Optimally Sparse Representations of 3D Data with $C^2$ Surface Singularities Using Parseval Frames of Shearlets
Kanghui Guo, Demetrio Labate · SIAM Journal on Mathematical Analysis · 2012
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 $\mathbb{R}^3$ which are $C^2$-regular away from discontinuities along $C^2$ surfaces. As a consequence, we show that, within this class of functions, the N-term approximation $f_N^S$ obtained by selecting the N largest coefficients of the shearlet expansion of f satisfies the asymptotic estimate $\|f-f_N^S\|_2^2 \asymp N^{-1} (\log N)^2$ as $N \to \infty$. 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 Candès and Donoho using curvelets and is the first nonadaptive construction to provide provably (nearly) optimal representations for a large class of 3D data.