Measuring and Representing Holes in Discrete Objectsaldo.gonzalez-lorenzo.perso.luminy.univ-amu.fr/downloads/dtmm-gb-slides.pdf · sdt O(x)= (−d(x,Oc) if x∈ O d(x,O) if x∈

Measuring and Representing Holes in Discrete Objects

Measuring and Representing

Holes in Discrete Objects

Aldo Gonzalez-Lorenzo

Aix-Marseille Université - CNRS, LIS UMR 7020

March 25, 2019

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Outline

1 Introduction

2 Background

3 Measuring Holes

4 Representing Holes

5 Applications

6 Conclusion

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Introduction

Outline

1 Introduction

2 Background

3 Measuring Holes


5 Applications

6 Conclusion

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Introduction

Holes ≃ homology

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Introduction

Holes ≃ homology

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Introduction

Holes ≃ homology

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Introduction

Holes ≃ homology

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Introduction

Holes ≃ homology

We can know how many holes there are in an object

We cannot know where or how they are

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Introduction

Size of a hole

The 1st one is bigger than the2nd one

The 2nd one is thicker than the1st one

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Introduction

Size of a hole



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Introduction

Size of a hole



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Introduction

Representing a hole

Homology Cohomology

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Introduction

Representing a hole

Do homology generators really represent holes?

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Introduction

Geometry + Topology↓ ↓

Signed distance transform Persistent homology

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Background

Outline

1 Introduction

2 BackgroundDigital Geometry

Cubical Complexes

Homology

Persistent Homology

3 Measuring Holes


5 Applications

6 Conclusion9 / 38


Background

Digital Geometry

Discrete object

A nD discrete object is a subset of Zn

Connectivity relation: 2n or the (3n − 1)-connectivity.10 / 38


Background

Digital Geometry

Discrete object




Background

Digital Geometry

Discrete object




Background

Digital Geometry

Signed distance transform

Let O be a discrete object,

sdtO(x) =

{

−d(x ,Oc) if x ∈ O

d(x ,O) if x /∈ O

Figure: Sublevel sets of the signed distance transform

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Background

Cubical Complexes

Cubical complex

Union1 of points, edges, squares, cubes, ... (cubes)

1with some conditions12 / 38


Background

Cubical Complexes

Cubical complex




Background

Cubical Complexes

Cubical complex




Background

Cubical Complexes

Cubical complex




Background

Cubical Complexes

Cubical complex




Background

Cubical Complexes

Discrete object −→ cubical complex ((3n − 1)-connectivity)

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Background

Cubical Complexes

Discrete object −→ cubical complex (2n-connectivity)

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Background

Homology

Blue: 1-cubeRed: its boundary (faces)

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Background

Homology

Blue: 2-cubeRed: its boundary (faces)

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Background

Homology

Blue: 1-chainRed: its boundary

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Background

Homology

Blue: 1-chain (1-cycle)Red: its boundary (= ∅)

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Background

Homology

Blue: 2-chainRed: its boundary (1-cycle)

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Background

Homology

Blue: 1-chain (1-cycle, but not boundary)Red: its boundary (= ∅)

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Background

Homology

K cubical complex

Chain complex of K

· · ·C3d3−−−→ C2

d2−−−→ C1d1−−−→ C0

d0−−−→ 0

where dqdq+1 = 0 ⇒ im(dq+1) ⊂ ker(dq)

q-dimensional homology groupHq(K ) := ker(dq)/ im(dq+1)

2 = (F2)βq

q-dimensional Betti number: βq

2∀x , y ∈ ker(dq), x ∼ y ⇔ x + y ∈ im(dq+1)

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Background

Homology

β0 = # connected components (0-holes)

β1 = # tunnels or handles (1-holes)

β2 = # cavities (2-holes)

Betti numbers are

Topological invariants → classification

Shape descriptors → understanding

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Background

Homology

β0 = 2, β1 = 2, β2 = 1, β3 = 0, . . .

