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

Coifman and Donoho (1995) 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 (1992), Coifman and Donoho also showed how the method could be implemented in O(Nlog 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 (1978), that with orthogonal wavelets it is equivalent to gradient descent in L ₂(I) along the semi-norm for the Besov space B₁ ¹(L₁(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