Numerical geometry of nonrigid shapes Numerical Geometry Numerical

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 1

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

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

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

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

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

Numerical geometry of non-rigid shapes Numerical Geometry Non-Euclidean case n Voronoi tessellation does not always exist in non-Euclidean case n Existence is guaranteed if the sampling is sufficiently dense (0. 5 convexity radius) 7

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

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

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

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

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

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

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