Linearized Riesz Transform and Quasi-Monogenic Shearlets

  • The only quadrature operator of order two on \(L_2 (\mathbb{R}^2)\) which covaries with orthogonal transforms, in particular rotations is (up to the sign) the Riesz transform. This property was used for the construction of monogenic wavelets and curvelets. Recently, shearlets were applied for various signal processing tasks. Unfortunately, the Riesz transform does not correspond with the shear operation. In this paper we propose a novel quadrature operator called linearized Riesz transform which is related to the shear operator. We prove properties of this transform and analyze its performance versus the usual Riesz transform numerically. Furthermore, we demonstrate the relation between the corresponding optical filters. Based on the linearized Riesz transform we introduce finite discrete quasi-monogenic shearlets and prove that they form a tight frame. Numerical experiments show the good fit of the directional information given by the shearlets and the orientation ob- tained from the quasi-monogenic shearlet coefficients. Finally we provide experiments on the directional analysis of textures using our quasi-monogenic shearlets.

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Author:Sören Häuser, Bettina Heise, Gabriele Steidl
URN (permanent link):urn:nbn:de:hbz:386-kluedo-35961
Document Type:Preprint
Language of publication:English
Publication Date:2013/08/21
Year of Publication:2013
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2013/08/22
Tag:Riesz Transform; Shearlets; instantaneous phase; local orientation; monogenic signals
Number of page:23
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):65-XX NUMERICAL ANALYSIS / 65Txx Numerical methods in Fourier analysis / 65T99 None of the above, but in this section
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 10.09.2012