Optimally sparse 3D approximations using shearlet representations
Demetrio Labate, Kanghui Guo · Electronic Research Announcements of the American Mathematical Society · 2010
This paper introduces a new Parseval frame, based on the 3-Dshearlet representation, which is especially designed to capturegeometric features such as discontinuous boundaries with very highefficiency. We show that this approach exhibits essentially optimalapproximation properties for 3-D functions $f$ which are smoothaway from discontinuities along $C^2$ surfaces. In fact, the $N$term approximation $f_N^S$ obtained by selecting the $N$ largestcoefficients from the shearlet expansion of $f$ satisfies theasymptotic estimate ||$f-f_N^S$||$_2^2$ ≍ $N^{-1} (\log N)^2, asN \to \infty.$ Up to the logarithmic factor,this is the optimal behavior for functions in this class andsignificantly outperforms wavelet approximations, which only yieldsa $N^{-1/2}$ rate. Indeed, the wavelet approximation rate was thebest published nonadaptive result so far and the result presented inthis paper is the first nonadaptive construction which is provablyoptimal (up to a loglike factor) for this class of 3-D data. Our estimate is consistent with the corresponding2-D (essentially) optimally sparse approximation results obtainedby the authors using 2-D shearlets and by Candès and Donoho usingcurvelets.