Rotation invariant, Riesz bases of directional wavelets
type: Item
instance of: scholarly article
title: Rotation invariant, Riesz bases of directional wavelets
zbMATH Open document ID: 1402.42045
author: Sylvain Durand
published in: Applied and Computational Harmonic Analysis
publication date: 2018-10-29T00:00:00Z
review text: Known directional wavelets are generated by a single mother wavelet in the Schwartz space \(S(\mathbb R^2)\). This article addresses the issue of designing bases for \(L^2(\mathbb R^2)\) that are generated by translations, rotations and dilations of a single mother wavelet. The results are generated with a new approach. The proposed construction simplifies previous ones by setting an odd number of directions and by choosing properly the phase of the Fourier transform of a single mother wavelet. A large part of the article is devoted to the proof of theorems that give sufficient conditions for a single mother wavelet to generate a Riesz sequence and a Riesz basis for \(L^2(\mathbb R^2)\). An example of a Riesz sequence whose restriction to each scale is orthonormal is given. The theoretical results are confirmed by numerical experiments where a discrete directional wavelet transform is introduced.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 46E20
zbMATH DE Number: 6967583
zbMATH Keywords: directional wavelets
zbMATH Keywords: Riesz basis
zbMATH Keywords: Schwartz space
zbMATH Keywords: space of square summable functions
zbMATH Keywords: Fourier transform
zbMATH Keywords: discrete directional wavelet transform
reviewed by: V. L. Leontiev
describes a project that uses: DT-CWT
MaRDI profile type: Publication
full work available at URL: j.acha.2017.04.001
OpenAlex ID: W2605893748
cites work: Two-dimensional directional wavelets and the scale-angle representation
cites work: Matricial filters and crystallographic composite dilation wavelets
cites work: Fast Discrete Curvelet Transforms
cites work: Ridgelets: a key to higher-dimensional intermittency?
cites work: Biorthogonal bases of compactly supported wavelets
cites work: Compactly supported bidimensional wavelet bases with hexagonal symmetry
cites work: Q3374125
cites work: Orthonormal Ridgelets and Linear Singularities
cites work: M-band filtering and nonredundant directional wavelets
cites work: On the Construction of Discrete Directional Wavelets: HDWT, Space-Frequency Localization, and Redundancy Factor
cites work: Q5486953
cites work: Wavelets with composite dilations and their MRA properties
cites work: Q5493732
cites work: Some simple Haar-type wavelets in higher dimensions
cites work: Wavelets with crystal symmetry shifts
cites work: Orthogonal bandelet bases for geometric images approximation
cites work: Multiresolution direction filterbanks: theory, design, and applications
cites work: The Double-Density Dual-Tree DWT
cites work: Directional wavelet bases constructions with dyadic quincunx subsampling
DOI: 10.1016/J.ACHA.2017.04.001
description: scientific article; zbMATH DE number 6967583