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A Study of Well-composedness in n -D Nicolas Boutry 1 , 2 - - PowerPoint PPT Presentation

Dec 28, 2023 285 likes •1.12k views

A Study of Well-composedness in n -D Nicolas Boutry 1 , 2 nicolas.boutry@lrde.epita.fr Advisors: Laurent Najman 2 & Thierry G eraud 1 (1) EPITA Research and Development Laboratory, LRDE, France e Paris-Est, LIGM, (2) Universit Equipe

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A Study of Well-composedness in n-D

Nicolas Boutry1,2

nicolas.boutry@lrde.epita.fr

Advisors: Laurent Najman2 & Thierry G´ eraud1

(1) EPITA Research and Development Laboratory, LRDE, France (2) Universit´ e Paris-Est, LIGM, ´ Equipe A3SI, ESIEE, France

2016-12-14

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 1

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Our quest

Digital topology has topological issues on cubical grids. These topological issues results from critical configurations: We are looking for a new representation of signals with no topological issues.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 2

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Outline

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 3

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Outline

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 3

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Outline

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 3

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Outline

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 3

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Outline

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 3

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Cubical grids in digital topology lead to topological issues

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 4

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Cubical grids in digital topology lead to topological issues

Our choice (1/2)

Simplicial complexes Cubical complexes Polyhedral complexes Triangular tilings Hexagonal tilings Cubical tilings/grids 9 11 15 7 1 13 3 5 3 Khalimsky tilings

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 5

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Cubical grids in digital topology lead to topological issues

Our choice (1/2)

Simplicial complexes Cubical complexes Polyhedral complexes Triangular tilings Hexagonal tilings Cubical tilings/grids 9 11 15 7 1 13 3 5 3 Khalimsky tilings

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 5

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Cubical grids in digital topology lead to topological issues

Our choice (2/2)

Cubical signals many sensors are cubical they are easy to process they are easy to store ...

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 6

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Cubical grids in digital topology lead to topological issues

How to get rid of critical configurations in 2D

2D digitization by intersection: low resolution high resolution there exists a small enough ρ in 2D.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 7

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Cubical grids in digital topology lead to topological issues

Any 3D digitization leads to critical configurations

⇒ even regular objects lead to critical configurations in 3D+.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 8

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Cubical grids in digital topology lead to topological issues

Critical configurations lead to topological issues

discrete topological issues

  • bject counting?

continuous topological issues manifoldness not preserved (“pinch”)

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 9

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Cubical grids in digital topology lead to topological issues

Cross-section topology

Threshold sets/binarizations of u : D → Z: ∀λ ∈ R, [u ≥ λ] = {x ∈ D ; u(x) ≥ λ}, ∀λ ∈ R, [u < λ] = {x ∈ D ; u(x) < λ}. → extension from set operators to graylevel operators (“stacking method”).

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 10

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Cubical grids in digital topology lead to topological issues

The Tree of Shape of an image

[Monasse & Guichard 2000, Caselles & Monasse 2009]: Shapes: US = {Sat(Γ) ; Γ ∈ CC([u ≥ λ], λ ∈ R)}, LS = {Sat(Γ) ; Γ ∈ CC([u < λ]), λ ∈ R}, Shape boundaries = level lines, To compute of the tree of shapes (ToS)...

D E B A C F O

  • r

D E B A C F O

→

A O F B C D E > 2

2

> 2 > 2 > 0

2 1

>

4

> > >

...a necessary condition is: level lines shall be Jordan curves.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 11

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Cubical grids in digital topology lead to topological issues

Ill-definedness of the ToS on cubical grids

ToS with the same connectivity for lower/upper shapes [G´ eraud et al. 2013]:

2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2

We have “intersecting nested” ⇒ the ToS does not exist.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 12

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Usual solutions to get rid of topological issues on cubical grids

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 13

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Usual solutions to get rid of topological issues on cubical grids

Solutions to get rid of topological issues

Many solutions exist: topological reparations interpolations mixed methods Their motivations: no “pinches” in the boundary (manifoldness) no connectivity ambiguity (determinism) both at the same time

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 14

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Usual solutions to get rid of topological issues on cubical grids

Topological reparations in Zn

Methodology: “remove” critical configurations. Problem: “propagation” of the critical configurations. [Latecki et al. 1998/2000] (2D, binary), minimal number of modifications (case-by-case study). [Siqueira et al. 2005/2008] (3D, binary). 3

2 × Card (CCs) modifications (randomized method).

