Interpreting translation-invariant wavelet shrinkage as a new image smoothing scale space

Coifman and Donoho suggested translation-invariant wavelet shrinkage as a way to remove noise from images. Basically, their technique applies wavelet shrinkage to a two-dimensional (2-D) version of the semi-discrete wavelet representation of Mallat and Zhong, Coifman and Donoho also showed how the method could he implemented in O(N log N) operations, where there are N pixels, In this paper, we provide a mathematical framework for iterated translation-invariant wavelet shrinkage, and show, using a theorem of Kato and Masuda, that with orthogonal wavelets it is equivalent to gradient descent in L-2(T) along the semi-norm for the Besov space B-1(1)(L-1(I)), which, in turn, can be interpreted as a new nonlinear wavelet-based image smoothing scale space. Unlike many other scale spaces, the characterization is not in terms of a nonlinear partial differential equation.