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The Construction of Smooth Parseval Frames ofShearlets

Published online by Cambridge University Press:  28 January 2013

K. Guo  and
D. Labate
K. Guo
Affiliation:
Department of Mathematics, Missouri State University, Springfield, Missouri 65804, USA
D. Labate*
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204, USA
*
∗ Corresponding author. E-mail: dlabate@math.uh.edu
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Abstract

The shearlet representation has gained increasing recognition in recent years as aframework for the efficient representation of multidimensional data. This representationconsists of a countable collection of functions defined at various locations, scales andorientations, where the orientations are obtained through the use of shear matrices. Whileshear matrices offer the advantage of preserving the integer lattice and being moreappropriate than rotations for digital implementations, the drawback is that the action ofthe shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, inthe standard construction, a Parseval frame of shearlets is obtained by combiningdifferent systems of cone-based shearlets which are projected onto certain subspaces ofL2(ℝD) with the consequence thatthe elements of the shearlet system corresponding to the boundary of the cone regions losetheir good spatial localization property. In this paper, we present a new constructionyielding smooth Parseval frame of shearlets forL2(ℝD). Specifically, allelements of the shearlet systems obtained from this construction are compactly supportedand C∞ in the frequency domain, hence ensuring that the systemhas also excellent spatial localization.

Keywords

affine systemsnonlinear approximationsshearletssparse approximationswavelets
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 82 - 105
Copyright
© EDP Sciences, 2013

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