However, modifying the data destroys the topology of the set/ binary image.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 15

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Usual solutions to get rid of topological issues on cubical grids

Topological reparation of cubical complexes

[Gonzalez-Diaz et al. 2011]: the topological reparation of cubical complexes in a homotopy equivalent polyhedral complex. Application: (co)homology computation and recognition tasks. However, the new structure is not cubical.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 16

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Usual solutions to get rid of topological issues on cubical grids

Interpolations with no topological issues (1/2)

[Rosenfeld et al. 1998] (2D): image magnification + C.C. elimination (simple deformations) Property: topology preserving (adjacency tree). [Latecki et al. 2000] (2D): resolution doubling + 0 → 1 Property: sets of black/white/boundary points are WC. [Stelldinger & Latecki 2006] (3D): “Majority Interpolation” (“counting process”) this techniques work on sets, not on graylevel images.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 17

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Usual solutions to get rid of topological issues on cubical grids

Interpolations with no topological issues (2/2)

[Latecki et al. 2000]: 2D, mean/median method (self-dual), [G´ eraud et al. 2015]: 2D, median method (self-dual), [Mazo et al. 2012]: n-D, min-/max-based interpolations (not self-dual),

9 11 15 7 1 13 3 5 3

  • 9

10 11 13 15 8 8 6 12 14 7 4 1 7 13 5 4 3 4 8 3 4 5 4 3 9 10 11 13 15 8 7 6 10 14 7 4 1 7 13 5 4 3 4 8 3 4 5 4 3 9 9 11 11 15 7 1 1 1 13 7 1 1 1 13 3 1 1 1 3 3 3 5 3 3 9 11 11 15 15 9 11 11 15 15 7 7 1 13 13 7 7 5 13 13 3 5 5 5 3

u Latecki G´ eraud Mazo(Imin) Mazo(Imax) Strong property: they “preserve” the topology of the initial image (“no new extrema”).

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 18

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Usual solutions to get rid of topological issues on cubical grids

State-of-the-Art

Latecki et al. 98 2D 3D n-D graylevel self-dual Siqueira et al. 2005 cubical Gonzalez-Diàz et al. 2011 Rosenfeld et al. 98 Latecki et al. 2000 (1) Stelldinger et al. 2006 Latecki et al. 2000 (2) Géraud et al. 2015 Mazo et al. 2012 topo.-pr.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 19

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How to make a self-dual representation in n-D without topological issues

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 20

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How to make a self-dual representation in n-D without topological issues

About self-duality

In practice, no contrast is known a priori: different objects of various contrasts in a same image more complex: nested objects need for a contrast-invariant representation.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 21

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How to make a self-dual representation in n-D without topological issues

Necessary properties of the new representation

Usual cubical signals present topological issues a new representation is needed: n-dimensionality (n ≥ 2), self-duality, no new extrema (in-between), no topological issues (no critical configurations).

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 22

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How to make a self-dual representation in n-D without topological issues

A generalization of DWCness to n-D

n-D blocks: ...

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 23

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How to make a self-dual representation in n-D without topological issues

A generalization of DWCness to n-D

Antagonists: ...

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 23

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How to make a self-dual representation in n-D without topological issues

A generalization of DWCness to n-D

Critical configurations: ...

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 23

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How to make a self-dual representation in n-D without topological issues

A generalization of DWCness to n-D

Critical configurations: ... Definition ([Boutry et al. ISMM 2015]) A digital set X ⊂ Zn, n ≥ 2, is said (digitally) well-composed (DWC) iff it does not contain any critical configuration.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 23

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How to make a self-dual representation in n-D without topological issues

Well-composedness for images

Definition ([Boutry et al. ISMM 2015]) A digital image u : D ⊂ Zn → Z, n ≥ 2, is said DWC iff its threshold sets are DWC. Theorem ([Boutry et al. ISMM 2015]) An image u : D → R is DWC iff ∀p, p′ ∈ D s.t. p′ = antagS(p): intvl(u(p), u(p′)) ∩ Span{u(q) ; q ∈ S \ {p, p′}} ∅.

