We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141--151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266--277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlap; it needs only log n level set functions for n phases in the piecewise constant case; it can represent boundaries with complex topologies, including triple junctions; in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.

: In this note, we extend the result described in a previous note to the case of non-smooth Hamiltonians for fully nonlinear stochastic partial differential equations. And we present some applications of our theory to pathwise stochastic control and to the propagation of fronts in random environments. R'esum'e : Dans cette note, nous 'etendons les r'esultats d'ecrits dans une note pr'ec'edente au cas d'Hamiltoniens non r'eguliers pour des 'equations aux d'eriv'ees partielles stochastiques compl`etement nonlin'eaires. Et nous pr'esentons quelques applications de notre th'eorie au controle stochastique trajectoriel et `a la propagation de fronts dans des environnements al'eatoires. Version Fran¸caise Abr'eg'ee : Nous 'etendons ici les r'esultats obtenus dans [LS1] `a des 'equations plus g'en'erales (pour des Hamiltoniens peu r'eguliers) et pr'esentons quelques applications de cette th'eorie. Plus pr'ecis'ement, nous consid'erons des 'equations paraboliques sltochastiques, 'eventuellement...

Properties of solutions of a bistable nonlinear convolution equation in higher space dimensions are studied. The nonlinearity is the derivative of a double well function. The theory of traveling waves for this equation was given in a previous publication [4]. Here we consider spreading phenomena for state regions, in some cases by means of the motion of domain walls, which are modeled by internal layers. These phenomena are analogous to those known for the bistable nonlinear diffusion equation, and in particular, for the Allen-Cahn equation, which is a model for the motion of some grain boundaries in solid materials. Cases when the two wells have unequal depth are considered, as well as when they have equal depth. In the latter case a motion-by-curvature law is derived formally in two space dimensions. 1 Introduction We consider the properties of solutions u(x; t) of the bistable nonlinear convolution equation u t = J u \Gamma u \Gamma f(u); x 2 R N ; t 0; (1) 1 in which the fu...

: An overview will be given of some nonlinear parabolic-like evolution problems which are off the classical beaten track, but have increased in importance during the past decade. The emphasis is on problems which are nonlocal, pattern-forming (including exhibiting propagative phenomena), and/or lead in some singular limit to free boundary problems. In all cases they have been proposed as models for phenomena in the natural sciences. Also emphasized are the relationships among the various classes of phenomena. 1

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this paper we prove the convergence of an approximation scheme for a global in time evolution of sets(

Abstract: Existence and uniqueness of solutions for Hele-shaw moving boundary problem for power-law fluid is established in the framework of viscosity solutions.

Abstract. This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phasefield theory for fractional reaction-diffusion equations. It is defined by using the level set method. The main results of this paper are: on one hand, the proper level set formulation of the geometric flow; on the other hand, stability and comparison results for the geometric equation associated with the flow.

Abstract. We give a simple proof of convergence of the anisotropic variant of a well-known algorithm for mean curvature motion, introduced in 1992 by Merriman, Bence, and Osher. The algorithm consists in alternating the resolution of the (anisotropic) heat equation, with initial datum the characteristic function of the evolving set, and a thresholding at level 1/2.