ShearLab : A rational design of a digital parabolic scaling algorithm

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Author(s)

Detail(s)

Original languageEnglish
Pages (from-to)1291-1332
Journal / PublicationSIAM Journal on Imaging Sciences
Volume5
Issue number4
Publication statusPublished - 2012
Externally publishedYes

Abstract

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is threefold: We first develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implies that shearlet theory provides a unified treatment of both the continuum and digital realms. Second, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying MATLAB toolbox called ShearLab (www.ShearLab.org) is provided. And, third, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform. © 2012 Society for Industrial and Applied Mathematics.

Research Area(s)

  • Curvelets, Digital shearlet system, Directional representation system, Fast digital shearlet transform, Parabolic scaling, Performance measures, Software package, Tight frames

Citation Format(s)

ShearLab : A rational design of a digital parabolic scaling algorithm. / Kutyniok, Gitta; Shahram, Morteza; Zhuang, Xiaosheng.

In: SIAM Journal on Imaging Sciences, Vol. 5, No. 4, 2012, p. 1291-1332.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review