4 2 [2,4] {0}

no need to check the DWCness of each threshold set.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 24

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How to make a self-dual representation in n-D without topological issues

Computation of a local self-dual DWC n-D interpolation? (1/2)

DWC = local phenomenon (local 2n-connectivity) a local interpolation should be adapted ... usual properties of a local DWC interpolation: locality, DWC,

  • rdered,

in-between, self-duality, translation-/π/2-rotation-invariance.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 25

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How to make a self-dual representation in n-D without topological issues

Computation of a local self-dual DWC n-D interpolation? (2/2)

[Boutry et al. DGCI 2014]:

2 4 4 4 2 3 2 3 2 1 1 2 2 2 1 3 2 3 1 1 3 3 1 m ≤ 1 m ≥ 3 m 2 4 4 4 2

No self-dual local interpolation can make images DWC in n-D (n ≥ 3).

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 26

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How to make a self-dual representation in n-D without topological issues

Threshold sets of Interval-valued maps

Let U : D → IR be an interval-valued map. We define its threshold sets s.t. ∀λ ∈ R: [U ⊲ λ] = {x ∈ D ; ∀v ∈ U(x), v > λ}, [U ⊳ λ] = {x ∈ D ; ∀v ∈ U(x), v < λ}, [U λ] = D \ [U ⊳ λ], [U λ] = D \ [U ⊲ λ].

[ U 7] [ U 6] [ U 5] [ U 4] [ U 3] [1,3] [2,4] [1,3] [4,5] [3,6] [4,5] [1,3] [2,4] [1,3] U Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 27

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How to make a self-dual representation in n-D without topological issues

DWC Interval-valued maps

Definition ([Boutry et al. 2015]) U : D → IR is said DWC iff its threshold sets are DWC. Proposition ([Boutry et al. 2015]) U is DWC iff ⌊U⌋ and ⌈U⌉ are both DWC.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 28

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How to make a self-dual representation in n-D without topological issues

Origin of the front-propagation algorithm

[G´ eraud et al. 2013] computation of the tree of shape: u

immersion − − − − − − − − − − → U sort − − − − → ( u♭, R ) union−find − − − − − − − − − − − → T (u♭) emersion − − − − − − − − − → T (u).

the sorting step “flattens” U into a temporary image u♭ (“front-propagation”)

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 29

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How to make a self-dual representation in n-D without topological issues

The front-propagation algorithm

Input: U (interval-valued); Output: u♭ (single-valued); begin forall the h do deja vu(h) ← false; push(Q[ℓ∞], p∞); deja vu(p∞) ← true; ℓ ← ℓ∞ while Q is not empty do h ← priority pop(Q, ℓ); u♭(h) ← ℓ; forall the n ∈ N2n(h) such as deja vu(n) = false do priority push(Q, n, U, ℓ); deja vu(n) ← true;

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 30

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

{9} {11} {15} {13} {1} {7} {3} {5} {3}

{8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8}

[9,11] [11,15] [7,9] [1,11] [1,11] [1,15] [13,15] [1,13] [1,7] [3,7] [1,7] [1,5] [1,13] [3,13] [3,5] [3,5]

Let us start with a DWC interval-valued map U.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

{9} {11} {15} {13} {1} {7} {3} {5} {3}

{8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8} {8}

[9,11] [11,15] [7,9] [1,11] [1,11] [1,15] [13,15] [1,13] [1,7] [3,7] [1,7] [1,5] [1,13] [3,13] [3,5] [3,5]

Propagation of value ℓ = 8.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

11 11 15 13 13 7 7 1 5 5 5 7 3 5 3

{9} {11} {15} {13} {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

[9,11] [11,15] 8 8 8 8 [13,15] 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5] 9 9

Propagation of value ℓ = 8.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

11 11 15 13 13 7 7 1 5 5 5 7 3 5 3

{9} {11} {15} {13} {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

[9,11] [11,15] 8 8 8 8 [13,15] 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5] 9 9

Propagation of value ℓ = 9.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

11 11 15 13 13 7 7 1 5 5 5 7 3 5 3

9 {11} {15} {13} {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 [11,15] 8 8 8 8 [13,15] 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 9.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

11 11 15 13 13 7 7 1 5 5 5 7 3 5 3

9 {11} {15} {13} {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 [11,15] 8 8 8 8 [13,15] 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 11.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

15 13 13 7 7 1 5 5 5 7 3 5 3

9 11 {15} {13} {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 [13,15] 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 11.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

