Math. Model. Nat. Phenom.
Volume 8, Number 1, 2013Harmonic analysis
|Page(s)||82 - 105|
|Published online||28 January 2013|
The Construction of Smooth Parseval Frames of Shearlets
1 Department of Mathematics, Missouri
State University, Springfield, Missouri
2 Department of Mathematics, University of Houston, Houston, Texas 77204, USA
∗ Corresponding author. E-mail: firstname.lastname@example.org
The shearlet representation has gained increasing recognition in recent years as a framework for the efficient representation of multidimensional data. This representation consists of a countable collection of functions defined at various locations, scales and orientations, where the orientations are obtained through the use of shear matrices. While shear matrices offer the advantage of preserving the integer lattice and being more appropriate than rotations for digital implementations, the drawback is that the action of the shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, in the standard construction, a Parseval frame of shearlets is obtained by combining different systems of cone-based shearlets which are projected onto certain subspaces of L2(ℝD) with the consequence that the elements of the shearlet system corresponding to the boundary of the cone regions lose their good spatial localization property. In this paper, we present a new construction yielding smooth Parseval frame of shearlets for L2(ℝD). Specifically, all elements of the shearlet systems obtained from this construction are compactly supported and C∞ in the frequency domain, hence ensuring that the system has also excellent spatial localization.
Mathematics Subject Classification: 42C15 / 42C40
Key words: affine systems / nonlinear approximations / shearlets / sparse approximations / wavelets
© EDP Sciences, 2013
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