1 numerical geometry of non-rigid shapes numerical geometry numerical geometry of non-rigid shapes...
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1Numerical 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
2Numerical geometry of non-rigid shapes Numerical Geometry
Sampling of surfaces
Sampled surface Geometry 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
3Numerical geometry of non-rigid shapes Numerical Geometry
Depth images
Sampled surface Depth image
Particular case: Monge parametrization
Can be represented as a single matrix (depth image)
Typical output of 3D scanners
4Numerical 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
5Numerical 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
6Numerical geometry of non-rigid shapes Numerical Geometry
Sampling = representation
Voronoi tesselation
Replace by the closest representative point (sample)
Voronoi region
Voronoi region (cell) Voronoi edge Voronoi vertex
7Numerical 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)
8Numerical geometry of non-rigid shapes Numerical Geometry
Voronoi tessellation in nature
Giraffa camelopardalis Testudo hermanii Honeycomb
9Numerical 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
11Numerical geometry of non-rigid shapes Numerical Geometry
Given a sampling and the Voronoi tessellation it produces
Define connectivity as
Delaunay tesselation
Voronoi regions Connectivity Delaunay tesselation
In the non-Euclidean case, does not always exist and not always
unique
adjacent iff share a common edge
12Numerical 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 triangles Euclidean triangles
13Numerical geometry of non-rigid shapes Numerical Geometry
Discrete representations of surfaces
Point cloud(0-dimensional)
Connectivity graph(1-dimensional)
Triangulation(2-dimensional)
14Numerical 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
15Numerical 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…