Sampling conditions and topological guarantees for shape reconstruction algorithms

Andre Lieutier, Dassault Sytemes

Thanks to Dominique Attali for some slides (the nice ones)Thanks to Dominique Attali, Fréderic Chazal, David Cohen-Steiner for joint work

Shape Reconstruction

INPUT OUTPUT

Surface ofphysical object

UNKNOWN

• geometrically accurate• topologically correct

TriangulationSample in R3

INPUT OUTPUT

Unordered sequenceof images varyingin pose and lighting

Low-dimensional complex

Shape Reconstruction(or manifold learning)

Shape Reconstruction(or manifold learning)

INPUT OUTPUT

Space with smallintrinsec dimension

Sample in Rd

UNKNOWN

• geometrically accurate• topologically correct

Simplicial complex

Algorithms in 2DAlgorithms in 2D

heuristics to select a subsetof the Delaunay triangulation

Algorithms in 3DAlgorithms in 3D

heuristics to select a subsetof the Delaunay triangulation

A Simple Algorithm

INPUT OUTPUT

-offset = union of ballswith radius centeredon the sample

Shape Sample

UNKNOWN

A Simple Algorithm

INPUT OUTPUT

Shape Sample

-offset -complex

UNKNOWN

From Nerve Theorem:

A Simple AlgorithmOUTPUTShape Sample

Reconstruction theorem

Reconstruction theorem

[Niyogi Smale Weinberger 2004]

Sampling conditions

Beyond the reach : WFS and -reach

1

reach -reach wfs

wfs

Beyond the reach : WFS and -reach

Previous best known result for faithful reconstruction of set with positive m-reach

(Chazal, Cohen-Steiner,Lieutier 2006)

Previous best known result for faithful reconstruction of set with positive m-reach

(Chazal, Cohen-Steiner,Lieutier 2006)

Best known result for faithful reconstruction of set with positive m-reach

Under the conditions of the theorem, a simple offset of the sample is a faithful reconsruction

Previous best known result for faithful reconstruction of set with positive m-reach

(Chazal, Cohen-Steiner,Lieutier 2006)

Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)

(Chazal, Lieutier 2006)

Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)

(Chazal, Lieutier 2006)

Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)

(Chazal, Lieutier 2006)

Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)

(Chazal, Lieutier 2006)

Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)

(Chazal, Lieutier 2006)

Rips Complex

Samplng condition for Cech and Rips(D. Attali, A. Lieutier 2011)

[CCL06]

[NSW04]

Questions ?

Cech / Rips

Rips and Cech complexes generally don’t share the same the same topology, topology, but ...but ...

Rips and Cech complexes generally don’t share the same the same topology, topology, but ...but ...

Cech / Rips

Cech / Rips

Possesses a spirituous cyclethat we want to kill !

Possesses a spirituous cyclethat we want to kill !

Cech / Rips

Had there been a point close to the center,it would have distroy spirituous cycles appearingin the Rips, without changing the Cech.

Convexity defects function

Large -reach => small convexity defect functions

Density authorized

[CCL06]

[NSW04]

Questions ?

Cech Complex

Nerve Theorem

Persistent homology

Persistent homology

Persistent homology

Persistent homology

Persistent homology

Persistent homology