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Spectrum of the Laplacian as a shape-matching tool *August 19, 2010*

*Posted by Sarah in Uncategorized.*

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I like the Laplacian. It’s funny but I just realized that every time I’ve gotten suddenly passionate about an area of mathematics, the Laplacian has been involved. So I enjoyed this paper by a couple of computer graphics guys in Hannover, using the spectrum of the Laplace-Beltrami operator as a “fingerprint” for shape matching.

Measuring the similarity between shapes is obviously a big deal for computer graphics or machine vision, and there are many competing similarity measures — you can use moments or spherical harmonics or Gromov-Hausdorff distance, among many others. But the spectrum of the Laplacian has the particularly neat property that for some shapes (for example, the disc) you can prove that it uniquely determines the shape — any shape with the same spectrum as the disc must in fact be a disc. The spectrum of the Laplacian also determines the dimension of a manifold (determining dimension is a common difficulty, as many procedures for fitting a manifold to data assume the dimension is known.) There are enough handy properties here that the spectrum of the Laplacian seems — in my untutored opinion — to be a very natural way to characterize shapes. (A propos of the previous post, and applications of the Selberg Trace Formula.)

(Possibly related: from the Geomblog.)

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