Optimally sparse image representation by the easy path wavelet transform
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.42012
DOI: 10.1142/S0219691311004493
author: Stefanie Tenorth
author: Armin Iske
author: Gerlind Plonka
published in: International Journal of Wavelets, Multiresolution and Information Processing
publication date: 2012-06-20T00:00:00Z
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 68U10
Mathematics Subject Classification ID: 94A08
zbMATH DE Number: 6048369
zbMATH Keywords: Spares data representation
zbMATH Keywords: wavelet transform along pathways
zbMATH Keywords: image data compression
zbMATH Keywords: adaptive wavelet bases
zbMATH Keywords: \(N\)-term approximation
describes a project that uses: EPWT
MaRDI profile type: Publication
cites work: Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques
cites work: New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities
cites work: Fast Discrete Curvelet Transforms
cites work: Ten Lectures on Wavelets
cites work: Adaptive Multivariate Approximation Using Binary Space Partitions and Geometric Wavelets
cites work: Image compression by linear splines over adaptive triangulations
cites work: Wedgelets: Nearly minimax estimation of edges
cites work: Optimally Sparse Multidimensional Representation Using Shearlets
cites work: Wavelets with composite dilations
cites work: Multiresolution Representation of Data: A General Framework
cites work: MULTISELECTIVE PYRAMIDAL DECOMPOSITION OF IMAGES: WAVELETS WITH ADAPTIVE ANGULAR SELECTIVITY
cites work: Nonlinear piecewise polynomial approximation beyond Besov spaces
cites work: Bandelet Image Approximation and Compression
cites work: Q3836380
cites work: Geometrical grouplets
cites work: The Easy Path Wavelet Transform: A New Adaptive Wavelet Transform for Sparse Representation of Two-Dimensional Data
cites work: Easy path wavelet transform on triangulations of the sphere
cites work: A New Hybrid Method for Image Approximation Using the Easy Path Wavelet Transform
title: Optimally sparse image representation by the easy path wavelet transform
review text: During the last few years an increasing effort to develop efficient representations of large high-dimensional data, especially for signals, has been done. In the 1D case, wavelets are particular efficient to represent piecewise smooth signals with point singularities. However, in the 2D case tensor product wavelet bases are no longer optimal for the representation of piecewise smooth functions with discontinuities along curves. Very recently, more sophisticated methods have been developed to design approximation schemes for efficient representations of 2D data, in particular for images, where correlations along curves are considered in order to recover the geometry of the given data. As examples for non-adaptive highly redundant function frames with strong anisotropic directional selectivity, curvelets, shearlets and directionlets, among others, have been introduced.NEWLINENEWLINEIn the paper under review, the authors deduce an optimal \(N\)-term approximation \(f_{N}\) for a piecewise Hölder continuous function \(f\) of order \(\alpha\), with \(0<\alpha\leq 1\) with singularities along curves using a locally adaptive Easy Path Wavelet Transform (EPWT) introduced in [\textit{G. Plonka}, Multiscale Model. Simul. 7, No. 3, 1474--1496 (2009; Zbl 1175.65158)]. Indeed, \(||f-f_{N}||_{2} ^{2} \leq C N^{-\alpha}\). The restriction concerning \(\alpha\) is due to the application of the adaptive Haar wavelet basis used for the EPWT. In order to achieve optimal approximation results, some specific side conditions for the path vectors that are used in the EPWT algorithm are required. A strategy for the construction of the path vectors satisfying such side conditions is presented. Finally, the authors deal with the error analysis and the above asymptotically optimal \(N\)-term error estimates for piecewise Hölder continuous functions is deduced.NEWLINENEWLINENotice that, for \(\alpha>1,\) the problem requires a complete different approach since the adaptive multiresolution anal…
reviewed by: Francisco Marcellán
description: scientific article; zbMATH DE number 6048369