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Polyharmonic B-Spline Wavelets: From Isotropy to Directionality

D. Van De Ville, T. Blu, B. Forster, M. Unser

Invited talk, Advanced Concepts for Intelligent Vision Systems (ACIVS'06), Antwerp, Kingdom of Belgium, September 18-21, 2006.



Polyharmonic B-splines are excellent basis functions to build multidimensional wavelet bases. These functions are nonseparable, multidimensional generators 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 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 until now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from the scaling functions, which are the polyharmonic B-splines and, as such, explicitly known, and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi-orthogonal wavelet transform; e.g., orthonormal, B-spline, and 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 fast Fourier transform-based implementation of the discrete wavelet transform based on polyharmonic B-splines. Finally, we propose a new way to build directional wavelets using modified polyharmonic B-splines. This approach benefits from the previous results (construction of the wavelet filters, fast implementation,…) but allows one to recover directional information about the edges from the (complex-valued) wavelet coefficients.


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