Wavelet steerability and the higher-order Riesz transform

Unser, Michael; Van De Ville, Dimitri

doi:10.1109/TIP.2009.2038832

Scientific article

English

Wavelet steerability and the higher-order Riesz transform

Contributors

Unser, Michael; Van De Ville, Dimitri

Published in

IEEE transactions on image processing, vol. 19, no. 3, p. 636-652

Publication date

2010

Abstract

Our main goal in this paper is to set the foundations of a general continuous-domain framework for designing steerable, reversible signal transformations (a.k.a. frames) in multiple dimensions ( d ≥ 2). To that end, we introduce a self-reversible, Nth-order extension of the Riesz transform. We prove that this generalized transform has the following remarkable properties: shift-invariance, scale-invariance, inner-product preservation, and steerability. The pleasing consequence is that the transform maps any primary wavelet frame (or basis) of [Formula: see text] into another "steerable" wavelet frame, while preserving the frame bounds. The concept provides a functional counterpart to Simoncelli's steerable pyramid whose construction was primarily based on filterbank design. The proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions; it also yields a perfect reconstruction filterbank algorithm. We illustrate the method with the design of a novel family of multidimensional Riesz-Laplace wavelets that essentially behave like the N th-order partial derivatives of an isotropic Gaussian kernel.

UNIGE affiliation entities

Faculté de médecine / Section de médecine clinique / Département de radiologie et informatique médicale

UNIGE research groups

Traitement d'images médicales (893)

Citation (ISO format)

UNSER, Michael, VAN DE VILLE, Dimitri. Wavelet steerability and the higher-order Riesz transform. In: IEEE transactions on image processing, 2010, vol. 19, n° 3, p. 636–652. doi: 10.1109/TIP.2009.2038832

Identifiers

PID : unige:20580
DOI : 10.1109/TIP.2009.2038832
PMID : 20031498

Journal ISSN

1057-7149

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29/10/2024 19:50:44

Wavelet steerability and the higher-order Riesz transform

Contributors

Published in

Publication date

Abstract

UNIGE affiliation entities

UNIGE research groups

Citation (ISO format)

Identifiers

Journal ISSN

Technical informations

Creation

First validation

Update

Status update

Last indexation