Optimally Sparse Representations of 3D Data with C2 Surface Singularities Using Parseval Frames of Shearlets

This paper introduces a Parseval frame of shearlets for the representation of 3D data, which is especially designed to handle geometric features such as discontinuous boundaries with very high efficiency. This system of 3D shearlets forms a multiscale pyramid of well-localized waveforms at various locations and orientations, which become increasingly thin and elongated at fine scales. We prove that this 3D shearlet construction provides essentially optimal sparse representations for functions on R3 which are C2-regular away from discontinuities along C2 surfaces. As a consequence, we show that within this class of functions the N -term approximation fS N obtained by selecting the N largest coefficients of the shearlet expansion of f satisfies the asymptotic estimate ∥f − f N∥ 2 2 ≍ N−1(logN)2, as N → ∞. This asymptotic behavior significantly outperforms wavelet and Fourier series approximations which only yield an approximation rate of O(N−1/2) and O(N−1/3), respectively. This result extends to the 3D setting the (essentially) optimally sparse approximation results obtained by the authors using 2D shearlets and by Candès and Donoho using curvelets and is the first nonadaptive construction to provide provably (nearly) optimal representations for a large class of 3-dimensional data.