C. R. Acad. Sci. Paris, t. 329, Shie I, p. 249-254, 1999 Calcul des variations/Calculus of Variations

The calibration method for the Mumford-Shah functional

Giovanni ALBERT1 a, Guy BOUCHITTI? b, Gianni DAL MASO ’

a Dipartimento di Matematica, via Buonarroti, 2, 56127 Pisa, Italy E-mail: albertiQdm.unipi.it

’ Universite de Toulon et du Var, B.P. 132, 83957 La Garde cedex, France E-mail: bouchitte@univ-tln.fr

’ SISSA, via Beirut 4, 34014 Trieste, Italy E-mail: dalmasoQsissa.it

(ReCu le

Abstract. In this Note we adapt the calibration method to functionals of Mumford-Shah type, and provide a criterion (Theorem 1) to verify that a given function is energy minimizing. Among other applications, we use this criterion to show that certain triple-junction configurations are minimizing (Example 3). 0 Academic des Sciences/Elsevier, Paris

Une m6thode de calibration pour la fonctionnelle de

Mumford-Shah

RCsumC. Dans cette Note, nous presentons une condition suffisante d’optimalite’ pour la minimisation de la fonctionnelle de Mumford-Shah. Cette condition (theoreme 1) resulte d’une variante de la methode de calibration et permet de montrer notamment que certaines lignes a jonction triple correspondent a des configurations optimales (exemple 3). 0 Academic des Sciences/Elsevier, Paris

Version francake abr6ge’e

La fonctionnelle de Mumford-Shah a Ct6 introduite dans le contexte de la dktection de contours d’images ([8], [7]). Elle s’Ccrit en dimension n sous la forme :

F(u) := l,,, lW2 + ~~FI”-lw + BL,s,, Iu - 912> (1)

oti R est un ouvert rkgulier de R”, g : C? -+ [0, l] est une donnte (niveau de gris), (I et ,0 des constantes positives et 7-P-l la mesure de Hausdorff de dimension (n - 1) ; l’inconnue u : R -+ W

Note prCsentt!e par Pierre-Louis LIONS.

0764~4442/99/03290249 0 Acadtmie des Sciences/Elsevier, Paris 249

C. Alberti et al.

est supposC a priori de classe C1 en dehors d’un ensemble singulier Su dont la forme et la position n’est pas donnCe. Minimiser F revient done 2 optimiser 2 la fois la function et l’ensemble singulier.

Une simplification usuelle du mod5le souvent utiliske pour Ctudier la r6gularitC des minimiseurs de F consiste g enlever le terme p J IZL - g12, ce qui en posant a = 1, conduit 2 :

F(u) := J

(Vu12 + 7-r1(Su). (2) n\su

On s’intkesse alors aux fonctions qui minimisent F(u) lorsque la trace de u est donnte au bord. En considkrant diffbrents types de variations, on montre aisCment qu’un minimiseur tr.5 rkgulier ‘ZL doit satisfaire certains c&&es : u doit &tre harmonique sur R \ SZL avec des dCrivt!es normales nulles de chaque c&C de Su et de plus la courbure moyenne de Su doit Ctre kgale 3 la diffkrence des car&s des normes du gradient de u de chaque c&C. Des conditions SupplCmentaires ont CtC don&es dans le cas 7~ = 2 et nous renvoyons B [7] et [3] pour une description plus prkcise des conditions d’kquilibre et les ksultats de r6gularitC correspondants.

11 est t&s important de noter que F n’est pas convexe et par consCquent aucun systkme de conditions rkcessaires dkduites de variations infinitksimales ne pourra conduire a une condition suffisante d’optimalitk. L’objet de cette Note est de prksenter une condition suffisante (thkorkme 1) et quelques applications (exemples l-3). Les preuves dCtaillCes paraitront dans un papier ultkrieur [l].

Avant de p&enter la mkthode, nous devons Ctendre la dkfinition de F ti une classe plus large de fonctions : l’espace SBV(fl) des fonctions spkciales h variations bornCe (voir [2], [3]). Brikvement, la fonction Vu dans (2) devient le gradient approximatif (dCfini Lebesque p.p.) et Su represente l’ensemble des points de discontinuitks (essentielles) de u, i.e. Su = {z E R ; U+(X) > u-(z)}, oti u+(z), U-(X) dksignent les limites (approximatives) infkrieure et sup&ewe de 21. L’appartenance de 1~ g SBV(0) se traduit par le fait que cet ensemble Su est n - 1 rectifiable et admet en W-l- presque tout point z E Su, un hyperplan tangent dont nous noterons v~(z) la normale unitaire dirigCe de U-(X) vers U+(X). On peut montrer que le problbme initial et celui dans la formulation SBV sont complktement kquivalents (cJ: [4] ou [3], Chapter 7).

