TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte1-20140113.pptx Author: Andrea Pedrini...
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The Hidden Topology of a Noisy Point Cloud (Part I) A cri&cal reading of “Geometric Inference for Probability Measures” by Chazal, Steiner & Merigot, 2011. A. Pedrini, M. Piastra
Transcript of TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte1-20140113.pptx Author: Andrea Pedrini...
The Hidden Topology of a Noisy Point Cloud
(Part I)
A cri&cal reading of “Geometric Inference for Probability Measures”
by Chazal, Steiner & Merigot, 2011.
A. Pedrini, M. Piastra
The hidden topology of a noisy point cloud 2
Bibliography
• F. Chazal, D. Cohen-‐Steiner, Q. Mérigot Geometric Inference for Probability Measures Founda&ons of Computa&onal Mathema&cs (2011)
• F. Chazal, D. Cohen-‐Steiner, A. LieuBer A Sampling Theory for Compact Sets in Euclidean Space Discrete Computa&onal Geometry (2009)
• J.D. Boissonnat, F. Chazal, M. Yvinec ComputaBonal Geometric Learning Lecture Notes (2012)
The hidden topology of a noisy point cloud 3
The problem • is a compact subset of
The hidden topology of a noisy point cloud 4
The problem • is a compact subset of • is a point cloud obtained by a noise sample from
The hidden topology of a noisy point cloud 5
The problem
Which of the topologic
proper&es of a are s&ll present
in ?
• is a compact subset of • is a point cloud obtained by a noise sample from
The hidden topology of a noisy point cloud 6
The problem • Previously, in this reading group (see webpage):
• Point cloud samples coming from shapes sBll contain traces of the original topology
• If the point cloud is dense enough and the outliers are not too far away from the original shape, the topology can be reconstructed
• This Bme: • What if the noise is not so well behaved?
The hidden topology of a noisy point cloud 7
Offsets • If , the offset of a compact set
is the compact set:
The hidden topology of a noisy point cloud 8
Offsets • If , the offset of a compact set
is the compact set:
The hidden topology of a noisy point cloud 9
Offsets • If , the offset of a compact set
is the compact set:
The hidden topology of a noisy point cloud 10
• The Euclidean distance of a point from a compact set is
Distance function
The hidden topology of a noisy point cloud
The hidden topology of a noisy point cloud 11
• The Hausdorff distance of two compact sets is:
Hausdorff distance
The hidden topology of a noisy point cloud 12
• The Hausdorff distance of two compact sets is:
Hausdorff distance
The hidden topology of a noisy point cloud 13
• The Hausdorff distance of two compact sets is:
Hausdorff distance
The hidden topology of a noisy point cloud 14
• The Hausdorff distance of two compact sets is:
Hausdorff distance
The hidden topology of a noisy point cloud 15
• The Hausdorff distance of two compact sets is:
Hausdorff distance
The hidden topology of a noisy point cloud 16
• The Hausdorff distance of two compact sets is (alterna&ve defini&on):
Hausdorff distance
The hidden topology of a noisy point cloud 17
Beyond homeomorphism • In general, , and are not homeomorphic:
The hidden topology of a noisy point cloud 18
Beyond homeomorphism • In general, , and are not homeomorphic:
The hidden topology of a noisy point cloud 19
Beyond homeomorphism • In general, , and are not homeomorphic:
The hidden topology of a noisy point cloud 20
• How does the homotopy of the offsets change?
Homotopy and offsets
The hidden topology of a noisy point cloud 21
• How does the homotopy of the offsets change?
Homotopy and offsets
The hidden topology of a noisy point cloud 22
• How does the homotopy of the offsets change?
First change of the
homotopy type
Homotopy and offsets
The hidden topology of a noisy point cloud 23
• How does the homotopy of the offsets change?
Second change of the
homotopy type
Homotopy and offsets
The hidden topology of a noisy point cloud 24
• How does the homotopy of the offsets change?
