Relations between Soft Wavelet Shrinkage and Total Variation Denoising

Soft wavelet shrinkage and total variation (TV) denoising are two frequently used techniques for denoising signals and images, while preserving their discontinuities. In this paper we show that – under specific circumstances – both methods are equivalent. First we prove that 1-D Haar wavelet shrinkage on a single scale is equivalent to a single step of TV diffusion or regularisation of two-pixel pairs. Afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularisation of the Laplacian pyramid of the signal.