thehiddentopology$ of$anoisypointcloud · seminario3-parte2-20140113.pptx author: andrea pedrini...
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The Hidden Topology of a Noisy Point Cloud
(Part II) A cri&cal reading of
“Geometric Inference for Probability Measures” by Chazal, Steiner & Merigot, 2011.
A. Pedrini, M. Piastra
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The hidden topology of a noisy point cloud 2
Generalizing: distance-like • A func2on is distance-‐like if
1)
2)
3) is semi-‐concave, that is is concave (this is the crucial property)
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The hidden topology of a noisy point cloud 3
Concavity • A func2on is concave if
for any and any
![Page 4: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/4.jpg)
The hidden topology of a noisy point cloud 4
Concavity • A func2on is concave if
for any and any
![Page 5: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/5.jpg)
The hidden topology of a noisy point cloud 5
Concavity • A func2on is concave if
for any and any
![Page 6: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/6.jpg)
The hidden topology of a noisy point cloud 6
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if is concave
![Page 7: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/7.jpg)
The hidden topology of a noisy point cloud 7
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
![Page 8: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/8.jpg)
The hidden topology of a noisy point cloud 8
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
![Page 9: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/9.jpg)
The hidden topology of a noisy point cloud 9
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
Given a point of coordinates
![Page 10: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/10.jpg)
The hidden topology of a noisy point cloud 10
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
Given a point of coordinates
![Page 11: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/11.jpg)
The hidden topology of a noisy point cloud 11
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
![Page 12: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/12.jpg)
The hidden topology of a noisy point cloud 12
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
is concave
obtained via direct computa&on
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The hidden topology of a noisy point cloud 13
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
![Page 14: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/14.jpg)
The hidden topology of a noisy point cloud 14
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
![Page 15: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/15.jpg)
The hidden topology of a noisy point cloud 15
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
![Page 16: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/16.jpg)
The hidden topology of a noisy point cloud 16
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
φ2(x) = (p− x)2
φ2(y) = (p− y)2
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The hidden topology of a noisy point cloud 17
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
φ2(x) = (p− x)2
φ2(y) = (p− y)2
φ2(z) ≥ (p− z)2
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The hidden topology of a noisy point cloud 18
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
φ2(z)− z2 ≥ λ(φ2(x)− x2) + (1− λ)(φ2(y)− y2)
(p− z)2 − z2 = λ(φ2(x)− x2) + (1− λ)(φ2(y)− y2)
φ2(x) = (p− x)2
φ2(y) = (p− y)2
φ2(z) ≥ (p− z)2
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The hidden topology of a noisy point cloud 19
Semi-concavity It will play a crucial role: where does it come from?
• is semi-‐concave if there exists a closed set such that for any
is concave, whence is semi-‐concave
φ2 − || · ||2φ2
φ2(λx+ (1− λ)y)− (λx+ (1− λ)y)2 ≥λ(φ2(x)− x2) + (1− λ)(φ2(y)− y2)
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The hidden topology of a noisy point cloud 20
Semi-concavity • Sample construc2on of such that for any
for each , take the open excluded ball x ∈ Rm
B|φ(x)|((x, 0)) =
{y ∈ Rm+1 | d2((x, 0), y) < φ2(x)}
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The hidden topology of a noisy point cloud 21
Semi-concavity • Sample construc2on of such that for any
for each , take the open excluded ball x ∈ Rm
B|φ(x)|((x, 0)) =
{y ∈ Rm+1 | d2((x, 0), y) < φ2(x)}
![Page 22: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/22.jpg)
The hidden topology of a noisy point cloud 22
Semi-concavity • Sample construc2on of such that for any
for each , take the open excluded ball x ∈ Rm
B|φ(x)|((x, 0)) =
{y ∈ Rm+1 | d2((x, 0), y) < φ2(x)}
![Page 23: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/23.jpg)
The hidden topology of a noisy point cloud 23
Semi-concavity • Sample construc2on of such that for any
for each , take the open excluded ball x ∈ Rm
B|φ(x)|((x, 0)) =
{y ∈ Rm+1 | d2((x, 0), y) < φ2(x)}
![Page 24: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/24.jpg)
The hidden topology of a noisy point cloud 24
Semi-concavity • Sample construc2on of such that for any
for each , take the open excluded ball x ∈ Rm
B|φ(x)|((x, 0)) =
{y ∈ Rm+1 | d2((x, 0), y) < φ2(x)}
![Page 25: TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini Created Date: 3/20/2015 11:03:32 AM](https://reader035.vdocuments.net/reader035/viewer/2022071103/5fdc538865524118c43161e0/html5/thumbnails/25.jpg)
The hidden topology of a noisy point cloud 25
Semi-concavity • Sample construc2on of such that for any
define as the complement of the union of such balls
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The hidden topology of a noisy point cloud 26
Semi-concavity • Sample construc2on of such that for any
contains the 3-‐points set we started from
KKmax
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The hidden topology of a noisy point cloud 27
Semi-concavity • Sample construc2on of such that for any
In general, does not preserve the cri&cal points of in the subspace
Kφ
Rm
{critical points of K in R} = ∅
To be defined later on
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The hidden topology of a noisy point cloud 28
Semi-concavity • Sample construc2on of such that for any
The cri2cal points of in the subspace are the non-‐zero cri&cal points of (this happens because is symmetric w.r.t. )
φ
RmKmax
RmKmax
{critical points of Kmax in R} = {critical points of φ}
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The hidden topology of a noisy point cloud 29
-critical point • Given a distance-‐like func2on a point is -‐cri2cal if
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The hidden topology of a noisy point cloud 30
-critical point • Given a distance-‐like func2on a point is -‐cri2cal if
Example: and distance-‐like
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The hidden topology of a noisy point cloud 31
-critical point • Given a distance-‐like func2on a point is -‐cri2cal if
Example: and distance-‐like
For
Impo
ssible
Non
-‐ cri&cal
Cri&cal
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The hidden topology of a noisy point cloud 32
-critical point • Given a distance-‐like func2on a point is -‐cri2cal if
Example: distance-‐like At all 0-‐cri&cal poinP of we have and
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The hidden topology of a noisy point cloud 33
-critical point • Given a distance-‐like func2on a point is -‐cri2cal if
Example: and distance-‐like
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The hidden topology of a noisy point cloud 34
-critical point • Given a distance-‐like func2on a point is -‐cri2cal if
• Alterna&ve defini&on: a point is -‐cri2cal if it is -‐cri2cal for
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The hidden topology of a noisy point cloud 35
Isotopy Lemma • Given a distance-‐like func2on and two numbers such that there are no -‐cri2cal points in Then all the sublevel sets are isotopic
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The hidden topology of a noisy point cloud 36
-reach • Given a distance-‐like func2on the -‐reach of is
• Alterna&ve defini&on: the -‐reach of is
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The hidden topology of a noisy point cloud 37
Sup norm • Given two func2ons
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The hidden topology of a noisy point cloud 38
Reconstruction theorem • Let be two distance-‐like func2ons having for which it exists a such that
Then, for all and all the sublevel sets
have the same homotopy type (Theorem 4.6)