L’idCe fondamentale de notre approche consiste 5 r&&ire F comme une fonction convexe de la nouvelle variable 1, : 0 x Iw -+ [0, l] dCd ul e ‘t en associant B ‘u. la fonction caractkistique de son hypographe. Plus prkcidment, posons pour tout u : R + W, lu(z, t) := 1 si t 5 u(z), lu(2, t) := 0 si t > U(X) ; si ‘u. est dans SBV(fl), il est facile de montrer que la d&&e distributionnelle D 1, est une mesure de Radon bornCe R x W. Notre reprksentation convexe de 1’Cnergie F(u) fait l’objet du

LEMME 1. - Soit F d&nie par (2) et 3 l’ensemble des champs borkliens born& C+?J = (c+P, q5”) : R X R + R” X R tels que : (a) jV~,t)12 5 4@(z, t) pour 2 E Q t E W ;

@I is #z(x, t) dt 5 1 pour x E R, tl, t2 E R.

Alors L a, pour tout u E SBV(n),

F(u) = sup J q5. D 1,. &F RXR

(3)

(4)

De plus, lorsque F(u) est finie, l’kgalite’

F(u) = J $.Dlu RXR

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a lieu pour un champ borelien g5 E .7= si (et seulement si) (a’) +ZLf;Z;(~)) = 284~) et $+(z, U(X)) = IVU(Z)~~ pour p.t. 2 E fl \ Su. ;

W s

@(~,t) dt = vu(x) pour IV-l-p.t. z E Su. u-(z)

On peut maintenant deduire le resultat principal de cette Note.

THBORI~ME 1. - Soit u E SBV(s2) et supposons qu’il existe un champ 4 de classe Cl sur a x !J3 h divergence nulle et ve’rijiant (a), (b), (a’), (b’). Alors u minimise la fonctionnelle F definie par (2) sur l’ensemble des fonctions v telles que v = u sur 6%.

Remarque 1. - Nous appellerons calibration pour u tout champ 4 verifiant les conditions du theoreme 1, ceci par analogie avec la methode bien connue utilisee dans le cas des surfaces minimales (c$ [6]). 11 est suppose ici que 4 est de classe C1 afin de pouvoir integrer par parties (voir (5)), mais cette hypothese peut Ctre affaiblie de differentes manibres (voir [l] pour plus de details). Par exemple, il suffit que 4 soit C1 par morceaux avec des discontinuites purement tangentielles (la condition de divergence nulle sera alors satisfaite au sens des distributions).

Quelques applications significatives du theoreme 1 consistent a exhiber des calibrations dans quelques cas particuliers (voir exemples l-3 ci-apres et [l]). Le probleme de savoir si de telles calibrations existent pour tout minimiseur est encore ouvert.

1. Introduction

The Mumford-Shah functional was introduced in [8] in the context of a variational approach to edge detection problems (c$, also, [7]), and can be written, in dimension n, as:

F(u) := l,,.. IVu12 + ~‘FI”-l(w + PL,s., lu - !?12> (1)

where R is a regular bounded domain in R”, g : R 4 [0, l] is a given function (input grey level), Q and p are positive constants, ‘FI”-l is the (TZ - 1)-dimensional Hausdorff measure (that is, the usual (n - 1)-dimensional volume in case of subsets of Lipschitz hypersurfaces, the length in the most relevant case n = 2); the unknown function u : R --) R is regular (say, of class Cl) out of a closed singular set Su, whose shape and location are not prescribed. Thus minimizing F means optimizing the function and the singular set, which is indeed often regarded as an independent unknown.

A relevant simplification of the previous functional, which occurs in the study of regularity properties of minimizers of F, is obtained by dropping the lower order term p s 1’~. - g12, and setting for simplicity Q = 1, i.e.,

F(u) := s

n \ su. IVu12 + T-F(Su). (2)

In this case, one is interested in minimizers of F with given boundary values, that will be simply called minimizers.

