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ShearLab: a rational design of a digital parabolic scaling algorithm. (English) Zbl 1257.42048

Summary: 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. 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.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
65K99 Numerical methods for mathematical programming, optimization and variational techniques
65T60 Numerical methods for wavelets
65T99 Numerical methods in Fourier analysis
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

Software:

Matlab; ShearLab
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