Wavelet Design Using Polyharmonic B-splines
Matthieu Guerquin-Kern, Département EEA, Ecole Normale Supérieure de Cachan - France

Seminar • 19 August 2005 • BM 4.235

Abstract
The Isotropic Polyharmonic B-spline family has recently been defined, and used for wavelet design with a two-dimensional quincunx subsampling scheme. In this presentation, we focus on the two-dimensional dyadic scheme, which reduces the number of samples at each iteration by a factor of 4. Therefore, we need to jointly design 3 wavelets instead of 1 in the quincunx case. First, we characterise the wavelet space by 3 so-called pre-wavelets when using the polyharmonic B-spline as a scaling function. Then, we clarify the orthonormality conditions on the wavelet base. We propose a matrix-based method to ensure these conditions in order to construct orthonormal bases for the wavelet space. The design can be guided by a desired tiling in the frequency domain for the support of wavelets. It can be imposed by well-chosen symmetry relations for the solution. According to these mathematical solutions, we propose a Matlab implementation to compute wavelet filters. Finally, we implement the wavelet transform using an iterated filterbank algorithm. We apply the transform to test images.