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In this paper we show that the shearlets, an a-ne-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2{dimensional functions f which are C2 except for discontinuities along C2 curves. More speciflcally, if fS N is the N{term reconstruction of f obtained by using the N largest coe-cients in the shearlet representation, then the asymptotic approximation error decays as kf ¡ fS Nk 2 2 ‡ N ¡2 (logN)3; N ! 1; which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate N¡1 associated with wavelet approximations. Unlike the curvelets, that have similar sparsity properties, the shearlets form an a-ne-like system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations and translations to a single well-localized window function.

Optimally Sparse Multidimensional Representation Using Shearlets