Considering different classes of infinitesimal variations, one can show that every minimizer must satisfy certain equilibrium conditions, which could be globally labelled Euler-Lagrange equations for F. For instance, u must be harmonic on s1 \ SU with vanishing normal derivatives on Su, and the mean curvature of Su must be equal to the difference of the squared norms of the traces of Vu on

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the two sides of Su (at those points where Su is sufficiently regular). Additional conditions have been derived for the two-dimensional case; we refer the reader to [7] and [3] for a precise description of these equilibrium conditions, and related regularity results.

However, since F is not convex, all conditions which can be derived by infinitesimal variations are necessary for minimality, but never sufficient. The purpose of this Note is precisely to present a sufficient condition for minimality (Theorem l), and give a few applications (Examples l-3). Detailed proofs and further results will be given in the forthcoming paper [ 11.

2. Description of the results

Before proceeding, it is convenient to define F on a larger, and more flexible, class of functions, namely the class SBV(0) of special functions of bounded variation ([2], we refer to [3] for further details). In this setting, SU becomes the set of the essential discontinuity points of U, while Vu is the approximate gradient of U, which is defined almost everywhere (with respect to Lebesgue measure).

The set Su is no longer closed, but can be covered (up to an ?f”-’ -negligible subset) by countably many hypersurfaces of class Cl, and admits, at ?P-‘-a.e. 11: E Szu, an approximate tangent hyperplane, on both sides of which the function u has approximate limits. We denote the larger one by U+(X) and the smaller one by u- (2); vu(z) is the normal to Su pointing from the side of u-(z) to the side of U+(X). It can be proved that the minimum problems in the original setting and in the SBV setting are completely equivalent (see [4], or [3], Chapter 7).

The crucial point of our approach is to rewrite F as a convex functional (of a different variable), as in identity (3) below. For every u : R -+ Iw we denote by 1, : R x R --+ [0, I] the characteristic function of the subgraph of u, that is, lU(z, t) := 1 if t 5 U(X), and lU(z, t) := 0 if t > u(z). If u belongs to SBV(n), then the subgraph of u has finite perimeter in R x Iw, i.e., the distributional derivative D 1, is a bounded Radon measure on R x R. We begin with a lemma which will be proved in [l].

LEMMA 1. - Let F be given in (2), and F be the class of all (bounded and Bore1 measurable) vector jields 4 = (@, 4”) : R x R -+ R” x R which satisjj: (4 I$“~,t)l” I 4@(z,t) for 32 E Q t E R

@I I./

$;c(z, t) dt 5 1 for 2 E R, tl, ta E R.

Then, $- every u E SBV(s2), we have

F(u) = sup J $.Dlu dEF nxfB Moreover, the equality

F(u) = J 4.Dlu RXR

(3)

(4)

is achieved if (and only if) 4 satis$es: (a’) $“if;,v;(z)) = 2 Vu(z) and $“(z, U(X)) = lVu(z)[' for a.e. 5 E R \ Su;

(b’) J c#Y(x, t) dt = vu(z) for EFlnP1-a.e. 5 E Su. u-(z)

We can now give the main result of this Note.

THEOREM 1. - Let u E SBV(Q) b e g iven, and assume that there exists a C1 vectorjeld 4 on G x R’ which satis$es (a), (b), (a’), (b’) b a ove, and is divergence-free. Then u minimizes the functional F in (2) among all functions v with the same boundary values.

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Take indeed any ‘u such that ZI = u on dR. Then

F(v) 2 J $.Dlv = 4. D 1, = F(u), (5) RXR J RXR

where the inequality follows from (3), which in turn is implied by (a) and (b), the first equality follows from the fact that 4 is divergence-free and z1 = u on dR, the second equality follows from (4), which is implied by (a’) and (b’).

Remarque 1. - The vector field 4 in Theorem 1 is called a calibration for U. This term is taken from the theory of minimal surfaces (cfi [6]), where a vector field 4 in R” is said to calibrate an oriented hypersurface S if it agrees on S with the normal vector field, is divergence-free, and satisfies ]$J] 5 1 everywhere; the existence of a calibration implies that 5’ minimizes the (n - 1)-dimensional volume among all oriented hypersurfaces with the same boundary (with a one-line proof like (5)).

