88,102 research outputs found
A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science
In this paper, we study damped second-order dynamics, which are quasilinear
hyperbolic partial differential equations (PDEs). This is inspired by the
recent development of second-order damping systems for accelerating energy
decay of gradient flows. We concentrate on two equations: one is a damped
second-order total variation flow, which is primarily motivated by the
application of image denoising; the other is a damped second-order mean
curvature flow for level sets of scalar functions, which is related to a
non-convex variational model capable of correcting displacement errors in image
data (e.g. dejittering). For the former equation, we prove the existence and
uniqueness of the solution. For the latter, we draw a connection between the
equation and some second-order geometric PDEs evolving the hypersurfaces which
are described by level sets of scalar functions, and show the existence and
uniqueness of the solution for a regularized version of the equation. The
latter is used in our algorithmic development. A general algorithm for
numerical discretization of the two nonlinear PDEs is proposed and analyzed.
Its efficiency is demonstrated by various numerical examples, where simulations
on the behavior of solutions of the new equations and comparisons with
first-order flows are also documented
Introduction to the Special Issue on Partial Differential Equations and Geometry-Driven Diffusion in Image Processing and Analysis
©1998 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TIP.1998.66117
- …