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1 Beyond wavelets : New image representation paradigms

It is by now a well-established fact that the usual two-dimensional tensor product wavelet bases are not optimal for representing images consisting of different regions of smoothly varying grey values, separated by smooth boundaries. The chapter starts with a discussion of this phenomenon from a nonlinear approximation point of view, and then proceeds to describe approaches that have been suggested as a remedy. The methods can be sorted roughly into two groups: Adaptive geometry-based approaches such as wedgelets and related constructions on one hand, and directional frames, such as curvelets or ridgelets, on the other. We discuss wedgelets and curvelets in more details, as representatives of the different branches. These systems are first described in the continuous setting, and their construction is motivated by a discussion of their nonlinear approximation properties. We then present digital implementations of the schemes. For wedgelets and related transforms, we present a new method which results in a significant speedup, in comparison to preexisting implementations. We also give a short description of the contourlet approach to discrete curvelets. In the last section, we present the results of nonlinear approximation experiments, comparing wedgelets, contourlets and wavelets, and comment on the potential of the new techniques for image coding.