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$Page 1: SIS Digital Geometry - Institut d'électronique et d ...igm.univ-mlv.fr/~kenmochi/ens/dg2016/dg2.pdf · UPEM Master 2 Informatique \SIS" Digital Geometry Topic 2: Digital topology:$

UPEM Master 2 Informatique “SIS”

Digital Geometry

Topic 2:Digital topology: object boundaries and

curves/surfaces

Yukiko Kenmochi

October 5, 2016Digital Geometry : Topic 2 1/34

Opening

Representations of discrete shapes

grid point set

graph

(grid points + adjacent relations)

complex

(grid cells + neighboring relations)

Digital Geometry : Topic 2 2/34

Opening

grid point set

graph (grid points + adjacent relations)

complex

(grid cells + neighboring relations)

Opening

grid point set

graph (grid points + adjacent relations)

complex (grid cells + neighboring relations)

Point set representation

Object boundary in the Euclidean space

For A ⊂ Rn, the set of interior points is defined by

Int(A) = {x ∈ A : ∃r ∈ R+,Ur (x) ⊆ A}where

Ur (x) = {y ∈ Rn : ‖x − y‖ < r}.

The set of border points is:

Br(A) = A \ Int(A).

Then we obtain the set of boundary points such that

Fr(A) = Br(A) ∪ Br(A) = Fr(A).

(Hausdorff, 1937)Digital Geometry : Topic 2 3/34

Ur (x) = {y ∈ Rn : ‖x − y‖ < r}.The set of border points is:

Br(A) = A \ Int(A).

Ur (x) = {y ∈ Rn : ‖x − y‖ < r}.The set of border points is:

Br(A) = A \ Int(A).

Object boundary in the nD discrete space

For A ⊂ Zn, the set of m-interior points is defined by

Intm(A) = {x ∈ A : Nm(x) ⊆ A}where

Nm(x) = {y ∈ Zn : dm(x , y) ≤ 1}where m = 4, 8 for n = 2 and m = 6, 18, 26 for n = 3.

The set of m-boundary points is:

Fr(A) = Brm(A) ∪ Brm(A)

whereBrm(A) = A \ Intm(A) m-interior border,Brm(A) = A \ Intm(A) m-exterior border.

A

Br4(A) Br4(A)

For A ⊂ Zn, the set of m-interior points is defined by

Intm(A) = {x ∈ A : Nm(x) ⊆ A}where

Nm(x) = {y ∈ Zn : dm(x , y) ≤ 1}where m = 4, 8 for n = 2 and m = 6, 18, 26 for n = 3.The set of m-boundary points is:

A Br4(A) Br4(A)

Neighborhoods in the 2D discrete space

Definition (m-neighborhood)

The m-neighborhood of a grid point x ∈ Z2 is defined by:

Nm(x) = {y ∈ Z2 : dm(x , y) ≤ 1}

where dm(x , y) = ‖x − y‖p for m = 4, 8 if p = 1,∞ respectively.

Norm on a d-dimensional vector space: ‖x‖p =

(d∑

i=1|xi |p

) 1p

(Manhattan norm for p = 1, Euclidean norm for p = 2, Maximum norm for p =∞)

Neighborhoods in the 3D discrete space

Definition (m-neighborhood)

The m-neighborhood of a grid point x ∈ Z3 is defined by:

Nm(x) = {y ∈ Z3 : dm(x , y) ≤ 1}

for m = 6, 18, 26 where

d6(x , y) = ‖x − y‖1,

d26(x , y) = ‖x − y‖∞,

d18(x , y) = max

{d26(x , y),

⌈d6(x , y)

2

⌉}.

In the discrete space, a set A and its complement A do not have the commonboundary. The boundary of A consists of elements in A, and that of Aconsists of elements in A. (Clifford, 1956)

Alternative definition of m-border points:

Brm(A) = {x ∈ A : Nm(x) ∩ A 6= ∅}.

A Br4(A) Br4(A)Digital Geometry : Topic 2 7/34

In the discrete space, a set A and its complement A do not have the commonboundary. The boundary of A consists of elements in A, and that of Aconsists of elements in A. (Clifford, 1956)

Alternative definition of m-border points:

Brm(A) = {x ∈ A : Nm(x) ∩ A 6= ∅}.

A Br4(A) Br4(A)Digital Geometry : Topic 2 7/34

Graph representation

Adjacency graph

Definition (m-adjancency)

If a grid point x is m-neighboring from another distinct grid point y , xand y are m-adjacent, denoted by x ∈ Am(y) and y ∈ Am(x).

