Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Optimally sparse 3D approximations using shearlet representations

Pages: 125 - 137, January 2010      doi:10.3934/era.2010.17.125

       Abstract        References        Full Text (228.1K)              Related Articles       

Kanghui Guo - Department of Mathematics, Missouri State University, Springfield, Missouri 65804, United States (email)
Demetrio Labate - Department of Mathematics, University of Houston, Houston, Texas 77204, United States (email)

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 $f_N^S$ obtained by selecting the $N$ largest coefficients from the shearlet expansion of $f$ satisfies the asymptotic estimate

||$f-f_N^S$||$_2^2$ ≍ $N^{-1} (\log N)^2, as N \to \infty.$

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 3-D 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.

Keywords:  Affine systems, curvelets, shearlets, sparsity, wavelets.
Mathematics Subject Classification:  42C15, 42C40.

Received: September 2010;      Available Online: October 2010.