalenly Ā = in A. Proof. By definiion for open b A = b d A = d and d = d A which is also equivalen o Ā = and = A Ā = and=ina. Hence he necessary and sufficien condiion finally reduces o in A = Ā. Lemma 4.2. Le R N be bounded open wih Lipschizian boundary. Then, Γ, m N (Γ) = 0, and m N () = m N () in =. Proof. By conradicion. If in, here exiss x such ha d in (x) = ρ>0 and hence B(x, ρ). Therefore B(x, ρ) Γandm N ( B(x, ρ)) m N (Γ) = 0. By assumpion m N () =m N () = m N () implies m N ( B(x, ρ)) = m N ( B(x, ρ)) = 0. Since, hereexissx such ha m N ( B(x, ρ)) = 0. Bu his is a conradicion since is an open se wih Lipschizian boundary Anoher compacness heorem. The compacness of he following special family of cracked ses conained in a frame is a corollary o he compacness Theorem 3.3.

17 THE NEW FAMILY OF CRACKE SETS 45 Theorem 4.2. Le be a bounded open subse of R N wih Lipschizian boundary and h>0 and α>0 be real numbers. Consider he family open and m N () = m N () F c (, h, α) def = : and x Γ, d, d =1, (4.4) d Γ (x + d) such ha α 0<<h C c,h,α b () def = {b : F c (, h, α)}. Then C c,h,α b () is compac in W 1,p (), 1 p<. Proof. By sandard argumens and Lemma 4.2. The conclusion follows from Theorem 3.3 by adding he consrain m N ( n )=m N () which will be verified for he limi se for which a subsequence of {b n } converges o b in W 1,1 () and hence {χ n } converges o χ in L 1 () Proof of Theorem 4.1. Proof. [Proof of Theorem 4.1] (i) For each open F c (, h, α), he problem F (,ϕ), def F(,ϕ) = ε ϕ 2 + ϕ f 2 dx ϕ H 1 () has a unique soluion y in H 1 () since he objecive funcion F (,ϕ) is coninuous and coercive on H 1 (). efine m def = F c (,h,α) ϕ H 1 () ε ϕ 2 + ϕ f 2 dx. (ii) The minimum is finie since he objecive funcion is posiive and open F c (, h, α), ε ϕ 2 + ϕ f 2 dx f 2 dx ϕ H 1 () by choosing ϕ =0. Le{ n } be a minimizing sequence of open subses of in F c (, h, α) andforeachn le y n H 1 ( n ) be he minimizing elemen of F ( n,ϕ) over H 1 ( n ). Therefore, ε y n 2 + y n f 2 dx m n c>0 such ha n, ε y n 2 + y n f 2 dx c. n By coerciviy he sequence {y n } is uniformly bounded in H 1 ( n ), ha is here exiss a consan c>0 such ha n, y n L 2 ( n) c and y n L 2 ( n) c. By Theorem 4.2 here exiss a subsequence of { n } andanopense F c (, h, α) such ha b n b in H 1 () andc(). In paricular d n d in C(). By he

18 46 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO compacivorous propery (cf. [11], Theorem 2.4 (iii), p. 162), K compac in, N such ha n N, K n. Moreover, (4.5) (4.6) χ n χ L 2 () χ n χ L ()-weak fχ n fχ L 2 ()-weak. efine he disribuions < ỹ n,ϕ> def = y n ϕdx, ϕ () n < y n, Φ > def = y n Φ dx, Φ () N n < ỹ n, Φ >= ỹ n div Φ dx, Φ () N. I is readily seen ha we can idenify ỹ n and y n wih he exensions of y n and y n by zero from n o. As a resul here exis subsequences, sill denoed {ỹ n } and { y n },andỹ L 2 () andy L 2 () N such ha ỹ n ỹ in L 2 ()-weak and y n Y in L 2 () N -weak. By he compacivorous propery, for all Φ (), here exiss N such ha n >N, supp Φ n Φ ( n ). Therefore, for all n>n, < ỹ n y n, Φ >= y n div Φ dx n y n Φ dx =0 n since y n H 1 ( n ). Bu ( n ) () and by leing n go o iniy, 0= y n div Φ dx y n Φ dx ỹ div Φ dx Y Φ dx n n Φ (), ỹ div Φ dx + Y Φ dx =0. efine he new disribuion <y,ϕ> def = ỹϕdx, ϕ (). I is easy o check ha y L 2 () and hence Φ (), 0 = ỹ div Φ dx + Y Φ dx = ỹ div Φ dx + Y Φ dx Φ (), < y, Φ >= ỹ div Φ dx = Y Φ dx y = Y L 2 () y H 1 ().

