In this paper, we present a method for computing the domain of attraction for non-linear dynamical systems. We propose a level-set method where sets are represented as sublevel sets of polynomials. The problem of flowing these sets under the advection map of a dynamical system is converted to a semidefinite program, which we use to compute the coefficients of the polynomials. We further address...

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 ...

We propose here the use of the variational level set methodology to capture Lagrangian vortex boundaries in 2D unsteady velocity fields. This method reformulates earlier approaches that seek material vortex boundaries as extremum solutions of variational problems. We demonstrate the performance of this technique for two different variational formulations built upon different notions of coherenc...

The physical model describing heat transfer and melting taking place during and after the interaction of a laser beam with a semi-infinite metal surface is based on the classical Stefan problem with appropriately chosen boundary conditions to reflect direct selective laser sintering of metals. A level set method for solving this problem is presented in this paper. From the results of these comp...

We analyze a multiple level-set method for solving inverse problems with piecewise constant solutions. This method corresponds to an iterated Tikhonov method for a particular Tikhonov functional Gα based on TV–H 1 penalization. We define generalized minimizers for our Tikhonov functional and establish an existence result. Moreover, we prove convergence and stability results of the proposed Tikh...

In this article, we discuss the question “What Level Set Methods can do for image science”. We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techniques that are potentially useful for this class of applications. We will show that image science demands multi-disciplinary knowledge and flexible but still robust m...

We present a multi-resolution space carving algorithm that reconstructs a 3D model of visual scene photographed by a calibrated digital camera placed at multiple viewpoints. Our approach employs a level set framework for reconstructing the scene. Unlike most standard space carving approaches, our level set approach produces a smooth reconstruction composed of manifold surfaces. Our method outpu...

In this work we discuss variants of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead use piecewise constant level set functions, and let interfaces be represented by discontinuities. Some of the properties of the standard level set function are preserved in the proposed method. Using the methods for interf...

Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsi...

In this article, we review past work on Level Set Methods, introduced by Osher and Sethian in [20], and Fast Marching Methods, introduced by Sethian in [25], for tracking propagating interfaces in two and three space dimensions. Both sets of techniques are based on a partial differential equations view of interface motion, and rely on the use of the theory of viscosity solutions, upwind finite ...