Can one hear Shape?.
M. Reuter.
PAMM Proceedings of GAMM07 and ICIAM07, Vol 7, Issue 1, 6th International Congress of Industrial and Applied Mathematics, SIAM, October 2008.
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The question "Can one hear the shape of a drum" has been asked in several contexts before (e.g., by Bers and Kac). It is a pictorial way of asking if the eigenvalues of the Laplacian on a given domain completely characterize its shape, in other words, if the spectrum is a complete shape descriptor (which it is not in general). In this talk we will give an overview on how the computation of the spectra can be accomplished using FEM for manifolds in 2D and 3D (e.g. iso-surfaces, boundary representations, solid bodies, vector fields...) with the Dirichlet and Neumann boundary condition. We demonstrate that it is computational feasible to numerically extract geometric properties (volume, area, boundary length and even the Euler characteristic) from the first eigenvalues. Since the spectrum contains geometrical information and since it is an isometry invariant and therefore independent of the object's representation, parametrization, spatial position, and optionally of its size, it is optimally suited to be used as a fingerprint (Shape-DNA) in contemporary computer graphics applications like database retrieval, quality assessment, and shape matching in fields like CAD, medicine or engineering. hide |