Definition (Adjacency graph (Rosenfeld 1970))

For a given grid point set X ⊆ Zn, the adjacency graph is defined by

G = (X,Em)

where Em = {(x , y) ∈ X× X : y ∈ Am(x)}.

Adjacency graph

Definition (m-adjancency)

If a grid point x is m-neighboring from another distinct grid point y , xand y are m-adjacent, denoted by x ∈ Am(y) and y ∈ Am(x).

Definition (Adjacency graph (Rosenfeld 1970))

For a given grid point set X ⊆ Zn, the adjacency graph is defined by

G = (X,Em)

where Em = {(x , y) ∈ X× X : y ∈ Am(x)}.

Path

Definition (m-Path)

Let X be a set of grid points. An m-path in X joining two points p andq of X is a sequence π = (p0, . . . ,pn) of points in X such thatp0 = p, pn = q and pi ∈ Am(pi−1) for i = 1, . . . , n.

(Klette, Rosenfeld, 2003)

In general, m = 4, 8 for 2D and m = 6, 18, 26 for 3D.

Path

Definition (m-Path)

Let X be a set of grid points. An m-path in X joining two points p andq of X is a sequence π = (p0, . . . ,pn) of points in X such thatp0 = p, pn = q and pi ∈ Am(pi−1) for i = 1, . . . , n.

(Klette, Rosenfeld, 2003)In general, m = 4, 8 for 2D and m = 6, 18, 26 for 3D.

Discrete object (connected component)

Definition (m-object)

A set X of grid points is an m-object if there exists an m-path in X forevery pair p and q of X .

In other words, an m-object is a connected component of a graphG = (X ,Em).

Discrete object (connected component)

Definition (m-object)

A set X of grid points is an m-object if there exists an m-path in X forevery pair p and q of X .

In other words, an m-object is a connected component of a graphG = (X ,Em).

Connected component labeling (of a graph)

Algorithm (Connected components)

Input: Graph G , starting vertex s

Put s in the queue (or stack) L.

while L 6= ∅ dopull a vertex t from L.Label all the neighbors of t that are not labelled and put them in L.

It allows to calculate the connected components of a graph in lineartime.

breadth-first search

depth-first search

(Hopcropft and Tarjan, 1973)

depth-first search

Discrete curve

An m-path π is also called an m-curve.

Definition (m-curve)

An m-path π is an m-curve if for all the elements pi of π, i = 1, . . . , n,pi has exactly two m-adjacent points in π, except for p0 and pn thathave only one.

Definition (closed m-curve)

An m-curve π = (p0, . . . ,pn) is a closed m-curve if p0 = pn.

Definition (simple closed m-curve)

An m-curve π is a simple closed m-curve if every element of π ism-adjacent to exactly two other points of π.

Discrete curve

An m-curve π that is not closed is also called an m-arc.

Discrete curve

Jordan curve theorem

Theorem (Jordan curve theorem (Jordan, 1887))

Let C be a simple closed curve in the plane R2, called a Jordan curve. Then,its complement R2 \ C consists of exactly two components, the interior andexterior, and C is their boundary.

Problem

The discrete version of Jordan theorem does not hold for simple closedm-curve.

If the curve is connected, it does not disconnect its interior from its exterior(8-connectedness); if it is totally disconnected it does disconnect them (4-connectedness).

(Rosenfeld, Pflatz, 1966)

Jordan curve theorem

Theorem (Jordan curve theorem (Jordan, 1887))

Let C be a simple closed curve in the plane R2, called a Jordan curve. Then,its complement R2 \ C consists of exactly two components, the interior andexterior, and C is their boundary.

Problem

The discrete version of Jordan theorem does not hold for simple closedm-curve.

If the curve is connected, it does not disconnect its interior from its exterior(8-connectedness); if it is totally disconnected it does disconnect them (4-connectedness).

(Rosenfeld, Pflatz, 1966)Digital Geometry : Topic 2 13/34

Good adjacency pairs for 2D binary images

Theorem (Separation theorem (Duda, Hart, Munson, 1967))

A simple closed m-curve C m′-separates all pixels inside C from all pixelsoutside C , for (m,m′) = (4, 8), (8, 4).

