|Polyharmonic Splines and Wavelets|
Investigators: Dimitri Van De Ville, Thierry Blu
Summary: Polyharmonic B-splines are localized versions of radial basis functions. They can be designed to be nearly isotropic and Gaussian-like. They can also generate fractional wavelet bases on arbitrary lattices. The corresponding wavelets behave like multiscale fractional iterates of the Laplacian operator.
Polyharmonic B-spline functions are non-separable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic B-splines do not converge to a Gaussian as the order parameter increases, as opposed to their separable B-spline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic B-splines.
Next, we use polyharmonic B-splines to build multidimensional wavelet bases. Therefore, we focus on the two-dimensional quincunx subsampling scheme. This configuration is of particular interest for image processing, because it yields a finer scale progression than the standard dyadic approach. However, up to now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from scaling functions that are the polyharmonic B-splines—thus their explicit form is known—to derive a family of polyharmonic spline wavelets that correspond to different flavors of the semi-orthogonal wavelet transform (e.g., orthonormal, B-spline, dual). The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic B-spline wavelet converges to a combination of four Gabor atoms which are well-separated in the frequency domain. We also show that these wavelets are nearly isotropic and that they behave as an iterated Laplacian operator at low frequencies. We describe an efficient FFT-based implementation of the discrete wavelet transform based on polyharmonic B-splines.
Collaboration: Prof. Michael Unser
D. Van De Ville, T. Blu, B. Forster, M. Unser, "Isotropic-Polyharmonic B-Splines and Wavelets," Proceedings of the 2004 IEEE International Conference on Image Processing (ICIP'04), Singapore, Singapore, October 24-27, 2004, pp. 661-664.
S. Tirosh, D. Van De Ville, M. Unser, "Polyharmonic Smoothing Splines for Multi-Dimensional Signals with 1 ⁄ ||ω||τ-Like Spectra," Proceedings of the Twenty-Ninth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'04), Montréal QC, Canada, May 17-21, 2004, pp. III-297-III-300.