Remarque 2. - In Theorem 1, 4 was assumed of class C1 in order to get the first equality in (5) but this assumption can be relaxed in various ways (see [l] for details). For instance, one may consider piecewise C1 vector fields, which may be discontinuous along sufficiently regular interfaces. In this case the divergence-free condition must be understood in the distributional sense, i.e., the pointwise divergence vanishes (where defined) and the normal component of 4 is continuous across the discontinuity surfaces (see the examples below).

Remarque 3. - It is not clear if every minimizer must admit a calibration. Notice that minimal surfaces of codimension one always admit a calibration of some sort (as proved in [5]), but similar conclusions fail to be true in higher codimension. A first step towards an existence result is the following remark: replacing 1, with an arbitrary BV function w on R x R, the functional defined by the right-hand side of (3) is convex in w, and it turns out that the existence of a calibration for u is roughly equivalent to the fact that 0 belongs to the subdifferential of this functional at 1,. This condition is necessary and sufficient for minimizing convex functionals. However it is not known whether 1, is a minimizer of this auxiliary convex funtional in BV(0 x R) whenever u is a minimizer of F in SBV(n).

We conclude with a few applications of Theorem 1 (see [l] for the details).

Example 1. - Let n = 1, R = (0, a); it can be easily verified by direct computations that the linear function U(X) := bx minimizes F in (2) with respect to its boundary values if and only if b2a 5 1. The same conclusion applies to the step function given by U(X) := 0 for x 5 c and U(X) := ba for z > c (with c E (0, u)) if and only if b2u 2 1. In the two cases a calibration is given, in term of b := inf {b, $=}, by the piecewise constant vector field 4 below:

4(x, t> := (2b,b2) (6)

(0,O) elsewhere.

Example 2. - Let R an open subset of W”, 7~ arbitrary. As pointed out to us by A. Chambolle, a harmonic function u on R minimizes F in (2) with respect to its boundary values if

( supu - inof u >

*sup~VzL~ 5 1, cl R

which for 72 = 1 reduces to the constraint ub2 5 1 in Example 1. Inspired by the one-dimensional case, we construct the following calibration (c$ (6)):

$5(x, t) := (2774x), ]Vu(x)12)

(0,O) elsewhere.

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Example 3. - Let n = 2, R the unit disk centered at the origin, and u be given, in polar coordinates, byu:=mforO<%<~~,u:=Ofor~~~%<~~,~:=-mfor~~~%<2~,andmis a positive real constant. Thus Su is given by three line segments meeting at the origin with equal angles, and since this triple junction is length minimizing, it is natural to conjecture that, for large values of m, u minimizes the functional F in (2) among all functions with the same boundary values.

Indeed we calibrate u for every m 2 a. Inspired by the one-dimensional case described in Example 1, we take e+ := ( f a/2,1/2), and X > 0 such that $ + i < m, and we define the calibration by (c$ the figure below):

I (Xe+,A2/4) if a(1 + x. e+) 5 t 5 t(l +x * e+) + i,

d(Z4)‘= (Xe-,A2/4)

(0,O) elsewhere.

References

[I] Alberti G., Bouchitd G., Dal Maso G., The calibration method for the Mumford-Shah functional, (in preparation). [Z] Ambrosio L., Existence theory for a new class of variational problems, Arch. Rational Mech. AnaI. 111 (1990) 291-322. [3] Ambrosio L., Fusco N., Pallara D., Special Functions of Bounded Variation and Free-Discontinuity Problems, Oxford Univ.

Press, (in preparation). [4] De Giorgi E., Carrier0 M., Leaci A., Existence theorem for a minimum problem with free discontinuity, Arch. Rational

Mech. Anal. 108 (1989) 195-218. [5] Federer H., Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974) 351407. [6] Morgan F., Calibrations and new singularities in area-minimizing surfaces: A survey, in: Variational Methods, Paris 1988,

A. Berestycki et al. (Eds.), Progr. Nonlin. Differ. Eq. Appl. 4, Birkhluser, Boston, 1990, pp. 329-342. [7] Morel J.-M., Solimini S., Variational Methods in Image Segmentation, Progr. Nonlin. Differ. Eq. Appl. 14, Birkhiuser,

Boston, 1995. [8] Mumford D., Shah J., Optimal approximation by piecewise smooth functions and associated variational problems, Commun.

Pure Appl. Math. 42 (1989) 577-685.

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