15 13 13 7 7 1 5 5 5 7 3 5 3

9 11 {15} {13} {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 [13,15] 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 13.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

15 7 7 1 5 5 5 7 3 5 3

9 11 {15} 13 {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 13.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

15 7 7 1 5 5 5 7 3 5 3

9 11 {15} 13 {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 15.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

7 7 1 5 5 5 7 3 5 3

9 11 15 13 {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 15.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

7 7 1 5 5 5 7 3 5 3

9 11 15 13 {1} {7} {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 [1,7] [3,7] [1,7] [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 7.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

1 5 5 5 3 5 3

9 11 15 13 {1} 7 {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 7 7 7 [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 7.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

1 5 5 5 3 5 3

9 11 15 13 {1} 7 {3} {5} {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 7 7 7 [1,5] 8 8 [3,5] [3,5]

Propagation of value ℓ = 5.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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SLIDE 53

How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

1 3 3

9 11 15 13 {1} 7 {3} 5 {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 7 7 7 5 8 8 5 5

Propagation of value ℓ = 5.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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SLIDE 54

How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

1 3 3

9 11 15 13 {1} 7 {3} 5 {3}

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 7 7 7 5 8 8 5 5

Propagation of value ℓ = 3.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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SLIDE 55

How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

1

9 11 15 13 {1} 7 3 5 3

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 7 7 7 5 8 8 5 5

Propagation of value ℓ = 3.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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SLIDE 56

How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

1

9 11 15 13 {1} 7 3 5 3

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 7 7 7 5 8 8 5 5

Propagation of value ℓ = 1.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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SLIDE 57

How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (1/3)

9 11 15 13 1 7 3 5 3

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

9 11 8 8 8 8 13 8 7 7 7 5 8 8 5 5

Result: u♭ is DWC.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 31

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SLIDE 58

How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (2/3)

[Boutry et al. ISMM 2015]:

∀ U DWC, u♭ = FP(U) is DWC.

Note: we proved it in n-D (n ≥ 2).

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 32

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SLIDE 59

How to make a self-dual representation in n-D without topological issues

Front-propagation preserves DWCness (3/3)

Intuition: FP chooses in a set of images one which is “regular” (DWC).

selection

2 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2

... ...

2 2

... ... ... ... ...

2 2

[0,2] [0,2] [0,2] [0,2] [0,2]

{

=

U u♭ ⌊U⌋ ⌈U⌉

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 33

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SLIDE 60

How to make a self-dual representation in n-D without topological issues

Our self-dual DWC n-D interpolation

[Boutry et al. ISMM 2015]:

2 2

selection

2 2 1 1 1 1 1 2 2 2 2 2 2 2 2 2 ... ... 2 2

... ... ... ... ...

Imin Imax

u

ISpan

→ U

FP

→ u♭.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 34

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SLIDE 61

How to make a self-dual representation in n-D without topological issues

Properties of u♭ (1/2)

u♭ is Alexandrov-well-composed (AWC), boundaries in Hn are discrete surfaces, u♭ is Continuously well-composed (CWC), boundaries in Rn are manifolds, u♭ is well-composed based on the Equivalence of connectivities (EWC), same components whatever the connectivity, the ToS of u♭ exists and is connectivity-invariant.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 35

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SLIDE 62

How to make a self-dual representation in n-D without topological issues

Properties of u♭ (2/2)

2D 3D n-D graylevel self-dual cubical topo.-pr. Boutry et al. 2015 (u )

♭

Latecki et al. 98 Siqueira et al. 2005 Gonzalez-Diàz et al. 2011 Rosenfeld et al. 98 Latecki et al. 2000 (1) Stelldinger et al. 2006 Latecki et al. 2000 (2) Géraud et al. 2015 Mazo et al. 2012

all goals have been reached!