Definition (Generarisation: good adjacency pairs (Kong, 2001))

(α, β) is called a good pair iff, for (m,m′) ∈ {(α, β), (β, α)}, any simpleclosed m-curve m′-separates its (at least one) m′-holes from the backgroundand any totally m-disconnected set cannot m′-separate any m′-hole from thebackground.

Good adjacency pairs for 2D binary images

Theorem (Separation theorem (Duda, Hart, Munson, 1967))

A simple closed m-curve C m′-separates all pixels inside C from all pixelsoutside C , for (m,m′) = (4, 8), (8, 4).

Definition (Generarisation: good adjacency pairs (Kong, 2001))

(α, β) is called a good pair iff, for (m,m′) ∈ {(α, β), (β, α)}, any simpleclosed m-curve m′-separates its (at least one) m′-holes from the backgroundand any totally m-disconnected set cannot m′-separate any m′-hole from thebackground.

Exercise: shape border

Question

Consider the set A of discrete points (black points) in the figure.

1 Illustrate the 4-interior border Br4(A).

2 Illustrate the 8-interior border Br8(A).

3 Does each Brm(A) for m = 4, 8 form a simple closed curve? If theanswer is positive, what adjacency pair (a, a′) should be used forthe curve and its complement so that the discrete version ofJordan separation theorem holds?

Exercise: shape boundary (answers)

1-2

3 Br4(A) does not form a simple closed curve; there is a point that hasmore than two adjacent points.

On the other hand, Br8(A) forms a simple closed 4-curve if theadjacency pair (4, 8) is used. (Similarly, the following answer is alsoaccepted: Br4(A) forms a simple closed m-curve if one of the adjacencypairs (m,m′) = (4, 4), (4, 8) is used.)

Exercise: shape boundary (answers)

1-2

3 Br4(A) does not form a simple closed curve; there is a point that hasmore than two adjacent points.

On the other hand, Br8(A) forms a simple closed 4-curve if theadjacency pair (4, 8) is used. (Similarly, the following answer is alsoaccepted: Br4(A) forms a simple closed m-curve if one of the adjacencypairs (m,m′) = (4, 4), (4, 8) is used.)

Inter-pixel boundary of a discrete shape

Let us consider a discrete space as a pair (V ,W ) where V is a countable set

and W is a symmetric relation on V × V .

For example: (V ,W ) = (Z2,E4), (Z2,E8).

Definition (Inter-pixel boundary)

Let (V ,W ) be a discrete space, and X be a subset of V . The boundary of Xand its complement X is defined by

∂(X,X) = {(u, v) ∈W : u ∈ X ∧ v ∈ X}.

Note that every element of ∂(X,X) is directed.

∂(X,X) = {(u, v) ∈W : u ∈ X ∧ v ∈ X}.

(Klette, Rosenfeld, 2003)Digital Geometry : Topic 2 17/34

Cell complex approach

Cell complex

Definition (Cell complex)

A cell complex is a set C of cells such that

the empty cell is included in C ,

all the faces of every cell of C also belong to C ,

the intersection of two cells is one of their common faces.

The r -cell is an r -dimensional convex polyhedron.

Simplicial complex Cubical complex

Face of complex

Definition (Face)

A face of an r -cell σ is an s-cell that is included in the boundary of σwith s < r .

3-cell its faces

Digital image and complex representations

Grid point set(digital image) Adjacency graph

Kovalevsky topology(cubical complex)

Kovalevsky topology

For a digital image,

we define cells as

Definition

Kovalevsky topology is defined by

C = (A,B)

such that A is a set of cells and B ⊂ A× A is a set of their orders.

Order of cells

cell order

If an r -cell σ is a face of s-cell τ , then

σ < τ.

Note: r < s.

nD discrete object and boundary (Kovalevsky topology)

Definition (nD discrete object (n-complex))

Given an m-object A ⊂ Zn (m = 4, 8 for n = 2 and m = 6, 18, 26 for n = 3)we obtain the set Kn of n-cells whose centroid are the points of A. The nDdiscrete object is then represented by the n-complex given by

K = Kn ∪ {τ < σ : σ ∈ Kn}.

Definition (Boundary ((n-1)-cell set))

The boundary of a nD discrete object in Kovalevky topology is the set of(n − 1)-cells located between the interior border and the exterior border.

The same boundary as the inter-pixel/voxel boundary in the dual sense.

K = Kn ∪ {τ < σ : σ ∈ Kn}.

