BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

Harmonic Singular Integrals and Steerable Wavelets in L2(ℝd)

J.P. Ward, M. Unser

Applied and Computational Harmonic Analysis, vol. 36, no. 2, pp. 183-197, March 2014.

Here we present a method of constructing steerable wavelet frames in L2(ℝd) that generalizes and unifies previous approaches, including Simoncelli's pyramid and Riesz wavelets. The motivation for steerable wavelets is the need to more accurately account for the orientation of data. Such wavelets can be constructed by decomposing an isotropic mother wavelet into a finite collection of oriented mother wavelets. The key to this construction is that the angular decomposition is an isometry, whereby the new collection of wavelets maintains the frame bounds of the original one. The general method that we propose here is based on partitions of unity involving spherical harmonics. A fundamental aspect of this construction is that Fourier multipliers composed of spherical harmonics correspond to singular integrals in the spatial domain. Such transforms have been studied extensively in the field of harmonic analysis, and we take advantage of this wealth of knowledge to make the proposed construction practically feasible and computationally efficient. 

@ARTICLE(http://bigwww.epfl.ch/publications/ward1401.html,
AUTHOR="Ward, J.P. and Unser, M.",
TITLE="Harmonic Singular Integrals and Steerable Wavelets in
	$L_{2}({\mathbb{R}}^{d})$",
JOURNAL="Applied and Computational Harmonic Analysis",
YEAR="2014",
volume="36",
number="2",
pages="183--197",
month="March",
note="")

© 2014 Elsevier. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from Elsevier. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.