Third change of the
homotopy type
Homotopy and offsets
The hidden topology of a noisy point cloud 25
• Comparing the changes in homotopy type
Homotopy types
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• is a compact set, is a point
Generalized gradient
The hidden topology of a noisy point cloud 27
• is a compact set, is a point
Generalized gradient
Note: point has been chosen so that it has two closest neighbors in
The hidden topology of a noisy point cloud 28
• is a compact set, is a point
• Closest neighbors:
Generalized gradient
The hidden topology of a noisy point cloud 29
• is a compact set, is a point
Generalized gradient
The hidden topology of a noisy point cloud 30
• is a compact set, is a point
Generalized gradient
The hidden topology of a noisy point cloud 31
• The generalized gradient of the distance funcBon of a compact set
at a point is
Generalized gradient
The hidden topology of a noisy point cloud 32
Generalized gradient • The generalized gradient of the distance funcBon of a compact set
at a point is
The hidden topology of a noisy point cloud 33
• The generalized gradient of the distance funcBon of a compact set
at a point is
Generalized gradient
( is the aperture of the smallest cone with apex containing )
The hidden topology of a noisy point cloud 34
Generalized gradient
The hidden topology of a noisy point cloud 35
• The medial axis of is the set
Medial axis
The hidden topology of a noisy point cloud 36
• The medial axis of is the set
Medial axis
The hidden topology of a noisy point cloud 37
• The medial axis of is the set
Medial axis
The hidden topology of a noisy point cloud 38
• The medial axis of is the set
Medial axis
The hidden topology of a noisy point cloud 39
• The reach of a compact is
Reach
The hidden topology of a noisy point cloud 40
• A cri6cal point of a compact set is a point such that
is cri&cal iff
Critical points and wfs
criBcal point
The hidden topology of a noisy point cloud 41
• A cri6cal point of a compact set is a point such that
Critical points and wfs
• A real is a cri6cal value of if there exists a cri&cal point
such that • A regular value of is a real which is not criBcal
• is always considered criBcal
The hidden topology of a noisy point cloud 42
• A cri6cal point of a compact set is a point such that
Critical points and wfs
The hidden topology of a noisy point cloud 43
• A cri6cal point of a compact set is a point such that
Critical points and wfs
The hidden topology of a noisy point cloud 44
• A cri6cal point of a compact set is a point such that
Critical points and wfs
The hidden topology of a noisy point cloud 45
• A cri6cal point of a compact set is a point such that
Critical points and wfs
criBcal point: the homotopy of the offset changes
The hidden topology of a noisy point cloud 46
• A cri6cal point of a compact set is a point such that
Critical points and wfs
The hidden topology of a noisy point cloud 47
• A cri6cal point of a compact set is a point such that
Critical points and wfs
The hidden topology of a noisy point cloud 48
• A cri6cal point of a compact set is a point such that
Critical points and wfs
criBcal point: the homotopy of the offset changes
The hidden topology of a noisy point cloud 49
• A cri6cal point of a compact set is a point such that
Critical points and wfs
criBcal point: the homotopy of the offset changes
The hidden topology of a noisy point cloud 50
• A cri6cal point of a compact set is a point such that
Critical points and wfs
• The weak feature size of is the Euclidean distance of to the set of its criBcal points
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• Theorem 1: Let be a compact set. If , then and are isotopic (and hence homeomorphic)
Offsets and Isotopy
The hidden topology of a noisy point cloud 52
• Theorem 1: Let be a compact set. If , then and are isotopic (and hence homeomorphic)
Offsets and Isotopy
The hidden topology of a noisy point cloud 53
• Theorem 1: Let be a compact set. If , then and are isotopic (and hence homeomorphic)
Offsets and Isotopy
The hidden topology of a noisy point cloud 54
• Theorem 1: Let be a compact set. If , then and are isotopic (and hence homeomorphic)
Theorem 1 does not hold
Offsets and Isotopy
The hidden topology of a noisy point cloud 55
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
The hidden topology of a noisy point cloud 56
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
The hidden topology of a noisy point cloud 57
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
The hidden topology of a noisy point cloud 58
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
The hidden topology of a noisy point cloud 59
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
The hidden topology of a noisy point cloud 60
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
The hidden topology of a noisy point cloud 61
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
The hidden topology of a noisy point cloud 62
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
cos(30°)-‐criBcal points
The hidden topology of a noisy point cloud 63
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
cos(35°)-‐criBcal points
The hidden topology of a noisy point cloud 64
• A -‐cri6cal point ( ) of a compact set is a point such that
µ-critical points
cos(45°)-‐criBcal points
The hidden topology of a noisy point cloud 65
• The -‐medial axis of is
µ-medial axis
The hidden topology of a noisy point cloud 66
• The -‐medial axis of is
µ-medial axis
The hidden topology of a noisy point cloud 67
• The -‐medial axis of is
µ-medial axis
The hidden topology of a noisy point cloud 68
• The -‐medial axis of is
µ-medial axis
The hidden topology of a noisy point cloud 69
• The -‐reach of is
µ-reach
The hidden topology of a noisy point cloud 70
• The -‐reach of is
µ-reach
The hidden topology of a noisy point cloud 71
• The -‐reach of is
µ-reach
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• The -‐reach of is
µ-reach
The hidden topology of a noisy point cloud 73
Reconstruction Theorem
• Theorem 8 : and compact subsets of , .
Then for any such that and for any and such that
and
we have that and are homotopy equivalent
The hidden topology of a noisy point cloud 74
• The assumpBon (with small) means that the point cloud must be close enough:
Main limitations: outliers
The hidden topology of a noisy point cloud 75
• The assumpBon (with small) means that the point cloud must be close enough:
Main limitations: outliers
The hidden topology of a noisy point cloud 76
• The assumpBon (with small) means that the point cloud must be close enough:
Main limitations: outliers
Just a few outliers can make the reconstrucBon impossible!
The hidden topology of a noisy point cloud 77
• The assumpBon (with small) means that the point cloud must be close enough:
Main limitations: outliers
Just a few outliers can make the reconstrucBon impossible!
