On Iterations and Scales of Nonlinear Filters

Methods modeled by partial differential equations (PDEs), and wavelet-based shrinkage procedures belong to the most successful approaches to nonlinear signal processing. The two groups of methods are quite different on the first glance: PDE based methods find the solution iteratively, while wavelet shrinkage performs a single step on multiple scales of the signal. On the examples of total-variation (TV) diffusion on one hand, and translation-invariant soft Haar wavelet shrinkage on the other, we study the role of iterations and multiple scales for nonlinear filtering. Iterations and multi-scale processing go in the same direction in the sense that they allow simple, local operations to lead to global effects. We demonstrate that it may be advantageous to combine both, and create a powerful and efficient iterative multi-scale nonlinear procedure.