TABLE OF CONTENTS Several problems have been raised in this chapter. The first consisted of defining the class of functions (signals, images) whose wavelet expansions are sparse, in one sense or another. These functions are adequately compressed with wavelets. Depending on the norm that was used to measure the appropriate approximation, several characterizations in terms of Besov spaces have been presented.
The second message of this chapter seems to be a success story for wavelet analysis: Whenever the a priori information on a given class of signals or images can be formulated as a bound on a Besov norm, then wavelet shrinkage provides an optimal denoising. On the other hand, if u is a smooth function inside finitely many domains with jump discontinuities across their boundaries, then one should shrink the ridgelet coefficients of f = u + ? v to recover u ( v is a standard Gaussian white noise).
These two statements seem to be contradictory, but they become consistent if one returns to the definition of the worst risk. This worst risk is the supremum of 
taken over the Besov ball ? u ? B ? C. Such a supremum can be attained for certain intricate functions u that do not correspond to our notion of a cartoon image. Besov balls are indeed very large sets. With the availability of ridgelets, new algorithms for optimal denoising should soon be available.
Another message is that there continues to be a...
