We consider the Cauchy problem for the one-dimensional PeronaMalik equation ut = 1− ux (1 + ux) 2 uxx in the interval [−1, 1], with homogeneous Neumann boundary conditions. We prove that the set of initial data for which this equation has a localin-time classical solution u : [−1, 1]× [0, T ] → R is dense in C1([−1, 1]). Here “classical solution” means that u, ut, ux and uxx are continuous func...

We propose the coarsening strategy for the finite volume computational method given by K. Mikula and N. Ramarosy (Numer. Math. 89, 2001, 561–590) for the numerical solution of the (modified in the sense of F. Catté et al. (SIAM J. Numer. Anal. 29, 1992, 182–193)) Perona–Malik nonlinear image selective smoothing equation (called anisotropic diffusion in image processing). The adaptive aproach is...

The aim of the work presented in this paper is to investigate experimentally the behavior of different types of anisotropic diffusion schemes. The efficiency of the filters based on the anisotropic diffusion is determined to large extent by the properties of the conductance function in the PDE equation, which describes the diffusion process. Changing the shape of the conductivity function, we c...

Image deblurring is a discrete ill-posed problem. This paper discusses cascadic multilevel methods designed for the restoration of images that have been contaminated by nonsymmetric blur and noise. Prolongation is carried out by nonlinear edge-preserving and noise-reducing operators, while restrictions are determined by weighted local least-squares approximation. The restoration problem is on e...

In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and Γ-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functiona...

This paper proposes an alternative to partial differential equations (PDEs) for the solution of diffusion (Perona and Malik scheme), using the heat transfer problem. Traditionally, the method for solving such physics-based problems is to discretize and solve a PDE by a mathematical process. We propose to use the global heat equation and decompose it into simpler laws. Some of these laws admit a...

Many reasons can be cited for the desire to harness the power of nonlinear anisotropic diffusion in image processing. Perona and Malik proposed one of the pioneering models which, while numerically viable, proves mathematically ill-posed. This discrepancy between its analytical properties and those of its numerical implementations spurred a significant amount of research in the past twenty year...

It is progressively realized that noise can play a constructive role in the domain of nonlinear information processing. This phenomenon, also known as stochastic resonance (SR) effect, has experienced large varieties of extensions with variations concerning the type of noise, the type of information carrying signal or the type of nonlinear system interacting with the signal-noise mixture. In th...

In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and Γ-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functiona...

This paper investigates the use of cascadic multiresolution methods for image deblurring. Iterations with a conjugate gradient-type method are carried out on each level, and terminated by a stopping rule based on the discrepancy principle. Prolongation is carried out by nonlinear edge-preserving operators, which are defined via PDEs associated with Perona–Malik or total variation-type models. C...