Optimally Sparse 3d Approximations Using Shearlet Representations

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 C2 surfaces. In fact, the N term approximation fS N obtained by selecting the N largest coefficients from the shearlet expansion of f satisfies the asymptotic estimate ‖f − f N‖2 3 N−1(log N), as N →∞. Up to the logarithmic factor, this is the optimal behavior for functions in this class and significantly outperforms 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.