VIVARIUM Seminar: Polyharmonic wavelets---from isotropy to steerability

July 11, 2008
Location: TASC 9204 West
Presenter: Dimitri Van De Ville, Radiology Department, CIBM Signal Processing Core, University Hospital Geneva


The aim of this presentation is to show the tight link between wavelets, some classical image processing operators and David Marr's theory of early vision. We start by illustrating a general wavelet basis design procedure. Starting from fundamental invariances principles, we identify the corresponding class of operators and the associated Green's functions. Next, we design a single-generator wavelet, in an analytical way, and we show that it yields a semi-orthogonal basis of L_2(\mathbb{R}^2)$, irrespective of the dilation matrix used. Moreover, the wavelet behaves as the chosen operator when applied to data. We also provide an efficient FFT-based filterbank implementation. The design procedure is illustrated for operators that are translation-, scale- and rotation-invariant, which relates to the (iterated) Laplacian. The isotropic polyharmonic B-spline wavelets are ideally suited for the analysis of multidimensional data with fractal characteristics (isotropic differentiation and whitening property) or for applications such as statistical resampling without directional bias.

Next, we consider the rotation covariant case that leads to the class of complex gradient-Laplace operators. Through generalized polyharmonic B-splines, we are able to construct complex-valued steerable wavelets; i.e., a suitable combination of real and imaginary part of the wavelet coefficients allows to obtain the response for any angle. We show how the transform's properties can be improved by introducing a pyramid with mild redundancy. The steerable polyharmonic wavelets has some striking similarities with Marr's theory of vision. We show an image reconstruction algorithm based on multiscale edge characterization only.

Dimitri Van De Ville (M’02) received the M.S. degree in engineering and computer sciences from Ghent University, Ghent, Belgium, in 1998, as well as the Ph.D. degree, in 2002. He obtained a grant as Research Assistant with the Fund for Scientific Research Flanders Belgium (FWO). In 2002, he joined Prof. M. Unser’s Biomedical Imaging Group at the École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland. In December 2005, he became responsible for the Signal Processing Antenna at the University Hospital of Geneva, Geneva, Switzerland, as part of the Centre d’Imagerie Biomédicale (CIBM). His research interests include wavelets, statistical analysis, multidimensional splines, and applications in biomedical imaging, such as functional magnetic resonance imaging, spectroscopy, electro- encephalography, and microscopy.

Dr. Van De Ville serves as an Associate Editor for the IEEE TRANSACTIONS ON IMAGE PROCESSING (since February 2006) and was previously an Associate Editor for IEEE SIGNAL PROCESSING LETTERS (2004–2006). Since 2003, he has also been an Editor and Webmaster of The Wavelet Digest. He is co-chair of the Wavelets XII international conference (August 2007), together with V. Goyal and M. Papadakis. More information can be found at: http://bigwww.epfl.ch/vandeville/.

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