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This paper introduces a Parseval frame of shearlets for the representation of three-dimensional (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 C²2 surfaces. As a consequence, we show that, within this class of functions, the N-term approximation fNS obtained by selecting the N largest coefficients of the shearlet expansion of f satisfies the asymptotic estimate ... This asymptotic behavior significantly outperforms wavelet and Fourier series approximations, which yield an approximation rate of only 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 Candes and Donoho using curvelets and is the first nonadaptive construction to provide provably (nearly) optimal representations for a large class of 3D data. [ABSTRACT FROM AUTHOR]

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 ƒ which are C2 except for discontinuities along C2 curves. More specifically, if ƒSN is the N-term reconstruction of ƒ obtained by using the N largest coefficients in the shearlet representation, then the asymptotic approximation error decays as ||ƒ-ƒNS||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. [ABSTRACT FROM AUTHOR]