Optimally sparse multidimensional representation using shearlets

In this paper we show that shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions f which are C-2 except for discontinuities along C-2 curves. More specifically, if f(N)(S) 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 parallel to f - f(N)(S)parallel to 2 asymptotic to 2 N-2 (logN)(3), N -> infinity, which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate N-1 associated with wavelet approximations. Unlike curvelets, which have similar sparsity properties, 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.