Of crucial importance in computer graphics, shape modeling, medical imaging, bioinformatics, and 3D face recognition is matching, retrieval, correspondence, and  segmentation of non-rigid, deformable shapes. Then, an interesting problem is to obtain a shape representation that is invariant under natural deformations, and, at the same time, contains enough information to perform these shape processing tasks.

To produce such a representation, previous methods used geodesic distances. Unfortunately, geodesic distances are sensitive to local topology changes. As a result, the representations based on them will have limited robustness. Can we avoid using the geodesic distances completely?

Our positive answer to this question is inspired by Bruno Levy's profound observation that the eigenfunctions of the Laplace-Beltrami differential operator “understand the geometry“ — in some sense, they capture the global properties of the surface. Another source of inspiration is the work of Reuter et al., where the eigenvalues of the same operator were used as a shape descriptor.

We combine the Laplace-Beltrami eigenvalues and eigenfunctions to construct the GPS embedding, a surface in the infinite-dimensional space, where the inner product and distance are related to the Green's function. The GPS embedding is invariant under natural deformations of the original surface, and can be used for deformable shape processing, including shape classification, segmentation, matching and correspondence.

For more see the paper  here.

You may also look at the talk slides — PPT and PDF.

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The watertight benchmark, including classification and thumbnails is courtesy of Dainiela Giorgi. See the report:

Daniela Giorgi, Silvia Biasotti, Laura Paraboschi: "Watertight Models Track", Technical Report IMATI-CNR-GE 9/07