Showing papers by "Stanley Osher published in 1999"

Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves

Abstract: 0. Introduction. In this paper, we obtain some new inequalities for integrals of convex functions of the curvature (resp. Wulff curvature) of convex plane curves. We also show that the difference between the two sides of our inequalities are monotone decreasing as the region flows under the unit-speed outward normal (resp. Wulff) flow. Given a bounded plane region K, the unit-speed outward normal flow has been highly studied, and is of interest in many applied problems, e.g. combustion. If instead K grows by varying the outward normal speed to be a function 7(0) of the direction of the unit normal, one has the WulfF flow, which is also of considerable interest, e.g. in studying the growth of crystals [O-M]. When the region K is convex, there is a simple closed-form expression which describes these flows, and the region converges to a disk in the first case and a Wulff shape in the second. The area of the region when the initial region K is convex is a polynomial in t, known respectively as the Steiner polynomial or WulfF-Steiner polynomial. A novel feature of our approach is that we study the roots of the Steiner and WulffSteiner polynomials, which occur at negative values of t. The classical isoperimetric inequality in both cases states that these polynomials have (negative) real roots ti >t2, and that they are distinct if and only if K is not a disc (respectively not a Wulff shape). Bonnesen's inequality states that the inradius and outradius r^ and re lie in the interval [—£1, —£2], and in the open interval if K is not a disk (respectively not a Wulff shape). Our inequalities are most naturally stated and proved in terms of the roots ti and £2We feel that this is a potentially quite fruitful approach to studying convex bodies in higher dimensions. In the context of this new approach, a very natural link between the outward normal and Wulff flows and the curvature integrals of the region appears. In important cases, the quantities that our inequalities state are positive are shown to be monotone decreasing as the region evolves under the flow. Particularly suggestive is the fact that the entropy of the curvature (respectively Wulff curvature) is bounded above in terms of the area and is monotone decreasing with time. The inequalities themselves are quite fascinating. It came as a surprise to us that there are interesting new things to be said about convex plane curves. We state our inequalities here for arbitrary smooth bounded convex plane regions K in the curvature case, and leave the Wulff case to the body of the paper. One of them is due to Gage [G], whose result was a source of inspiration to us. Gage's result is

A New Time Dependent Model Based on Level Set Motion for Nonlinear Deblurring and Noise Removal

Abstract: In this paper we summarize the main features of a new time dependent model to approximate the solution to the nonlinear total variation optimization problem for deblurring and noise removal introduced by Rudin, Osher and Fatemi. Our model is based on level set motion whose steady state is quickly reached by means of an explicit procedure based on an ENO Hamilton-Jacobi version of Roe's scheme. We show numerical evidence of the speed, resolution and stability of this simple explicit procedure in two representative 1D and 2D numerical examples.