Optimally sparse 3D approximations using shearlet representations

Abstract

This paper introduces a new Parseval frame, based on the 3-D shearlet representation, which is especially designed to capture geometric features such as discontinuous boundaries with very high efficiency. We show that this approach exhibits essentially optimal approximation properties for 3-D functions f which are smooth away from discontinuities along C 2 surfaces. In fact, the N term approximation f S N obtained by selecting the N largest coefficients from the shearlet expansion of f satisfies the asymptotic estimate f − f S N 2 2 N −1 (log N) 2 , as N → ∞. Up to the logarithmic factor, this is the optimal behavior for functions in this class and significantly outper-forms wavelet approximations, which only yields a N −1/2 rate. Indeed, the wavelet approximation rate was the best published nonadaptive result so far and the result presented in this paper is the first nonadaptive construction which is provably optimal (up to a loglike factor) for this class of 3D data. Our estimate is consistent with the corresponding 2-D (essentially) optimally sparse approximation results obtained by the authors using 2-D shearlets and by Candès and Donoho using curvelets.