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Background

Persistent Homology

Filtration F : K1 ⊂ K2 ⊂ K3 ⊂ · · ·

K1ι−−−−→ K2

ι−−−−→ K3ι−−−−→ · · ·

↓ ↓ ↓H(K1)

ι∗−−−−−→ H(K2)ι∗−−−−−→ H(K3)

ι∗−−−−−→ · · ·

βi ,j = dim(ι : H(Ki ) → H(Kj))number of holes in Ki still in Kj

µi ,j = βi ,j − βi ,j+1 − βi−1,j + βi−1,j+1

number of holes born in Ki and dying in Kj

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Background

Persistent Homology

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Background

Persistent Homology

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Background

Persistent Homology

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Background

Persistent Homology

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Background

Persistent Homology

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Background

Persistent Homology

Persistence pairs

PD(F ) = {(i , j) with multiplicity µi ,j}

We represent PD(F ) with a persistence diagram.

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Measuring Holes

Outline

1 Introduction

2 Background

3 Measuring Holes


5 Applications

6 Conclusion

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Measuring Holes

β2 = 10

“Full” of holes?

All holes have similar size?

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Measuring Holes

β2 = 10

“Full” of holes?

All holes have similar size?

Dilate... and erode

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Measuring Holes

Persistent homology with signed distance transform

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Measuring Holes


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Measuring Holes


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Measuring Holes


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Measuring Holes


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Measuring Holes


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Measuring Holes


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Measuring Holes

Thickness and breadth

Let O be a discrete object and F the filtration defined by thesublevel sets of its signed distance transform.Let TB(O) = {(−x , y) ∈ PD(F ) | x ≤ 0, y ≥ 0}.Its intervals are the thickness-breadth pairs of O.

One thickness-breadth pair (t, b) for each hole of O

t is the thickness of the hole and b, its breadth

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Measuring Holes

Thickness-breadth diagram

Thickness-breadth pairs can be represented like persistencediagrams

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Measuring Holes



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Measuring Holes



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Measuring Holes

Theorem

Let X and Y be two 3D discrete objects. Let us call

δ = dH(X ,Y ) + dH(Z3 \ X ,Z3 \ Y ) + 2

√3

Thus, for every thickness-breadth pair pX = (x , y) of X such thatx , y > δ, there exists another thickness-breadth pair pY = (x ′, y ′)of Y such that

||pX − pY ||∞ ≤ δ

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Representing Holes

Outline

1 Introduction

2 Background

3 Measuring Holes


5 Applications

6 Conclusion

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Representing Holes

Sketch of persistent homology computation:

Sort cells according to the filtration

For each cell, associate it with one of the previous ones

Each of these pairs makes a persistence pair

Thickness and breadth ball

Let (t, b) be a TB-pair and (σ, τ) its pair of cells

The thickness ball of (t, b) is the ball of radius t centered at σ

The breadth ball of (t, b) is the ball of radius b centered at τ

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Representing Holes

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Representing Holes

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Representing Holes

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Representing Holes

We can identify each hole by

Its thickness-breadth pair (unique)

The center of its thickness ball (non unique)

The center of its breadth ball (non unique)

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Applications

Outline

1 Introduction

2 Background

3 Measuring Holes


5 Applications

6 Conclusion

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Applications

Given a discrete object O and a TB pair (t, b)

Open the hole: remove voxels from O

Close the hole: add voxels to O to remove the hole

Compute a homology generator

Compute a cohomology generator

Why?

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Applications

Topological correction

Example

1 Scan an object, segmentate it and computeits TB diagram.

2 Identify “wrong” holes using TB pairs andballs.

3 Open or close them.

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Applications

Extract relation between holes (1/2)

Example

1 For each hole, close it and see which holes vanish.

2 Represent this with a graph.

3 Compare objects using these graphs.

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Applications

Extract relation between holes (2/2)

Example

1 For each hole, compute its homology generator.

2 Transform these generators into discrete objects.

3 Extract the relation between these holes.

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Conclusion

Outline

1 Introduction

2 Background

3 Measuring Holes


5 Applications

6 Conclusion

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Conclusion

Conclusion:

Topological-geometrical signature of objects.

Robust to noise → suitable for real applications.

Alternative visualization of holes.

Heuristics for minimal openings, closings and (co)homologygenerators.

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Conclusion

Conclusion:

Topological-geometrical signature of objects.

Robust to noise → suitable for real applications.

Alternative visualization of holes.

Heuristics for minimal openings, closings and (co)homologygenerators.

Thanks! Questions?

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Documents

Measuring and Representing Holes in Discrete Objectsaldo.gonzalez-lorenzo.perso.luminy.univ-amu.fr/downloads/dtmm-gb-slides.pdf · sdt O(x)= (−d(x,Oc) if x∈ O d(x,O) if x∈