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 36

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SLIDE 63

How to make a self-dual representation in n-D without topological issues

Take-home message

We developed a new representation on cubical grids which is: self-dual, n-D, with no topological issues (DWC), topology-preserving (in-between interpolation), deterministic, π/2-rotation-/translation-invariant, in linear time, ... Bonus: many powerful topological properties.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 37

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SLIDE 64

Theoretical Results and Applications

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 38

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SLIDE 65

Theoretical Results and Applications

Theoretical result: “Pure” self-duality

Self-duality equation [G´ eraud et al. 2015]: ToSca,cb(−u) = ToScb,ca(u). we had to switch the connectivities. u♭ DWC ⇒ we obtain “pure” self-duality: ToS(−u♭) = ToS(u♭). Note: any self-dual operator become “purely” self-dual on u♭.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 39

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SLIDE 66

Theoretical Results and Applications

Applications: DWC Laplacian (1/2)

Zero-crossings of the Laplacian ≡ boundaries of objects (image processing). Remark: a hierarchical representation of the Z.-C.’s could be useful: shape recognition, text detection, ... BUT boundaries must be Jordan curves/surfaces: [Huynh et al. 2016] ToS ◦ Sign ◦ IDWC ◦ L.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 40

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SLIDE 67

Theoretical Results and Applications

Applications: DWC Laplacian (2/2)

⇒ ⇒ )

surrounding 0-crossing background

s 0-crossing s pixels e hole 0-crossing e hole interior Y 0-crossing Y pixels e

0-crossing external

epixels

( ( ( ) ))

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 41

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SLIDE 68

Conclusion

Outline

1 Cubical grids in digital topology lead to topological issues 2 Usual solutions to get rid of topological issues on cubical grids 3 How to make a self-dual representation in n-D without topological issues 4 Theoretical Results and Applications 5 Conclusion

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 42

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SLIDE 69

Conclusion

What we did not speak about (1/2)

The different “flavors” of well-composedness and their relationship on cubical grids

2D: EWC ⇔ DWC ⇔ AWC ⇔ CWC [Latecki 1995] 3D: EWC ⇐ DWC ⇔ AWC ⇔ CWC [Latecki 1997] nD: EWC ⇐ DWC

HAL

⇔ AWC

Conj.

⇔ CWC [Boutry et al. 2015] [Boutry et al. 2015] [Najman et al. 2013] [Latecki et al. 2000]

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 43

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SLIDE 70

Conclusion

What we did not speak about (2/2)

n-D topological reparation of graylevel images [Boutry et al. ICIP 2015], n-D reformulation of DWCness for sets (2n-connectivity), hierarchical subdivision on orders, bordered discrete surfaces in polyhedral complexes, AWC interpolation(s) on polyhedral complexes.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 44

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SLIDE 71

Conclusion

Our self-dual DWC interpolation is “central”

Equivalence of connectivities (EWCness) No topological paradoxes / no ambiguities TOPOLOGICAL PROPERTIES Boundaries are discrete surfaces (AWCness)

  • -->Separation property

Boundaries are manifolds (CWCness)

  • --> Surface Parameterization (coordinate charts)
  • --> Calculus on Manifolds (Integrals)
  • --> unicity of the T
  • S
  • --> (2n,2n)-connectivity (faster algorithms)
  • --> Results are connectivity invariant

Continuity (set-valued sense)

  • --> intermediate value theorem

APPLICATIONS Tree of shapes well-defined T ext Detection (DWC Laplacian Sign T

  • S)

Pure self-duality of self-dual MM operators No irreducible thickness problem (thinning) Faster Euler number computation Shapings (grain filters, ...) Monotone plannings Geodesic Dilation/Erosion DWC-PRESERVING TRANSFORMS VISUALIZATION No pinch in the bdCA of threshold sets No crack in the extracted isosurface (M.C.) Our DWC self-dual representation

  • n cubical grids

Interpolation (Imm+FP) Easy to program Linear time COMPUTATION Coherent topological maps (watershed) Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 45

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SLIDE 72

Conclusion

Our self-dual DWC interpolation is “central”

Equivalence of connectivities (EWCness) No topological paradoxes / no ambiguities TOPOLOGICAL PROPERTIES Boundaries are discrete surfaces (AWCness)

  • -->Separation property

Boundaries are manifolds (CWCness)

  • --> Surface Parameterization (coordinate charts)
  • --> Calculus on Manifolds (Integrals)
  • --> unicity of the T
  • S
  • --> (2n,2n)-connectivity (faster algorithms)
  • --> Results are connectivity invariant

Continuity (set-valued sense)

  • --> intermediate value theorem

APPLICATIONS Tree of shapes well-defined T ext Detection (DWC Laplacian Sign T

  • S)

Pure self-duality of self-dual MM operators No irreducible thickness problem (thinning) Faster Euler number computation Shapings (grain filters, ...) Monotone plannings Geodesic Dilation/Erosion DWC-PRESERVING TRANSFORMS VISUALIZATION No pinch in the bdCA of threshold sets No crack in the extracted isosurface (M.C.) Our DWC self-dual representation