19 THE NEW FAMILY OF CRACKE SETS 47 (iv) Coming back o our objecive funcion ε ϕ 2 + ϕ f 2 dx = ε y n 2 + y n f 2 dx ϕ H 1 ( n) n n = ε y n 2 + ỹ n fχ n 2 dx. By convexiy and coninuiy of he objecive funcion wih respec o he pair (ỹ n fχ n, y n )inl 2 () L 2 () N and he fac ha, from (4.6), (ỹ n fχ n, y n ) (ỹ fχ,y)inl 2 () L 2 () N -weak, ε Y 2 + ỹ fχ 2 dx lim ε y n 2 + ỹ n fχ n 2 dx n = lim ε y n 2 + y n f 2 dx = m n n ε y 2 + y f 2 dx = ε y 2 + y fχ 2 dx m. By definiion of he minimum we have he equaliy and here exis an open se F c (, h, α) andy H 1 () soluion of he segmenaion problem Exisence of a cracked se wih minimum densiy perimeer. Theorem 4.1 gives an exisence resul for he family F(, h, α) ofopensessuchha m N () = m N () wihou consrain on he perimeer of. enoe by F (, h, α) he se of soluions o problem (4.3). In general, he perimeer can be inie as can be seen from he following example. Example 4.1. The funcion f : R is defined as follows 1, if x 1 (4.7) f(x) = 0, if x 2 where = {(x, y) : 2 <x<3, 1 <y<3}, = 1 2, 2 = \ 1,andhe open se 1 is consruced below (see Figure 5). The se wih y = f is a soluion of problem (4.3) wih inie perimeer. The se 1 is a wo-dimensional example consruced by Nicolas oyon 11 of an open domain saisfying he uniform cusp condiion of [11] for he funcion h(θ) =θ α, 0 < α < 1. I can easily be generalized o an N-dimensional example. Consider he open domain 1 in R 2 def 1 = {(x, y) : 1 <x 0 and 0 <y<2} {(x, y) :0<x<1 and f(x) <y<2} {(x, y) :1 x<2 and 0 <y<2}, 11 éparemen de Mahémaiques e de saisique, Universié demonréal.

20 48 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO where f :[0, 1] R is defined as follows f(x) def = f k (x), f k : k=0 [ k, 1 1 ] 2 k+1 R. Associae wih 0 <α<1 and k 0 he even ineger η k =2[(2 k+1 ) α 1 α ],where[β] is 2 f =0 frame 1 f =1 Fig. 5. The wo open componens 1 and 2 of he open domain for N =2. he smalles ineger greaer or equal o β. Assume ha for each k 0 η k /2 def f k = g k,j :[x k,j 1,x k,j 1 + δ k ] R j=1 g k,j def x k,j =1 1 2 k +(j 1)2δ def 1 k, 1 j η k /2, δ k = η k 2 k+1 and ha he funcion g k,j is given by he expression 0, 0 x<x k,j 1 g k,j (x) def (x x k,j 1 ) α, x k,j 1 x x k,j 1 + δ k = (2x k,j 1 x) α, x k,j 1 + δ k x x k,j 1 +2δ k 0, x > x k,j 1 +2δ k. Noe ha g k,j (x k,j 1 + δ k )=(δ k ) α is independen of j and is he maximum of he funcion g k,j. The uniform cusp propery is verified for ρ =1/8, λ = h(ρ), andh(θ) =θ α.the boundary of 1 is made up of sraigh lines of oal lengh 9 plus he lengh of he curve C def = {(x, f(x)) : 0 x<1}, C = k=0 C k, C k = η k/2 j=1 C k,j C k,j def = {(x, f(x)) : x k,j 1 x<x k,j 1 + δ k }.

