Sampling conditions and topological guarantees for shape reconstruction algorithms
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Transcript of Sampling conditions and topological guarantees for shape reconstruction algorithms
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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
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Shape Reconstruction
INPUT OUTPUT
Surface ofphysical object
UNKNOWN
• geometrically accurate• topologically correct
TriangulationSample in R3
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INPUT OUTPUT
Unordered sequenceof images varyingin pose and lighting
Low-dimensional complex
Shape Reconstruction(or manifold learning)
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Shape Reconstruction(or manifold learning)
INPUT OUTPUT
Space with smallintrinsec dimension
Sample in Rd
UNKNOWN
• geometrically accurate• topologically correct
Simplicial complex
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Algorithms in 2DAlgorithms in 2D
heuristics to select a subsetof the Delaunay triangulation
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Algorithms in 3DAlgorithms in 3D
heuristics to select a subsetof the Delaunay triangulation
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A Simple Algorithm
INPUT OUTPUT
-offset = union of ballswith radius centeredon the sample
Shape Sample
UNKNOWN
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A Simple Algorithm
INPUT OUTPUT
Shape Sample
-offset -complex
UNKNOWN
From Nerve Theorem:
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A Simple AlgorithmOUTPUTShape Sample
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Reconstruction theorem
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Reconstruction theorem
[Niyogi Smale Weinberger 2004]
Sampling conditions
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Beyond the reach : WFS and -reach
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1
reach -reach wfs
wfs
Beyond the reach : WFS and -reach
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Previous best known result for faithful reconstruction of set with positive m-reach
(Chazal, Cohen-Steiner,Lieutier 2006)
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Previous best known result for faithful reconstruction of set with positive m-reach
(Chazal, Cohen-Steiner,Lieutier 2006)
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Best known result for faithful reconstruction of set with positive m-reach
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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)
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Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)
(Chazal, Lieutier 2006)
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Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)
(Chazal, Lieutier 2006)
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Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)
(Chazal, Lieutier 2006)
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Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)
(Chazal, Lieutier 2006)
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Recovering homology(Cohen-Steiner,Edelsbrunner,Harer 2006)
(Chazal, Lieutier 2006)
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Rips Complex
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Samplng condition for Cech and Rips(D. Attali, A. Lieutier 2011)
[CCL06]
[NSW04]
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Questions ?
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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 ...
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Cech / Rips
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Cech / Rips
Possesses a spirituous cyclethat we want to kill !
Possesses a spirituous cyclethat we want to kill !
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Cech / Rips
Had there been a point close to the center,it would have distroy spirituous cycles appearingin the Rips, without changing the Cech.
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Convexity defects function
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Large -reach => small convexity defect functions
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Density authorized
[CCL06]
[NSW04]
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Questions ?
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Cech Complex
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Nerve Theorem
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Persistent homology
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Persistent homology
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Persistent homology
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Persistent homology
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Persistent homology
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Persistent homology