We consider the Cauchy problem for the one-dimensional Perona-Malik equation

in the interval , with homogeneous Neumann boundary conditions.

We prove that the set of initial data for which this equation has a local-in-time classical solution is dense in . Here ``classical solution'' means that , , and are continuous functions in .

RESUMEN

In this article we design the semiimplicit finite volume scheme for coherence enhancing diffusion in image processing and prove its convergence to the weak solution of the problem. The finite volume methods are natural tools for image processing applications since they use piecewise constant representation of approximate solutions similarly to the structure of digital images. They have been successfully applied in image processing, e.g., for solving the Perona–Malik equation or curvature-driven level set equations, where the nonlinearities are represented by a scalar function dependent on a solution gradient. Design of suitable finite volume schemes for tensor diffusion is a nontrivial task here we present the first such scheme with a convergence proof for the practical nonlinear model used in coherence-enhancing image smoothing. We provide basic information about this type of nonlinear diffusion including a construction of its diffusion tensor, and we derive a semiimplicit finite volume scheme for this nonlinear model with the help of covolume mesh. This method is well known as the diamond-cell method owing to the choice of covolume as a diamond-shaped polygon. Further, we prove a convergence of a discrete solution given by our scheme to the weak solution of the problem. The proof is based on Kolmogorov’s compactness theorem and a bounding of a gradient in the tangential direction by using a gradient in the normal direction. Finally computational results illustrated in figures are discussed.