1 Digital topology and cross-section topology for grayscale image processing M. Couprie A 2 SI lab., ESIEE Marne-la-Vallée, France

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Digital topology andcross-section topology for grayscale image processing

M. Couprie A2SI lab., ESIEEMarne-la-Vallée, Francewww.esiee.fr/a2si

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What is topology ?

« A topologist is interested in those properties of a thing that, while they are in a sense geometrical, are the most permanent- the ones that will survive distortion and stretching. »

Stephen Barr,"Experiments in Topology",1964

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Aims of this talk

Present the main notions of digital topologyShow their usefulness in image processing

applicationsStudy some topological properties of digital

grayscale images (ie. functions from Zn to Z)Topology-preserving image transformsTopology-altering image transformsApplications to image processing (filtering,

segmentation…)

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Some milestones

J.C. Maxwell (1870): ‘On hills and dales ’A. Rosenfeld (1970’s): digital topologyA. Rosenfeld (1980’s): fuzzy digital

topologyS. Beucher (1990): definition of

homotopy between grayscale imagesG. Bertrand (1995): cross-section

topology

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Contributors

Gilles BertrandMichel CouprieLaurent PerrotonZouina Aktouf (Ph.D student)Francisco Nivando Bezerra (Ph.D

student) Xavier Daragon (Ph.D student)Petr Dokládal (Ph.D student)Jean-Christophe Everat (Ph.D student)

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Outline of the talk

Digital topology for binary imagesApplications to graylevel image

processing

Cross-section topology for grayscale images

Topology-altering transforms, applications

Part 1

Part 2

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Digital topology for binary images

8-adjacency, 4-adjacency

in black: X (object)

in white: X (background)

.

path, connected components

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.


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More examples

Black:4, white:8 Black:8, white:4

3 1 2 2

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Jordan property

Any simple closed curve divides the plane into 2 connected components.

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Jordan property (cont.)

A subset X of Z2 is a simple closed curve if each point x of X has exactly two neighbors in X

4-curve 8-curve

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Not a 4-curve

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Not an 8-curve

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The Jordan property does no hold if X and its complement have the same adjacency

8-adjacency 4-adjacency

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Topology preservation - notion of simple point

A topology-preserving transform must preserve the number of connected components of both X and X.

Definition (2D): A point p is simple (for X) if its modification (addition to X, withdrawal from X) does not change the number of connected components of X and X.

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Simple point: examples

Set X (black points)

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Simple point of X

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Simple point of X

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Simple point: counter-examples

Non-simple point (background component creation if deleted)

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Simple point: counter-examples

Non-simple point (component splitting if deleted)

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Simple point: parallel deletion

Deleting simple points in parallel may change the topology

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Simple point: local characterization ?

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Simple point: local characterization ?

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Connectivity numbers

T(p)=number of Connected Components of (X - {p}) N(p) where N(p)=8-neighborhood of p

T(p)=number of Connected Components of (X - {p}) N(p)A

Characterization of simple points (local): p is simple iff T(p) = 1 and T(p) = 1 A

T=2,T=2 T=1,T=1 T=1,T=0 T=0,T=1

U

U

Interior point Isolated point

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Skeletons

We say that Y is a skeleton of X if Y may be obtained from X by sequential deletion of simple points

If Y is a skeleton of X and if Y contains no simple point, then we say that Y is an ultimate skeleton of X

If Y is a skeleton of X and if Y contains only non-simple points and ‘end points’, then we say that Y is a curve skeleton of X

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End points

We say that p is an end point for X if (X - {p}) N(p) contains exactly one point

End point End point (8) Non-end point Non-end point

U

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Skeletons (examples)

Original image Curve skeleton Ultimate skeleton

End points

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Homotopy

We say that X and Y are homotopic (i.e. they have the same topology) if Y may be obtained from X by sequential addition or deletion of simple points

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Basic thinning algorithms

Ultimate thinning(X)Repeat until stability:

Select a simple point p of XRemove p from X

Curve thinning(X)Repeat until stability:

