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 C² except for discontinuities along C² curves. More specifically, if fNS 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-fNS₂² N⁻² (log N)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.