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…