1 Numerical geometry of non-rigid shapes Projects Quasi-isometries Project 1 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry.

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1 Numerical geometry of non-rigid shapes Projects Quasi-isometries Project 1 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

2 Numerical geometry of non-rigid shapes Projects Relaxation of isometry We have seen two relaxations of the notion of isometry Relative distortion (bi-Lipschitz map): Absolute distortion (almost-isometry): Tool to quantify shape dissimilarity Can be computed using generalization of MDS

3 Numerical geometry of non-rigid shapes Projects Quasi-isometries Another relaxation: quasi-isometry Goal: compute shape dissimilarity in the sense of quasi-isometry Multi-criteria trade-off Set-valued distance

4 Numerical geometry of non-rigid shapes Projects Shape matching under the influence of noise Project 2 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

5 Numerical geometry of non-rigid shapes Projects Sensitivity to noise Comparison of intrinsic geometries = invariant shape similarity Goal: study how intrinsic geometry of a shape is sensitive to noise Perturbation of vertices Perturbation of triangulation & topology How do they influence geodesic distances? How are different stress functions sensitive to noise?

6 Numerical geometry of non-rigid shapes Projects Intersection-free as-isometric-as-possible shape morphing Project 3 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

7 Numerical geometry of non-rigid shapes Projects Intersection-free as-isometric-as-possible morph Given two nearly isometric shapes and find a continuous family of as isometric as possible shapes. Define a Riemannian metric on space of shapes quantifying local distortion Find geodesic connecting two shapes = minimum distortion morph Caveat: intermediate shapes might self-intersect On the other hand: there exists parametrizations guaranteeing intersection-free morphing Goal: combine two approaches in simplified 2D setting

8 Numerical geometry of non-rigid shapes Projects Sensitivity of Laplace-Beltrami operator to boundary conditions Project 4 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009