On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs

Soft wavelet shrinkage, total variation (TV) diffusion, TV regularization, and a dynamical system called SIDEs are four useful techniques for discontinuity preserving denoising of signals and images. In this paper we investigate under which circumstances these methods are equivalent in the one-dimensional case. First, we prove that Haar wavelet shrinkage on a single scale is equivalent to a single step of space-discrete TV diffusion or regularization of two-pixel pairs. In the translationally invariant case we show that applying cycle spinning to Haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of TV diffusion. We prove that space-discrete TV diffusion and TV regularization are identical and that they are also equivalent to the SIDEs system when a specific force function is chosen. Afterwards, we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the Laplacian pyramid of the signal. We analyze possibilities to avoid Gibbs-like artifacts for multiscale Haar wavelet shrinkage by scaling the thresholds. Finally, we present experiments where hybrid methods are designed that combine the advantages of wavelets and PDE/variational approaches. These methods are based on iterated shift-invariant wavelet shrinkage at multiple scales with scaled thresholds.