Select a simple, non-end point p of XRemove p from X

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Centering skeletons

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Breadth-first thinning

Simple pointsdetectedduring the 1stiterationCandidatesfor the 2nditeration

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Breadth-first thinning

Breadth-first ultimate thinning(X)T := X ; T’ := Repeat until stability

While p T do T := T \ {p} If p is a simple point for X then

X := X \ {p} For all q neighbour of p, q X,

do T’ := T’ {q}

T := T’ ; T’ :=

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Directional thinning

(only in 2D)

NorthSouthEastWest

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Parallel directional algorithm

In 2D, simple and non-end points of the same type (N,S,E,W) can be removed in parallel

Directional Curve thinning(X) Repeat until stability:

For dir =N,S,E,W Let Y be the set of all simple, non-end points of X of type dir X = X \ Y

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Does not work in 3D

In 3D, there are 6 principal directions: N,S,E,W,Up,Down

Up

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Thinning guided by a distance map

Let X be a subset of Z2, the distance map DMX is defined by, for all point p:

DMX(p) = min{dist(p,q), q not in X}

where dist is a distance (e.g. city block, chessboard, Euclidean)

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Discrete distances

City block distance (d4)d4(p,q) = length of a shortest 4-

pathbetween p and q

Chessboard distance (d8)d8(p,q) = length of a shortest 8-

pathbetween p and q

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Discrete distance maps

1 11 1 1 111111111

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11111111

11 1 11 1

1 111

11

22222222

22222222

2222

22 22

222233

3 3 3 3

3 3 3 333

33

44

44

22222222

2 2222222

2222

22 22

33 3

3

3 3 3 333

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4 4

1 11 1 1 111111111

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11111111

11 1 11 1

1 111

11

1 1

11

11

11

d4 d8

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Discrete and Euclidean distance maps

d8

d4de

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Thinning guided by a distance map: algorithm

Ultimate guided thinning(X)Compute the distance map DMX

Repeat until stability:Select a simple point p of X

such that DMX(p) is minimalRemove p from X

Curve guided thinning(X)Id., replace « simple » by « simple non end »

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Thinning guided by a distance map: results

d8

d4de

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The 3D case

A topology-preserving transform must preserve:

- number of connected components of X

- number of connected components of X

- number of « holes » (or « tunnels »)

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3D hole (tunnel)

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3D hole (tunnel)

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3D-skeletons

Originalobject

Surfaceskeleton

Curveskeleton

Ultimateskeleton

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Topological classification of points (G. Bertrand)

T=1,T=1 T=1,T=2 T=3,T=1(1D isthmus)(simple point) (2D isthmus)

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3D hole closing

G. BertrandZ. Aktouf (Ph. D)L. Perroton

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3D hole closing

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3D hole closing

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3D hole closing

T=2,T=1 (1D isthmus)

T=1,T=2 (2D isthmus)

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3D hole closing

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3D hole closing algorithm

Let X be an object and Y be a simply connected set containing X

We remove iteratively all points x of Y \ X such that T(x) = 1(i.e. simple points and 1D isthmus)

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3D hole closing

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3D hole closing using a distance map

The points are processed in an order depending on their distance from X (points far from X are selected first)

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When a point p with T=2 is selected, its removal would create a hole. The value of the distance map for this point (DMX(p)) gives an information on the « size » of the corresponding hole in X.

We can decide to let a hole open or to close it according to this criterion.

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3D hole closing: illustration

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Outline of the talk


processing



Part 1

Part 2

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Liver vascular system segmentation

Contrast-enhanced tomography scan of liver

(2-D cut from a 3-D image)

G. BertrandP. Dokladal (Ph. D)

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Intensity-guided skeletonization

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Segmentation results

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Skeleton Filtering (1)

Noisy image Noisy skeleton

Identification of relevant

components

Filtered skeleton

Filtering criterion based on minimum accepted meanluminosity of each skeleton segment.

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Skeleton Filtering (2)

Two criteria based on local information:

•mean luminosity

•amplitude difference

Filtering criterion based on detection of irrelevant end points

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Skeleton Filtering (results)

Non filtered Filtered

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Segmentation of the brain cortex

M. CouprieX. Daragon (PhD)

Basic assumption: the geometry of the brain is complex, but its topology is simple

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Segmentation of the brain cortex

Cortex (gray matter)

White matter

Cerebro-spinal fluid

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Segmentation of the brain cortex: main steps

Pre-segmentation

Extension to the cortex

White matter extraction

Cortex extraction

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Extraction of the white matter

Morphological closing with topological control

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Morphological closing with topological control

A: original image B: opening of Awith a small disc

C: ultimate skeletonof A in B

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Extension to the cortex

W.M.

