Optimally sparse 3D approximations using shearlet representations
Kanghui Guo - Department of Mathematics, Missouri State University, Springfield, Missouri 65804, 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.
Keywords: Affine systems, curvelets, shearlets, sparsity, wavelets.
Received: September 2010; Available Online: October 2010. |