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.

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 re-tains some of the attractive features of existing methods and over-comes some of their limitations. Our techniques can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori as-sumption about the object’s topology is made. A single instance of our 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 to model propagating solidhiquid interfaces with curva-ture-dependent speeds. The interface (front) is a closed, noninter-secting, hypersurface flowing along its gradient field with con-stant speed or a speed that depends on the curvature. It is moved by solving a “Hamilton-Jacob? ’ type equation written for a func-tion in which the interface is a particular level set. A speed term synthesizpd from the image is used to stop the interface in the vi-cinity of object boundaries. The resulting equation of motion is solved by employing entropy-satisfying upwind finite difference schemes. We present a variety of ways of computing 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. Index Terms- Shape modeling, shape recovery, interface mo-tion, level sets, hyperbolic conservation laws, Hamilton-Jacobi

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.

We present a fast marching level set method for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial dierential equation for a propagating level set function, and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. In this paper, we describe a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithog...

A method is introduced to decrease the computational labor of the standard level set method for propagating interfaces. The fast approach uses only points close to the curve at every time step. We describe this new algorithm and compare its efficiency and accuracy with the standard level set approach. 1 A Fast Level Set Implementation The level set technique was introduced in [9] to track moving interfaces in a wide variety of problems. It relies on the relation between propagating interfaces and propagating shocks. The equation for a front propagating with curvature dependent speed is linked to a viscous hyperbolic conservation law for the propagating gradients of the fronts. The central idea is to follow the evolution of a function OE whose zero--level set always corresponds to the position of the propagating interface. The motion for this evolving function OE is determined from a partial differential equation in one higher dimension which permits cusps, sharp corners, and changes i...

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 paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel edge-detection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edge-seeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the Allen-Cahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3-D active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a level-set approach. Key words: Active vision, antiphase boundary, visual tracking, edge detection, segmentation, gradient flows, Riemannian metrics, viscosity solutions, geometric heat equ...

In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, and/or curves, and/or surface patches. We do not assume any knowledge of the ordering, connectivity or topology of the data sets or of the true shape. Only the location of each point or general Hausdorff distance to the data set is known. The key idea in our approach is to find an implicit nonparametric representation of the curve or surface on a fixed rectangular grid. With this representation of surfaces we can easily (a) find the closest point and distance from any point to the surface (useful in illumination and many other applications), (b) find the intersection curve of two surfaces which is guaranteed to lie on both surfaces in our representation, and (c) perform any Boolean operatio...

The level set method was devised by Osher and Sethian in [64] as a simple and versatile method for computing and analyzing the motion of an interface Γ in two or three dimensions. Γ bounds a (possibly multiply connected) region Ω. The goal is to compute and analyze the subsequent motion of Γ under a velocity field �v. This velocity can depend on position, time, the geometry of the interface and the external physics. The interface is captured for later time as the zero level set of a smooth (at least Lipschitz continuous) function ϕ(�x,t), i.e., Γ(t)={�x|ϕ(�x,t)=0}. ϕ is positive inside Ω, negative outside Ω andiszeroonΓ(t). Topological merging and breaking are well defined and easily performed. In this review article we discuss recent variants and extensions, including the motion of curves in three dimensions, the Dynamic Surface Extension method, fast methods for steady state problems, diffusion generated motion and the variational level set approach. We also give a user’s guide to the level set dictionary and technology, couple the method to a wide variety of problems involving external physics, such as compressible and incompressible (possibly reacting) flow, Stefan problems, kinetic crystal growth, epitaxial growth of thin films,

We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The absolute minimal Lipschitz extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range.