1. 1 Numerical geometry of non-rigid shapes Numerical Geometry Numerical geometry of non-rigid shapes Numerical geometry Alexander Bronstein, Michael Bronstein, Ron Kimmel © 2007 All rights reserved

2. 2 Numerical geometry of non-rigid shapes Numerical Geometry Sampling of surfaces Sampled surfaceGeometry image Represent a surface as a cloud of points Parametric surface can be sampled in parametrization domain Cartesian sampling of parametrization domain Surface represented as three matrices

3. 3 Numerical geometry of non-rigid shapes Numerical Geometry Depth images Sampled surfaceDepth image Particular case: Monge parametrization Can be represented as a single matrix (depth image) Typical output of 3D scanners

4. 4 Numerical geometry of non-rigid shapes Numerical Geometry Regular sampling in parametrization domain may be irregular on the surface Depends on geometry and parametrization A sampling is said to be an -covering if Measures sampling radius In order to be efficient, sampling should contain as few points as possible A sampling is -separated if Sampling quality

5. 5 Numerical geometry of non-rigid shapes Numerical Geometry Farthest point sampling Start with arbitrary point kth point is the farthest point from the previous k-1 Sampling radius: -separated, -covering

6. 6 Numerical geometry of non-rigid shapes Numerical Geometry Sampling = representation Voronoi tesselation Replace by the closest representative point (sample) Voronoi region Voronoi region (cell)Voronoi edgeVoronoi vertex

7. 7 Numerical geometry of non-rigid shapes Numerical Geometry Voronoi tessellation does not always exist in non-Euclidean case Non-Euclidean case Existence is guaranteed if the sampling is sufficiently dense (0.5 convexity radius)

8. 8 Numerical geometry of non-rigid shapes Numerical Geometry Voronoi tessellation in nature Giraffa camelopardalis Testudo hermaniiHoneycomb

9. 9 Numerical geometry of non-rigid shapes Numerical Geometry Point cloud represents only the structure of Does not represent the relations between points Neighborhood Connectivity Two neighboring points are called adjacent Adjacency can be represented as a graph K nearest neighbors

10. 10 Numerical geometry of non-rigid shapes Numerical Geometry Connectivity on Cartesian grids 4-neighborhood8-neighborhood

11. 11 Numerical geometry of non-rigid shapes Numerical Geometry Given a sampling and the Voronoi tessellation it produces Define connectivity as Delaunay tesselation Voronoi regionsConnectivityDelaunay tesselation In the non-Euclidean case, does not always exist and not always unique adjacent iffshare a common edge

12. 12 Numerical geometry of non-rigid shapes Numerical Geometry Geodesic triangles cannot be represented by a computer Replace geodesic triangles by Euclidean triangles Triangular mesh : collection of triangular patches glued together Triangular mesh Geodesic trianglesEuclidean triangles

13. 13 Numerical geometry of non-rigid shapes Numerical Geometry Discrete representations of surfaces Point cloud (0-dimensional) Connectivity graph (1-dimensional) Triangulation (2-dimensional)

14. 14 Numerical geometry of non-rigid shapes Numerical Geometry Triangular mesh = polyhedral surface Any point on triangular mesh falls into some triangle Barycentric coordinates: local representation for the point as a convex combination of the triangle vertices Barycentric coordinates

15. 15 Numerical geometry of non-rigid shapes Numerical Geometry Objects can be sampled and represented as clouds of points connectivity graphs triangle meshes This approximates the extrinsic geometry of the object In order to approximate the intrinsic metric we need numerical tools to measure shortest path lengths Conclusions so far…