21 THE NEW FAMILY OF CRACKE SETS 49 The lengh of he curve C k,j is bounded below by H N 1 (C k,j ) 2 (δ k ) 2 +(δ k ) 2α 2(δ k ) α H N 1 (C k ) η k /2 H N 1 (C k ) η ( ) α k 1 2 2(δ k) α = η k (δ k ) α 1 α = η k η k 2 k+1 = η k j=1 H N 1 (C k,j ) ( 1 2 k+1 ) α ( ) α 1 α 1 H N 1 (C k ) η k 2 k+1 = ( ( ) 2[(2 k+1 ) α ) α 1 α 1 α 1 ] 2 k+1 ( ) α α (2 k+1 ) α 2 k+1 =2 1 α H N 1 (C) = H N 1 (C k ) α =+. k=0 When a leas one soluion has a bounded perimeer, i is possible o show ha here is one ha minimizes he h-densiy perimeer. Theorem 4.3. Assume ha he assumpions of Theorem 4.1 are verified. There exiss an in F (, h, α) which minimizes he h-densiy perimeer. Proof. If for all in F (, h, α) heh-densiy perimeer is + he heorem is rue. If for some F (, h, α), P h (Γ) c, hen here exiss a sequence { n } in F (, h, α) such ha P h (Γ n ) P h(γ). F (,h,α) By he compacness Theorems 3.1 and 4.2, here exis a subsequence and,γ, such ha b n b in W 1,p (), F(, h, α), m N () = m N (), and P h (Γ ) lim n P h(γ n ) c. Finally by going back o he proof of Theorem 4.1 and using he fac ha all he n s are already minimizers in F (, h, α), i can be shown ha is indeed one of he minimizers in he se F (, h, α) Uniform bound or penalizaion erm in he objecive funcion on he densiy perimeer. To complee he resuls on he segmenaion problem, we urn o he exisence of a segmenaion for a family of ses wih a uniform bound or wih a penalizaion erm in he objecive funcion on he h-densiy perimeer. Theorem 4.4. Given a bounded open frame R N wih a Lipschizian boundary and real numbers h>0 and c>0 12, here exiss an open subse of, Γ, 12 Noe ha he consan c mus be large enough o ake ino accoun he conribuion of he boundary of.

22 50 MICHEL C. ELFOUR AN JEAN-PAUL ZOLÉSIO wih finie h-densiy parameer such ha m N ( )=m N () (P h (Γ ) c for (4.8)), and y H 1 ( ) soluions of he respecive problems (4.8) (4.9) open, P h (Γ) c m N ()=m N () open m N ()=m N () ε ϕ 2 + ϕ f 2 dx ϕ H 1 () ε ϕ 2 + ϕ f 2 dx + cp h (Γ). ϕ H 1 () Proof. The proof for he objecive funcion (4.8) is exacly he same as he one of Theorem 4.1. I uses Lemma 4.2 o show ha he minimizing se has a hin boundary and he compacness Theorem 3.1. The proof for he objecive funcion (4.9) uses he fac ha here is a minimizing sequence for which he h-densiy perimeer is uniformly bounded and he lower semiconinuiy of he densiy perimeer in he W 1,p -opology given by Corollary 3.1. Problem (4.9) was he one originally considered in [3]. The above wo idenificaion problems can be furher specialized o he family of cracked ses F(, h, α). Corollary 4.1. Given a bounded open frame R N wih a Lipschizian boundary and real numbers h>0, α>0 and c>0 13, here exiss an open subse of in F(, h, α) such ha m N ( )=m N () (P h (Γ ) c for (4.10)), and y H 1 ( ) soluions of he problem (4.10) (4.11) open, F(,h,α) P h (Γ) c, m N ()=m N () open, F(,h,α) m N ()=m N () ε ϕ 2 + ϕ f 2 dx ϕ H 1 () ε ϕ 2 + ϕ f 2 dx + cp h (Γ). ϕ H 1 () Proof. Since he minimizing sequence {b n } consruced in he proof of Theorem 4.1 srongly converges o b in W 1,p () for all p, 1 p<, from propery (3.4) in Theorem 3.1, we have P h (Γ ) lim n P h(γ n ) c and he opimal consruced in he proof of he heorem saisfies he addiional consrain on he densiy perimeer. 13 Noe ha he consan c mus be large enough o ake ino accoun he conribuion of he boundary of.

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