Integrodifferential Equations for Multiscale Wavelet Shrinkage: The Discrete Case

Abstract

We investigate the relations between wavelet shrinkage and integrodifferential equations for image simplification and denoising in the discrete case. Previous investigations in the continuous one-dimensional setting are transferred to the discrete multidimentional case. The key observation is that a wavelet transform can be understood as a derivative operator in connection with convolution with a smoothing kernel. In this paper, we extend these ideas to a practically relevant discrete formulation with both orthogonal and biorthogonal wavelets. In the discrete setting, the behaviour of smoothing kernels for different scales is more complicated than in the continuous setting and of special interest for the understanding of the filters. With the help of tensor product wavelets and special shrinkage rules, the approach is extended to more than one spatial dimension. The results of wavelet shrinkage and related integrodifferential equations are compared in terms of quality by numerical experiments.