The starting point for this paper is the well known equivalence between convolution filtering with a rescaled Gaussian and the solution of the heat equation. In the first sections we analyze the equivalence between multiscale convolution filtering, linear smoothing methods based on continuous wavelet transforms and the solutions of linear diffusion equations. I.e. we determine a wavelet ψ, resp. a convolution filter φ, which is associated
with a given linear diffusion equation ut = Pu and vice versa. This approach has an extension to non-linear smoothing techniques. The main result of this paper is the derivation of a differential equation, whose solution is equivalent to non-linear multi-scale smoothing based on soft shrinkage methods applied to Fourier or continuous wavelet transforms.
The common way to process radar wind profiler (RWP) data by moments estimation of the Fourier power spectrum fails in presence of transient intermittent clutter contributions. Wavelets are especially suitable for detecting and removing transient components because of their high localization in time and frequency domain. We give an overview on the wavelet filtering of contaminated discrete RWP signals and introduce a new technique involving the wavelet packet decomposition and a splitting in progressive and regressive signal components. This technique has been successfully tested on severely real-data sets where classical wavelet routines fail.
We discuss a wavelet based treatment of variational problems arising in the context of image processing, inspired by papers of Vese-Osher and Osher-Sole-Vese, in particular, we introduce a special class of variational functionals, that induce a decomposition of images in oscillating and cartoon components. Cartoons are often modeled
by BV functions. In the setting of Vese et.el. and Osher et.al. the incorporation of BV penalty terms leads to PDE schemes that are numerically intensive. We propose to embed the problem in a wavelet framework. This provides us with elegant and numerically efficient schemes even though a basic requirement, the involvement of the space BV , has to be softened slightly. We show results on test images of our wavelet algorithm with a B11 (L1) penalty term, and we compare them with the BV restorations of Osher-Sole-Vese.