  • n cubical grids

Interpolation (Imm+FP) Easy to program Linear time COMPUTATION Coherent topological maps (watershed) Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 45

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SLIDE 73

Conclusion

Our self-dual DWC interpolation is “central”

Equivalence of connectivities (EWCness) No topological paradoxes / no ambiguities TOPOLOGICAL PROPERTIES Boundaries are discrete surfaces (AWCness)

  • -->Separation property

Boundaries are manifolds (CWCness)

  • --> Surface Parameterization (coordinate charts)
  • --> Calculus on Manifolds (Integrals)
  • --> unicity of the T
  • S
  • --> (2n,2n)-connectivity (faster algorithms)
  • --> Results are connectivity invariant

Continuity (set-valued sense)

  • --> intermediate value theorem

APPLICATIONS Tree of shapes well-defined T ext Detection (DWC Laplacian Sign T

  • S)

Pure self-duality of self-dual MM operators No irreducible thickness problem (thinning) Faster Euler number computation Shapings (grain filters, ...) Monotone plannings Geodesic Dilation/Erosion DWC-PRESERVING TRANSFORMS VISUALIZATION No pinch in the bdCA of threshold sets No crack in the extracted isosurface (M.C.) Our DWC self-dual representation

  • n cubical grids

Interpolation (Imm+FP) Easy to program Linear time COMPUTATION Coherent topological maps (watershed) Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 45

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SLIDE 74

Conclusion

Our self-dual DWC interpolation is “central”

Equivalence of connectivities (EWCness) No topological paradoxes / no ambiguities TOPOLOGICAL PROPERTIES Boundaries are discrete surfaces (AWCness)

  • -->Separation property

Boundaries are manifolds (CWCness)

  • --> Surface Parameterization (coordinate charts)
  • --> Calculus on Manifolds (Integrals)
  • --> unicity of the T
  • S
  • --> (2n,2n)-connectivity (faster algorithms)
  • --> Results are connectivity invariant

Continuity (set-valued sense)

  • --> intermediate value theorem

APPLICATIONS Tree of shapes well-defined T ext Detection (DWC Laplacian Sign T

  • S)

Pure self-duality of self-dual MM operators No irreducible thickness problem (thinning) Faster Euler number computation Shapings (grain filters, ...) Monotone plannings Geodesic Dilation/Erosion DWC-PRESERVING TRANSFORMS VISUALIZATION No pinch in the bdCA of threshold sets No crack in the extracted isosurface (M.C.) Our DWC self-dual representation

  • n cubical grids

Interpolation (Imm+FP) Easy to program Linear time COMPUTATION Coherent topological maps (watershed) Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 45

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SLIDE 75

Conclusion

Our self-dual DWC interpolation is “central”

Equivalence of connectivities (EWCness) No topological paradoxes / no ambiguities TOPOLOGICAL PROPERTIES Boundaries are discrete surfaces (AWCness)

  • -->Separation property

Boundaries are manifolds (CWCness)

  • --> Surface Parameterization (coordinate charts)
  • --> Calculus on Manifolds (Integrals)
  • --> unicity of the T
  • S
  • --> (2n,2n)-connectivity (faster algorithms)
  • --> Results are connectivity invariant

Continuity (set-valued sense)

  • --> intermediate value theorem

APPLICATIONS Tree of shapes well-defined T ext Detection (DWC Laplacian Sign T

  • S)

Pure self-duality of self-dual MM operators No irreducible thickness problem (thinning) Faster Euler number computation Shapings (grain filters, ...) Monotone plannings Geodesic Dilation/Erosion DWC-PRESERVING TRANSFORMS VISUALIZATION No pinch in the bdCA of threshold sets No crack in the extracted isosurface (M.C.) Our DWC self-dual representation

  • n cubical grids

Interpolation (Imm+FP) Easy to program Linear time COMPUTATION Coherent topological maps (watershed) Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 45

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SLIDE 76

Conclusion

Our self-dual DWC interpolation is “central”

Equivalence of connectivities (EWCness) No topological paradoxes / no ambiguities TOPOLOGICAL PROPERTIES Boundaries are discrete surfaces (AWCness)

  • -->Separation property

Boundaries are manifolds (CWCness)

