Showing papers by "Stanley Osher published in 1990"
TL;DR: In this article, shock filters for image enhancement are developed, which use new nonlinear time dependent partial differential equations and their discretizations, which satisfy a maximum principle and the total variation of the solution for any fixed fixed $t > 0$ is the same as that of the initial data.
Abstract: Shock filters for image enhancement are developed. The filters use new nonlinear time dependent partial differential equations and their discretizations. The evolution of the initial image $u_0 (x,y)$ as $t \to \infty $ into a steady state solution $u_\infty (x,y)$ through $u(x,y,t)$, $t > 0$, is the filtering process. The partial differential equations have solutions which satisfy a maximum principle. Moreover the total variation of the solution for any fixed $t > 0$ is the same as that of the initial data. The processed image is piecewise smooth, nonoscillatory, and the jumps occur across zeros of an elliptic operator (edge detector). The algorithm is relatively fast and easy to program.
850 citations
01 Feb 1990
TL;DR: In this paper, high order essentially non- oscillatory (ENO) schemes for H-J equations are investigated, which yield uniform high order accuracy in smooth regions and resolve discontinuities in the derivatives sharply.
Abstract: : Hamilton-Jacobi (H-J) equations are frequently encountered in applications, e.g. in control theory and differential games. H-J equations are closely related to hyperbolic conservation laws -- in one space dimension the former is simply the integrated version of the latter. Similarity also exists for the multi-dimensional case, and this is helpful in the design of difference approximations. In this paper we investigate high order essentially non- oscillatory (ENO) schemes for H-J equations, which yield uniform high order accuracy in smooth regions and resolve discontinuities in the derivatives sharply. The ENO scheme construction procedure is adapted from that for hyperbolic conservation laws. We numerically test the schemes on a variety of one-dimensional and two-dimensional problems, including a problem related to control optimization, and observe high order accuracy in smooth regions, good resolution of discontinuities in the derivatives, and convergence to viscosity solutions. (edc)
143 citations
01 Jan 1990
TL;DR: In this article, a triangle-based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures.
Abstract: A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.
11 citations
01 Jan 1990
TL;DR: In this paper, the authors demonstrate that they can go beyond the second order accurate TVD barrier and suppress spurious numerical oscillations near discontinuities and other steep gradients, using a simple expository article in this volume that can be easily read and used by a novice to the field of shock capturing.
Abstract: Professor Roe’s beautiful expository article in this volume can be easily read and used by a novice to the field of shock capturing. Our purpose here is to demonstrate that we can go beyond the second order accurate TVD barrier [7] and still suppress spurious numerical oscillations near discontinuities and other steep gradients.