Optimally Sparse Multidimensional Representation Using Shearlets

In this paper we show that the shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2–dimensional functions f that are C2 except for discontinuities along C2 curves. More specifically, if fS N is the N–term reconstruction of f obtained by using the N largest coefficients in the shearlet representation, then the asymptotic approximation error decays as ‖f − f N‖2 3 N−2 (log N), N →∞, 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 affine-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.