W.M.

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Extension to the cortex: use of a distance map

Section of the 3D segmentation of the white matter

Section of its 3D distance map (limited to distances < 1cm)

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Constraint set: centers of maximal balls

Distance map (section)

Centers of maximal balls

A maximal ball (for a set X) is a ballincluded in X which is not included

in any other ball included in X.The set of the centers of maximal balls

for X can be computed fromthe distance map.

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Results (sections of 3D images)

Without constraint set With constraint set

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Results (3D view)

Without constraint set With constraint set

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Results (3D view)

White matter Cortex

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Conclusion (1st part)

Image simplification with topology preservation

Controled topology modificationBinary topological operators guided

by a distance map or by a grayscale image

Efficient implementations

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Outline of the talk


processing



Part 1

Part 2

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Cross-section topology

G. BertrandM. CouprieJ.C. Everat (PhD)F.N. Bezerra (PhD)

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Cross-section topology

Basic idea: consider the topology of each cross-section (threshold) of a function.

Given a function F (Z2 Z) and k in Z, we define the cross-section Fk as the set of points p of Z2 such that F(p) k.

We say that two functions F and G are homotopic if, for every k in Z, Fk and Gk are homotopic (in the binary sense)

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Homotopy: an illustration

x

yF(x,y)

F3

F2

F1

x

G(x,y)

G3

G2

G1

y

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Destructible point(G. Bertrand)

Definition: a point p is destructible (for F) if it is simple for Fk, with k = F(p) .

Property: p is destructible iff its value may be lowered by one without changing the topology of any cross-section.

Definition: a point p is constructible (for F) if it is destructible for -F (duality)

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Destructible point: examples

x

yF(x,y)

F3

F2

F1

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Destructible point: examples

x

yF(x,y)

F3

F2

F1

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Destructible point: counter-examples

x

yF(x,y)

F3

F2

F1

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Destructible point: counter-examples

x

yF(x,y)

F3

F2

F1

Componentdeleted

Componentsplitted

Backgroundcomponentcreated

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Connectivity numbers

N+(p) = {q in N(p), F(q) F(p)}T+(p) = number of Conn. Comp. of N+(p)

.

N--(p) = {q in N(p), F(q) < F(p)}T--(p) = number of Conn. Comp. of N--(p)

.

N++, T++, N-, T- : similar If an adjacency relation (eg. 4) is chosen

for T+, T++, then the other adjacency (8) must be used for T-, T--

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Connectivity numbers: examples

1 2 19 5 19 9 9

T+ = 1T-- = 1

1 2 89 5 89 1 1

T+ = 2T-- = 2

1 2 11 5 12 2 1

T+ = 0T-- = 1

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Destructible point: local characterization

Property: the point p is destructible iff T+(p) = 1 and T--(p) = 1

1 2 19 5 19 9 9

T+ = 1T-- = 1

1 2 89 5 89 1 1

T+ = 2T-- = 2

1 2 11 5 12 2 1

T+ = 0T-- = 1

destructible non-destructible non-destructible

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Classification of points(G. Bertrand)

The local configuration of a point p corresponds to exactly one of the eleven following cases:

well (T- = 0) minimal constructible

(T++ = T- = 1, T-- = 0) minimal convergent

(T++ > 1, T-- = 0) constructible divergent

(T++ = T- = 1, T-- > 1)

peak (T+ = 0) maximal destructible

(T+ = T-- = 1, T++ = 0) maximal divergent

(T-- > 1, T++ = 0) destructible convergent

(T+ = T-- = 1, T++ > 1) interior (T++ = T-- = 0) simple side (T+ = T-- = T++ = T- =

1) saddle (T++ > 1, T-- > 1)

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Classification of points: examples

1 2 19 5 19 9 9

1 2 89 5 89 1 1

1 2 11 5 12 2 1

Simple side Saddle

Peak

1 2 19 9 19 9

1 29 9 9

1 1

1 2 15

Maximaldestructible

Maximaldivergent

Destructibleconvergent

5 5 55 5 55 5 5

Interior

1

2

199

995

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Grayscale skeletons

We say that G is a skeleton of F if G may be obtained from F by sequential lowering of destructible points. .

