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Sparsity Equivalence of Anisotropic Decompositions

Anisotropic decompositions using representation systems such as curvelets, contourlet, or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the concept of sparsity equivalence, we aim to provide a framework which allows categorization of the ability for sparse approximations of representation systems. This framework, in particular, enables transferring results on sparse approximations from one system to another. We demonstrate this concept for the example of curvelets and shearlets, and discuss how this viewpoint immediately leads to novel results for both systems.