Homogeneous approximation property for continuous shearlet transforms in higher dimensions

Yu Su,Wenting Su,Wanchang Zhang

doi:10.1186/s13660-016-1116-y

Abstract

This paper is concerned with the generalization of the homogeneous approximation property (HAP) for a continuous shearlet transform to higher dimensions. First, we give a pointwise convergence result on the inverse shearlet transform in higher dimensions. Second, we show that every pair of admissible shearlets possess the HAP in the sense of $L^{2}(R^{d})$ . Third, we give a sufficient condition for the pointwise HAP to hold, which depends on both shearlets and functions to be reconstructed.

Highlights

Modern technology allows for easy creation, transmission, and storage of huge amounts of higher-dimensional data
The aim of this paper is threefold. (i) We give a pointwise convergence result on the inverse shearlet transform in higher dimensions. (ii) We show that every pair of admissible shearlets possess the homogeneous approximation property (HAP) in the sense of L (Rd )
(iii) We give a sufficient condition for the pointwise HAP to hold, which depends on both shearlets and functions to be reconstructed in higher dimensions

Summary

Introduction

Modern technology allows for easy creation, transmission, and storage of huge amounts of higher-dimensional data. The key problem is to extract the relevant information from these huge amounts of higher-dimensional data. In this context, one particular problem which is currently in the center of interest is the analysis of directional information in higher dimensions. New direction representation systems have to be developed, such as ridgelets [ ], curvelets [ ], contourlets [ ], shearlets [ , ], and many others. Among all these approaches, the shearlet transform stands out because it is related to group theory, and it has a flexible enough extension to precisely detect the position and orientation of singularities and to provide optimally sparse representations

Objectives

Conclusion