If G is a skeleton of F and if G contains no destructible point, then we say that G is an ultimate skeleton of F

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Ultimate skeleton: 1D example

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Ultimate skeleton: illustration

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Ultimate skeleton: illustration

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Regional minima

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Original image F

Regionalminimaof F(white)

UltimateskeletonG of F

Regionalminimaof G(white)

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(

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Watersheds

Powerful segmentation operator from the field of Mathematical Morphology

Introduced as a tool for segmenting grayscale images by S. Beucher and C. Lantuejoul in the 70s

Efficient algorithms based on immersion simulation were proposed by L. Vincent, F. Meyer, P. Soille (and others) in the 90s

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Watersheds

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Watersheds: illustration

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)

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Ultimate skeleton vs. Watershed line

1 1 111 3

1 1 1

31 311

3 3

333

3

3 3 3 3

3

32

1 1 1

111 1 1 1 1 1 1 1 1

1111111

Ultimate skeleton(non-minimal points)

1 1 11 6 61 6 3

1 1 16 63

1 6 31 61

73 3

37 7

777 7 73

3

3

3 3 3 3

3

3

6 6 6

2 5666 6 6 6 6 6 6

66666

1 1 1

111 1 1 1 1 1 1 1 1

1111111

Watershed

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3

3

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66666

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66666

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666 6 6 6 6 6 6

66666

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Ultimate skeleton vs. Watershed line

1 1 11 6 61 6 1

1 1 16 61

1 6 11 61

71 1

17 7

777 7 71

1

1

1 1 1 1

1

1

6 6 6

2 6666 6 6 6 6 6 6

66666

1 1 1

111 1 1 1 1 1 1 1 1

1111111

1

1

1

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1

1

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6 66

6 6

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7 7 777

7 7 7

6 6 6

666 6 6 6 6 6 6

66666

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6

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6A watershed point cannotbe characterized locally

Central point: belongsto the watershed line

Central point: does notbelong to the watershed line

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Thinness

In the previous examples, the set of non-minimal points of an ultimate skeleton was « thin » (a set X is thin if it contains no interior point). Is it always true ?

The answer is no, as shown by the following counter-examples.

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Thinness

11

31

3 3 3 3 3 3 3 31 1

111 1 1

13 3 3 3 3 3 3 3 3

3333 3

33

33

3 133

333

111 3 33 13 31 1 1 133 33 3

1 1 1 133 3 3 33 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 33333 3

33

33

333

333

3 333 333 33 333 3 3 3

3 3 3 3 3 3 3 3 3

3

1 1 1 1 11 11 1 1 1 1 1

1 1 11 1 11 1 11 1 1

1 1 11 1 11 1 11 1 1

3 3 3 3 3 33

333

333

3

333

333

3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3

322222

3 3 3 3 3 33

333

333

3

333

333

3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3

322222

2

111

2 22

22 2

Thinness

1 1 1 1 11 11 1 1

1 1 11 1 11 11 1 1

1 1 1

1 1 11

11 1 1 1

1 1 1 1 11 1 1 1 111

1 1 11 1 1

11

1 1 11 1 11 1 11 1 11 1 11 1 1

1 11 1 1

1 11 1 1

3 3 3 3 3 3 3 3 33 3 3 3 3

3 3 3

33

333

33 3

33 3

32 2

2 2 22 2 22 2 2

222

3

333

33333 3 3 3 3 3 3 3 33 3 3 3

3

333

33333

3 3 3 3 3

3

3

3

333

133

333

3 3 3 3

22

22

221

11

3 3 3 3 3

3

3

3

333

33

333

3 3 3 3

2 2 22 22

22 2

2 2 2

2 22

22 2

3 3 3 3 3 3 3 3 33 3 3 3 3

3 3 3

33

333

33 3

33 3

3

2 2 22 2 22 2 2

222

3

333

33333 3 3 3 3 3 3 3 33 3 3 3

3

333

33333

2 2 22 2 22 2 2

2

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Basic algorithm

Basic ultimate grayscale thinning(F)

Repeat until stability: Select a destructible point p for F F(p) := F(p) – 1

Inefficient: O(n.g), where: n is the number of pixels g is the maximum gray level

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Lowest is best

The central point is destructible: it can thus be lowered down to 5 without changing the topology.