  • --> Surface Parameterization (coordinate charts)
  • --> Calculus on Manifolds (Integrals)
  • --> unicity of the T
  • S
  • --> (2n,2n)-connectivity (faster algorithms)
  • --> Results are connectivity invariant

Continuity (set-valued sense)

  • --> intermediate value theorem

APPLICATIONS Tree of shapes well-defined T ext Detection (DWC Laplacian Sign T

  • S)

Pure self-duality of self-dual MM operators No irreducible thickness problem (thinning) Faster Euler number computation Shapings (grain filters, ...) Monotone plannings Geodesic Dilation/Erosion DWC-PRESERVING TRANSFORMS VISUALIZATION No pinch in the bdCA of threshold sets No crack in the extracted isosurface (M.C.) Our DWC self-dual representation

  • n cubical grids

Interpolation (Imm+FP) Easy to program Linear time COMPUTATION Coherent topological maps (watershed) Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 45

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Conclusion Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 46

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Conclusion

Thierry G´ eraud, Yongchao Xu, Edwin Carlinet, and Nicolas Boutry. Introducing the dahu pseudo-distance (submitted). In International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, 2017. Nicolas Boutry, Thierry G´ eraud, and Laurent Najman. Digitally well-composed sets and functions on the n-D cubical grid (in preparation). In Journal of Mathematical Imaging and Vision, 2017. Nicolas Boutry, Laurent Najman, and Thierry G´ eraud. About the equivalence between AWCness and DWCness. Research report, LIGM/LRDE, October 2016. Nicolas Boutry, Thierry G´ eraud, and Laurent Najman. How to make n-D functions digitally well-composed in a self-dual way. In International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, pages 561–572. Springer, 2015. Nicolas Boutry, Thierry G´ eraud, and Laurent Najman. How to make n-D images well-composed without interpolation. In Image Processing (ICIP), 2015 IEEE International Conference on, pages 2149–2153. IEEE, 2015. Nicolas Boutry, Thierry G´ eraud, and Laurent Najman. Une g´ en´ eralisation du bien-compos´ e ` a la dimension n. Communication at Journ´ ee du Groupe de Travail de G´ eometrie Discr` ete (GT GeoDis, Reims Image 2014), November 2014. Nicolas Boutry, Thierry G´ eraud, and Laurent Najman. On making n-D images well-composed by a self-dual local interpolation. In International Conference on Discrete Geometry for Computer Imagery, pages 320–331. Springer, 2014.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 47

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Conclusion

Context: rigid transformations

Topological properties should be preserved under rigid tranformations (continuous VS discrete): “well-composedness”? (no ambiguity) adjacency tree? Methodology [Ngo et al. 2013]): (simply) forbid some critical patterns.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 48

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Conclusion

Context: well-composed segmentations

[Tustison et al. 2011]: front-propagation method s.t.: adds only simple points, does not create any C.C. in the expanded seeds. ⇒ topology- and WCness-preserving FP method. ⇒ boundary of the final segmentation is a manifold (glamorous glue by Jordan arcs).

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 49

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Conclusion

Context: thin topological maps of grayscale images

[Marchadier et al. 2004]: “A discrete image I is the digitization of a piecewise continuous function f.” Methodology: (1) Computation of the gradient (made WC) of I, (2) WC thinning ⇒ WC irreducible image, Note: No ambiguity ⇒ well-defined crest network. (3) Case-by-case study → coherent topological map (representing f).

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 50

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Conclusion

Context: Euler characteristic

ξ(∅) = 0, ξ(S) = 1 if S non-empty and convex, ξ(S1 ∪ S2) = ξ(S1) + ξ(S2) − ξ(S1 ∩ S2). S ⊂ R3 polyhedral ⇒ ξ = η0 − η1 + η2 − η3 (∀ triangulation). ⇒ ξ = b0 − b1 + b2 (topological invariant) License Plates Recognition tasks, Object Counting, ... BUT depends on the connectivity: No critical configuration ⇒ ξ(4,8) = ξ(8,4) ⇒ ξ well-defined Bonus of WCness: in 2D, ξ is locally computable (and then faster).

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 51

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Conclusion

Context: Well-composed Jordan Curves

In Z2 In H2 [Wang & Bhattacharya 1997] Jordan curve theorem holds for WC curves.

Nicolas Boutry (LRDE/LIGM) A Study of Well-composedness in n-D 2016-12-14 52