Abstract
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 × 2 matrices M 2 × 2 . 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 , p (or W − 1 , ∞ ) 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 , ∞ perturbations.
Published Version
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