It can obviously be lowered more:- down to 3 (since there is no value

between 6 and 3 in the neighborhood)- down to 1 (we can check that once at

level 3, the point is still destructible)

3 1 19 6 19 9 9

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Two special values

If p is destructible, we define:

-(p)=highest value strictly lower than F(p) in the neighborhood of p

-(p)=lowest value down to which F(p) can be lowered without changing the topology

3 1 19 6 19 9 9

Here: -(p)=3

-(p)=1

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Better but not yet good

If we replace: F(p) := F(p) – 1 in the basic

algorithm by: F(p) := -(p), we get a faster algorithm. But its complexity is still bad. Let us show why:

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Fast algorithm

Fast ultimate grayscale thinning(F)

Repeat until stability: Select a destructible point p for F of minimal graylevel

F(p) := -(p)

- Can be efficiently implemented thanks to a hierarchical queue- Execution time roughly proportional to n (number of pixels)

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Complexity analysis: open problem

1 1 1 1 11 11 1 1

1 1 11 1 11 11 1 1

1 1 1

1 1 11

11 1 1 1

1 1 1 1 11 1 1 1 111

1 1 11 1 1

11

1 1 11 1 11 1 11 1 11 1 11 1 1

1 11 1 1

1 11 1 1

3 3 3 3 3 3 3 3 33 3 3 3 3

422

3

333

33333 3 3 3 3 3 3 3 33 3 3 3

3

333

33333

1 11

1 11

33

333

333

34

4 4 4444

444

3

2 2 22 2 22 2 2

3241

1 1 1

1 1 11

1 1 1

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Outline of the talk


processing



Part 1

Part 2

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G. BertrandM. CouprieJ.C. Everat (PhD)F.N. Bezerra (PhD)

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Altering the topology

Control over topology modification.

Criteria: Local contrast : notion of -skeleton Regional contrast : regularization Size : topological filtering Topology : crest restoration

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Non-homotopic operators

Topology preservation: strong restriction

Our goal: Change topology in a controlled way

over segmentation

regional minima

segmented regions

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-destructible point

-destructible

not -destructible

Illustration (1D profile of a 2D image)

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-destructible point

Definition:

Let X be a set of points, we define F-(X)=min{F(p), p

in X}

Let be a positive integer

A destructible point p is -destructible A k-divergent point p is -destructible if at least k-1

connected components ci (i=1,…,k-1) of N--(p) are such that F(p) - F-(ci)

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-skeleton

.

We say that G is a -skeleton of F if G may be obtained from F by sequential lowering of -destructible or peak points

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-skeleton : example

1 1 12 9 91 9 2

1 2 19 9 11 9 21 9 19 9 11 1 1

2 9 11 9 91 1 2

Original image

1 25 41 41 54 41 1

1 1 12 9 91 9 1

1 2 19 9 11 9 21 9 19 9 11 1 1

2 9 11 9 91 1 2-skeleton (=3)

1 21 11 11 11 11 1

The order of operations matters

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-skeleton : examples

= 0 = 15 = 30

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Leveled skeleton

We say that G is a leveled skeleton of F if G may be obtained from F by sequential lowering of destructible or peak points.

