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Séminaire du département Images et Signal du 07/12/2016 à 15h00


Kernel-based Image Reconstruction from Scattered Radon Data

Intervenant : Stefano DE MARCHI, Department of Mathematics, University of Padova

Lieu : Salle Chartreuse


Résumé :

In this talk we present a novel kernel-based reconstruction method for image reconstruction from scattered Radon data. Our reconstruction relies on generalized Hermite-Birkhoff interpolation by positive definite kernel functions. For radially symmetric kernels, however, a straightforward application of generalized Hermite-Birkhoff interpolation fails to work (cf. [1, 2]). For the wellposedness of the reconstruction scheme, we introduce anisotropic positive definite kernels, which are symmetric but not radially symmetric. We prove the well-posedness of the resulting reconstruction scheme.  Moreover, we introduce a novel concept for the construction of anisotropic positive definite kernels, before we develop concrete examples for suitable combinations of radial weight functions and commonly used positive definite kernels. This leads to a very flexible image reconstruction method, which works for arbitrary distributions of Radon lines and allows also to select the most significant ones by a thinning approach based on Newton’s bases. The good performance of the proposed kernel-based image reconstruction method is supported by numerical examples and comparisons. This is a joint work with A. Iske (Hamburg), G. Santin (Stuttgart) and A. Sironi (Lousanne).

[1] S. De Marchi, A. Iske and A. Sironi: Kernel-based Image Reconstruction from Scattered Radon Data, Dolomites Res. Notes Approx. 9(2016), special issue on Kernel-based Methods and Function Approximation 2016, pp. 19–31. 
[2] S. De Marchi, A. Iske and G. Santin: Kernel-based Image Reconstruction from Scattered Radon Data by Anisotropic Positive Definite Functions, in preparation.

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