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Stable Spectral Mesh Filtering
Artiom Kovnatsky1, Michael M. Bronstein1, and Alexander M. Bronstein2
1Institute of Computational Science, Faculty of Informatics, Università della Svizzera Italiana, Lugano, Switzerland
artiom.kovnatsky@usi.ch
michael.bronstein@usi.ch
2School of Electrical Engineering, Tel Aviv University, Israel
bron@eng.tau.ac.il
Abstract. The rapid development of 3D acquisition technology has brought with itself the need to perform standard signal processing operations such as filters on 3D data. It has been shown that the eigenfunctions of the Laplace-Beltrami operator (manifold harmonics) of a surface play the role of the Fourier basis in the Euclidean space; it is thus possible to formulate signal analysis and synthesis in the manifold harmonics basis. In particular, geometry filtering can be carried out in the manifold harmonics domain by decomposing the embedding coordinates of the shape in this basis. However, since the basis functions depend on the shape itself, such filtering is valid only for weak (near all-pass) filters, and produces severe artifacts otherwise. In this paper, we analyze this problem and propose the fractional filtering approach, wherein we apply iteratively weak fractional powers of the filter, followed by the update of the basis functions. Experimental results show that such a process produces more plausible and meaningful results.
Keywords: Computational Geometry and Object Modeling, Hierarchy and geometric transformations, Laplace-Beltrami operator, 3D Mesh filtering
LNCS 7583, p. 83 ff.
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© Springer-Verlag Berlin Heidelberg 2012