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Leveled skeleton: example

1 1 12 7 61 9 2

1 2 16 8 11 9 21 7 19 8 11 1 1

2 8 11 9 91 1 2

1 1 12 6 61 8 1

1 2 16 6 11 8 21 7 18 8 11 1 1

2 8 11 8 91 1 2

1 1 12 6 61 6 1

1 2 16 6 11 7 21 7 18 7 11 1 1

2 8 11 8 81 1 2

Original image After 7 steps After 11 steps

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1 1 12 6 61 6 1

1 2 16 6 11 7 21 7 18 7 11 1 1

2 8 11 8 81 1 2

1 1 12 6 61 6 1

1 2 16 6 11 6 21 6 16 6 11 1 1

2 6 11 6 81 1 2

1 1 12 6 61 6 1

1 2 16 6 11 6 21 6 16 6 11 1 1

2 6 11 6 61 1 2After 18 steps(final result)

After 11 steps(reminder)

After 17 steps

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Original image

Regional minima Regional minima

Leveled skeleton

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Regularization

In a leveled skeleton, the regional minima are separated by ‘thin crest lines’ .

The graylevel on these lines correspond to the ‘altitude of the lowest pass connecting two neighboring minima’ (in the skeleton, and in the original image as well) .

This allows to detect and modify ‘irregular crest points’

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Regularization (cont.)

a

b

x

If a > b then x is irregular

a

bx

If a > b then x is irregular

Leveled skeleton

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Regularization: example

Original image Regularized leveled skeleton

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Binary reconstruction

Original image Regularized skeleton

a

b

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Regularization and reconstruction: segmentation without any parameter

Original image Regularized leveled skeleton

Binaryreconstruction

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Topological filtering

A: original C: reconstruction ofB under A

B: thinning+peak deletion

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Topological filtering (cont.)

Original image

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Homotopic thinning (n steps)

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Peak deletion

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Homotopic reconstruction

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Final result

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Comparison with the morphological approach

Morphological opening (erosion,dilation): makes no difference between a disk of diameter d and an elongated object of thickness d.

Area opening: makes no difference between a disk and an elongated object having the same area

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Crest restoration

Motivation

Thinning + thresholding

GradientOriginal image

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Crest restoration (cont.)

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p is a separating point if there is k such that T(p, Fk)=2

p is extensible if p is a separating point, and

p is a constructible or saddle point, and

there is a point q in its neighborhood that is an end point or an isolated point for Fk, with k=F(p)+1

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p is a separating point if there is k such that T(p, Fk)=2

p is extensible if p is a separating point, and

p is a constructible or saddle point, and

there is a point q in its neighborhood that is an end point or an isolated point for Fk, with k=F(p)+1

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Thinning +thresholding

Thinning +crest restoration +

thresholdingGradient

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Thinning +crest restoration +

thresholding(1 parameter)

Thinning+hysteresis thresholding

(2 parameters)

Thinning+hysteresis thresholding

(2 parameters)

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Crest Restoration: result

‘Significant’ crests have been highlighted (in green)

Before crest restoration:

After crest restoration:

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Centering grayscale skeletons

Breadth-first ultimate thinning(F)T := domain of F ; T’ := Repeat until stability

While p T do T := T \ {p} If p is a destructible point for F then

F(p) := -(p) For all q neighbour of p, do T’ := T’ {q}

T := T’ ; T’ :=

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Breadth-first ultimate thinning : example

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New strategies to reduce anisotropy

Generalized Euclidean distance map(computes one distance map for each cross-section)

Dynamic estimation of Euclidean distance(uses the Danielson’s principle and approximation)

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Reducing anisotropy: results

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Conclusion (2nd part)

Strict preservation of both topological and grayscale information

Combining topology-preserving and topology-altering operators

Control based on several criteria (contrast, size, topology)

Strategies to reduce anisotropy

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Perspectives

Study of complexityExtension to 3DTopology in orders (G. Bertrand)

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Perspectives

G. Bertrand, J. C. Everat and M. Couprie: "Image segmentation through operators based upon topology", Journal of Electronic Imaging, Vol. 6, No. 4, pp. 395-405, 1997.

M. Couprie, F.N. Bezerra, Gilles Bertrand: "Topological operators for grayscale image processing", Journal of Electronic Imaging, Vol. 10, No. 4, pp. 1003-1015, 2001.

www.esiee.fr/~coupriem/Sdi/publis.html

http://www.esiee.fr/~info/a2si/Ps/elimag97.ps.gz




Documents

1 Digital topology and cross-section topology for grayscale image processing M. Couprie A 2 SI lab., ESIEE Marne-la-Vallée, France