Polyharmonic Splines and Wavelets 
Investigators: Dimitri Van De Ville, Thierry Blu 

Summary: Polyharmonic Bsplines are localized versions of radial basis functions. They can be designed to be nearly isotropic and Gaussianlike. They can also generate fractional wavelet bases on arbitrary lattices. The corresponding wavelets behave like multiscale fractional iterates of the Laplacian operator. 

Polyharmonic Bspline functions are nonseparable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic Bsplines do not converge to a Gaussian as the order parameter increases, as opposed to their separable Bspline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic Bsplines.
Next, we use polyharmonic Bsplines to build multidimensional wavelet bases. Therefore, we focus on the twodimensional 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 Bsplines—thus their explicit form is known—to derive a family of polyharmonic spline wavelets that correspond to different flavors of the semiorthogonal wavelet transform (e.g., orthonormal, Bspline, dual). The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic Bspline wavelet converges to a combination of four Gabor atoms which are wellseparated 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 FFTbased implementation of the discrete wavelet transform based on polyharmonic Bsplines. 

Collaboration: Prof. Michael Unser 



[2]  D. Van De Ville, T. Blu, B. Forster, M. Unser, "IsotropicPolyharmonic BSplines and Wavelets," Proceedings of the 2004 IEEE International Conference on Image Processing (ICIP'04), Singapore, Singapore, October 2427, 2004, pp. 661664.

[3]  S. Tirosh, D. Van De Ville, M. Unser, "Polyharmonic Smoothing Splines for MultiDimensional Signals with 1 ⁄ ω^{τ}Like Spectra," Proceedings of the TwentyNinth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'04), Montréal QC, Canada, May 1721, 2004, pp. III297III300.


