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Young measure solutions and instability of the one-dimensional Perona-Malik equation
journal contributionposted on 2023-06-08, 06:26 authored by Shahnez Taheri, Qi Tang, Kewei Zhang
We apply a variational approach to the one-dimensional version of the widely used Perona-Malik equation in image processing. We rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2 x 2 matrices M-2x2. We then use the solutions of some heat equations as the centre of the mass for the Young measure-valued solutions to construct the approximate solutions by using simple laminates. The approximate solutions can be viewed as solutions of a perturbation problem by W--1,W- p (or W--1,W- infinity) functions. The sequences of the approximate solutions generates Young measure-valued solutions. Our results also show that the solutions of the one-dimensional Perona-Malik equation are unstable under small W--1,W- infinity perturbations.
JournalJournal of Mathematical Analysis and Applications
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- Mathematics Publications
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