TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini...

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

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)


Concavity •  A func2on is concave if

for any and any



for any and any


Semi-concavity It will play a crucial role: where does it come from?

•  is semi-‐concave if is concave



•  is semi-‐concave if there exists a closed set such that for any




Given a point of coordinates




is concave

obtained via direct computa&on




φ2(x) = (p− x)2

φ2(y) = (p− y)2




φ2(x) = (p− x)2

φ2(y) = (p− y)2

φ2(z) ≥ (p− z)2




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




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)


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)}




B|φ(x)|((x, 0)) =

{y ∈ Rm+1 | d2((x, 0), y) < φ2(x)}



define as the complement of the union of such balls



contains the 3-‐points set we started from

KKmax



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



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}


-critical point •  Given a distance-‐like func2on a point is -‐cri2cal if



Example: and distance-‐like




For

Impo

ssible

Non

-‐ cri&cal

Cri&cal



Example: distance-‐like At all 0-‐cri&cal poinP of we have and



•  Alterna&ve defini&on: a point is -‐cri2cal if it is -‐cri2cal for


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


-reach •  Given a distance-‐like func2on the -‐reach of is

•  Alterna&ve defini&on: the -‐reach of is


Sup norm •  Given two func2ons


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)

TheHiddenTopology$ of$aNoisyPointCloud · Seminario3-Parte2-20140113.pptx Author: Andrea Pedrini...

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