Optimally sparse multidimensional representations using shearlets

Abstract. In this paper we show that shearlets, an affine-like system of functions recently intro-duced 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 specifically, if fSN 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 − fSN‖22 N−2 (logN)3, N → ∞, 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