TABLE OF CONTENTS One can argue that functions of bounded variation do not adequately model images. Indeed, a function of bounded variation is either the characteristic function of a domain whose boundary has a finite length, or it is an average of such functions. This atomic decomposition is provided by the co-area identity. Modeling objects with characteristic functions of domains with finite length boundaries may be inappropriate, since the objects we have in mind are probably not that complicated. Donoho decided to describe an image as a collection of objects delimited by smooth boundaries instead of merely rectifiable ones.
If one wants to efficiently represent (or compress) smooth domains, standard isotropic wavelets are not optimal. A better algorithm relies on an efficient description of the boundary, and this calls for orthonormal bases that can efficiently represent elongated objects, such as the arc of a circle. No one knew how to do this until Donoho constructed a remarkable orthonormal basis that was designed to provide a sparse representation for objects having arbitrary large eccentricities. Donoho's construction improved previous work by E. Cand s. We are going to describe this basis, and we begin with one of our main themes.
When constructing a wavelet basis, we should return to the issue raised by Jean Ville: Should we first segment the frequency domain, or should we use some bases that are built on a segmentation of the time (or space) domain? The construction of Donoho's basis uses both strategies.
Let ? = ( ?
