Showing papers by "Stanley Osher published in 2001"

The digital TV filter and nonlinear denoising

Abstract: Motivated by the classical TV (total variation) restoration model, we propose a new nonlinear filter-the digital TV filter for denoising and enhancing digital images, or more generally, data living on graphs. The digital TV filter is a data dependent lowpass filter, capable of denoising data without blurring jumps or edges. In iterations, it solves a global total variational (or L/sup 1/) optimization problem, which differs from most statistical filters. Applications are given in the denoising of one dimensional (1-D) signals, two-dimensional (2-D) data with irregular structures, gray scale and color images, and nonflat image features such as chromaticity.

Fast surface reconstruction using the level set method

Abstract: We describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from a scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surface-like model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly nonuniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.

Diffusion-Generated Motion by Mean Curvature for Filaments

Abstract: Diffusion-generated motion by mean curvature is a simple algorithm for producing motion by mean curvature of a surface, in which the motion is generated by alternately diffusing and renormalizing a characteristic function In this paper, we generalize diffusion-generated motion to a procedure that can be applied to the curvature motion of filaments, ie, curves in R ^3, that may initially consist of a complex configuration of links The method consists of applying diffusion to a complex-valued function whose values wind around the filament, followed by normalization We motivate this approach by considering the essential features of the complex Ginzburg-Landau equation, which is a reaction-diffusion PDE that describes the formation and propagation of filamentary structures The new algorithm naturally captures topological merging and breaking of filaments without fattening curves We justify the new algorithm with asymptotic analysis and numerical experiments

Variational problems and PDEs on implicit surfaces

Abstract: A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is introduced. The key idea is to implicitly represent the surface as the level set of a higher dimensional function, and solve the surface equations in a fixed Cartesian coordinate system using this new embedding function. The equations are then both intrinsic to the surface and defined in the embedding space. This approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces, as is commonly done in the literature. We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, as well as image and vector field intrinsic regularization for data defined on 3D surfaces.

Variational Problems and PDE's on Implicit Surfaces

Abstract: A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is introduced in this paper. The key idea is to implicitly represent the surface as the level set of a higher dimensional function, and solve the surface equations in a fixed Cartesian coordinate system using this new embedding function. The equations are then both intrinsic to the surface and defined in the embedding space. This approach thereby eliminates the need for performing complicated and not-accurate computations on triangulated surfaces, as it is commonly done in the literature. We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, as well as image and vector field intrinsic regularization for data defined on 3D surfaces.

A level set approach for computing discontinuous solutions of a class of Hamilton-Jacobi equations

Abstract: We introduce two types of finite difference methods to compute the Lsolution [15] and the proper viscosity solution [14] recently proposed by the second author for semi-discontinuous solutions to a class of Hamilton-Jacobi equations. By regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher, we can treat the corresponding level set equation using the viscosity theory introduced by Crandall and Lions [7]. However, we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function. We demonstrate our Lax-Friedrichs type numerical methods for computing the L-solution using its original level set formulation. In addition, we couple our numerical methods with a singular diffusive term which is essential to computing solutions to a more general class of HJ equations that includes conservation laws. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using WENO Local Lax-Friedrichs methods [22]. We verify that our numerical solutions approximate the proper viscosity solutions of [14]. Finally, since the solution of scalar conservation law equations can be Research supported by ONR N00014-97-1-0027, DARPA/NSF VIP grant NSF DMS 9615854 and ARO DAAG 55-98-1-0323 yDepartment of Mathematics, University of California Los Angeles, Los Angeles, California 90095, email:ytsai@math.ucla.edu zDepartment of Mathematics, Hokkaido University, Sapporo 060-0810, Japan, email: giga@math.sci.hokudai.ac.jp xDepartment of Mathematics, University of California Los Angeles, Los Angeles, California 90095, email:sjo@math.ucla.edu

Numerical Methods for Computing Discontinuous

Abstract: In this article, we first introduce aLax-Friedrichs type finite dierence method to compute the $ \mathrm{L}$ -solution, following its original definition recently proposed by the second auther in(12) using level sets. We then generalize our numerical methods to compute the proper viscosity solution proposed in (11) for amore general class of HJ equations that includes conservation laws. We couple our numerical methods with asingular diusive term of essential im- portance. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using WENO Local Lax-Friedrichs methods (17). We verify that