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 INPUTOUTPUT Surface of physical object UNKNOWN geometrically accurate topologically correct Triangulation Sample in R 3

INPUTOUTPUT Unordered sequence of images varying in pose and lighting Low-dimensional complex Shape Reconstruction (or manifold learning)

INPUTOUTPUT Space with small intrinsec dimension Sample in R d UNKNOWN geometrically accurate topologically correct Simplicial complex

Algorithms in 2D heuristics to select a subset of the Delaunay triangulation

Algorithms in 3D heuristics to select a subset of the Delaunay triangulation

A Simple Algorithm INPUTOUTPUT  -offset = union of balls with radius centered on the sample ShapeSample UNKNOWN

A Simple Algorithm INPUTOUTPUT ShapeSample  -offset  -complex UNKNOWN From Nerve Theorem:

A Simple Algorithm OUTPUT ShapeSample

Reconstruction theorem

[Niyogi Smale Weinberger 2004] Sampling conditions

Beyond the reach : WFS and  -reach

1  reach  - reach 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)

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

Cech / Rips

Possesses a spirituous cycle that we want to kill ! Possesses a spirituous cycle that we want to kill !

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

Convexity defects function

Large  -reach => small convexity defect functions

Density authorized [CCL06] [NSW04]

Questions ?

Cech Complex

Nerve Theorem

Optimal sampling condition is tigth

Persistent homology

Simplicial complexes and Nerve Theorem Nerve theorem (Leray, Weyl, others,..1940’ s ): If any non empty intersection of sets of F is contractible, then the nerve N(F) has the same homotopy type as the union of all the sets of the familly. A familly of sets F Its nerve N(F) (Lets say finite familly of compacts sets)

Nerve Theorem Delauney complex The Delauney complex is the nerve of (closures of full dimensional) Voronoi cells  -complex The  -complex is the nerve of the intersections of (closures of full dimensional) voronoi cells with the  -offset of the point set

Generalized gradient, WFS and  -reach 1  reach  - reach wfs

Topological stability of offsets 1  reach  - reach wfs

Reconstruction from  -reach based condition and topological persistence..