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
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Numerical geometry of non-rigid shapes Numerical Geometry
Sampling of surfaces
 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
Sampled surface
Geometry image
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Numerical geometry of non-rigid shapes Numerical Geometry
Depth images
 Particular case: Monge parametrization
 Can be represented as a single matrix
(depth image)
 Typical output of 3D scanners
Sampled surface
Depth image
Numerical geometry of non-rigid shapes Numerical Geometry
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Sampling quality
 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
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
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Numerical geometry of non-rigid shapes Numerical Geometry
Voronoi tesselation
 Sampling = representation
 Replace
by the closest representative point (sample)
 Voronoi region
Voronoi region (cell)
Voronoi edge
Voronoi vertex
Numerical geometry of non-rigid shapes Numerical Geometry
Non-Euclidean case
 Voronoi tessellation does not always exist in non-Euclidean case
 Existence is guaranteed if the sampling is sufficiently dense (0.5
convexity radius)
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Numerical geometry of non-rigid shapes Numerical Geometry
Voronoi tessellation in nature
Giraffa camelopardalis
Testudo hermanii
Honeycomb
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Numerical geometry of non-rigid shapes Numerical Geometry
Connectivity
 Point cloud represents only the structure of
 Does not represent the relations between points
 Neighborhood
K nearest neighbors
 Two neighboring points are called adjacent
 Adjacency can be represented as a graph
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Numerical geometry of non-rigid shapes Numerical Geometry
Delaunay tesselation
 Given a sampling and the Voronoi tessellation it produces
 Define connectivity as
adjacent iff
share a common edge
 In the non-Euclidean case, does not always exist and not always unique
Voronoi regions
Connectivity
Delaunay tesselation
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Numerical geometry of non-rigid shapes Numerical Geometry
Triangular mesh
 Geodesic triangles cannot be represented by a computer
 Replace geodesic triangles by Euclidean triangles
 Triangular mesh
Geodesic triangles
: collection of triangular patches glued together
Euclidean triangles
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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
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Barycentric coordinates
 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
Numerical geometry of non-rigid shapes Numerical Geometry
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Conclusions so far…
 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