Joel Spruck

Bio: Joel Spruck is an academic researcher from Johns Hopkins University. The author has contributed to research in topics: Curvature & Mean curvature. The author has an hindex of 35, co-authored 89 publications receiving 9658 citations. Previous affiliations of Joel Spruck include Brooklyn College & University of Minnesota.

Motion of Level Sets by Mean Curvature. I

Abstract: We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set ⌈0 a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.

Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Abstract: On etudie les solutions regulieres non negatives de l'equation conformement invariante −Δu=u (n+2)/(n−2) , u>0 dans une boule perforee, B 1 (0)\{0}⊂R n , n≥3, avec une singularite isolee a l'origine

The dirichlet problem for nonlinear second‐order elliptic equations I. Monge‐ampégre equation

Abstract: On considere le probleme de Dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture Ω d'un domaine borne Ω dans R n avec une frontiere ∂Ω C ∞

The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian

Abstract: On etudie le probleme de Dirichlet dans un domaine borne Ω de R n a frontiere lisse ∂Ω:F(D 2 u)=ψ dans Ω, u=φ sur ∂Ω

Geodesic active contours

Abstract: A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. >

User’s guide to viscosity solutions of second order partial differential equations

Abstract: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

Geodesic Active Contours

Abstract: A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

Shape modeling with front propagation: a level set approach

Abstract: Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods and overcomes some of their limitations. The authors' techniques can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object's topology is made. A single instance of the authors' model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian (1988) to model propagating solid/liquid interfaces with curvature-dependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature. It is moved by solving a "Hamilton-Jacobi" type equation written for a function in which the interface is a particular level set. A speed term synthesized from the image is used to stop the interface in the vicinity of object boundaries. The resulting equation of motion is solved by employing entropy-satisfying upwind finite difference schemes. The authors present a variety of ways of computing the evolving front, including narrow bands, reinitializations, and different stopping criteria. The efficacy of the scheme is demonstrated with numerical experiments on some synthesized images and some low contrast medical images. >