Optimally sparse approximations of 3D functions by compactly supported shearlet frames

TL;DR

This paper introduces a new 3D shearlet frame system that achieves near-optimal sparse approximation of anisotropic features in 3D data, advancing the efficiency of representing complex multidimensional structures.

Contribution

It develops a pyramid-adapted, hybrid shearlet system for 3D data that provides nearly optimal sparse approximations for a new class of anisotropic cartoon-like images.

Findings

Shearlet systems achieve near-optimal sparsity for 3D anisotropic data.

The introduced shearlet frames effectively capture discontinuities on smooth surfaces.

The method improves data compression and analysis of 3D structures.

Abstract

We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized three-dimensional cartoon-like images. This function class will have two smoothness parameters: one parameter \beta controlling classical smoothness and one parameter \alpha controlling anisotropic smoothness. The class then consists of piecewise C^\beta-smooth functions with discontinuities on a piecewise C^\alpha-smooth surface. We introduce a pyramid-adapted, hybrid shearlet system for the three-dimensional setting and construct frames for L^2(R^3) with this particular shearlet structure. For the smoothness range 1<\alpha =< \beta =< 2 we show that pyramid-adapted shearlet systems provide a nearly optimally sparse approximation rate within the generalized cartoon-like image model class measured by means of non-linear N-term approximations.