Inter-pixel boundary following

Algorithm: Inter-pixel boundary following (Aztzy et al., 1981)

Input: Binary image in a discrete space (Z 2,m), starting 1-cell sOutput: Set F of 1-cells that form the boundary containing s

Put s in a list F and in a queue Q.

while Q 6= ∅ doPull f from Q.for each successor neighbor g of f do

if g is in F , put g in F and in Q.

1-cells which form the boundary are also called bel.

The graph structure and the similar idea to the graph traversal are used.

Inter-pixel boundary following

Algorithm: Inter-pixel boundary following (Aztzy et al., 1981)

Input: Binary image in a discrete space (Z 2,m), starting 1-cell sOutput: Set F of 1-cells that form the boundary containing s

while Q 6= ∅ doPull f from Q.for each successor neighbor g of f do

if g is in F , put g in F and in Q.

1-cells which form the boundary are also called bel.

The graph structure and the similar idea to the graph traversal are used.

Adjacency of 2-cells

Definition (2-cell adjacency)

Given a complex C , two distinct 2-cells of C are adjacent if they havethe common 1-face.

(Lachaud, Malgouyres, 2003)

Property (Voss, 1993)

In the boundary of a 6-object, each 2-cell is adjacent toexactly four neighboring 2-cells such that two of them areits successors and the others are its predecessors.

Adjacency of 2-cells

Definition (2-cell adjacency)

Given a complex C , two distinct 2-cells of C are adjacent if they havethe common 1-face.

Property (Voss, 1993)

In the boundary of a 6-object, each 2-cell is adjacent toexactly four neighboring 2-cells such that two of them areits successors and the others are its predecessors.

3D boundary following algorithm

Algorithm: 3D boundary following (Aztzy et al., 1981)

Input: 6-object, starting 2-cell sOutput: Set F of 2-cells that form the boundary

while Q 6= ∅ do

Pull f from Q.for each successor neighbor g of f do

if g is not in F , put g in F and in Q.

The similar idea to the algorithm for connected component labeling is used.

Data structure for Kovalevsky topology

Data structure

If the size of input 3D image is N × N × N (3D array size), the number ofelements of its Kovalevsky topology is 2N × 2N × 2N = 8N3.

The dimension of each cell σ is determined by the 3D array index (i , j , k):

if all of the integers i , j , k are even numbers, then σ is a 3-cell;

if one of the integers i , j , k is odd, then σ is a 2-cell;

if one of the integers i , j , k is even, then σ is a 1-cell;

if all of the integers i , j , k are odd numbers, then σ is a 0-cell.

The order between two cells is arithmetically defined depending on their 3D

array indices.

Data structure

array indices.

Data structure

array indices.Digital Geometry : Topic 2 27/34

Exercise: connected components and inter-pixel boundary

Question

Let X be the set of discrete points (black points) in the figure.

1 What are the 8-connected components of X and the 4-connectedcomponents of X?

3 What are the inter-pixel boundary between X and X?

Question

The boundary does not form a manifold!

Well-composed images

Adjacency pair

Definition (Good adjacency pair (Duda et al., 1967))

An image I is a (4, 8)- (resp. (8, 4)-) image if

the 4- (resp. 8-) adjacency is used for the black pixels and

the 8- (resp. 4-) adjacency is used for the white pixels.

(4, 8)-image Connected components

Adjacency pair

Definition (Good adjacency pair (Duda et al., 1967))

An image I is a (4, 8)- (resp. (8, 4)-) image if

the 4- (resp. 8-) adjacency is used for the black pixels and

the 8- (resp. 4-) adjacency is used for the white pixels.

(8, 4)-image Connected components

Well-composed image

Definition (Well-composed image (Latecki, 1998))

An image is well-composed if each 8-connected component for theblack (and white) points is also a 4-connected component.

Well-composed image 8-connected components

Well-composed image

Ill-composed image 8-connected components

Well-composed image

Properties of well-composed images

Property (Well-composed image (Latecki, 1998))

Any well-composed image

is characterized by 2× 2 local configuration, and

has no topological paradox related to Jordan Theorem.

Properties of well-composed images

Property (Well-composed image (Latecki, 1998))

Any well-composed image

is characterized by 2× 2 local configuration, and

has no topological paradox related to Jordan Theorem.

Repairing images to be well-composed

If a given image is not well-composed, it can be repaired.(Latecki, 1998; Rosenfeld et al., 1998)

Iterative method: modify every critical configuration iterativelysuch that the well-composedness is satisfied.

Up-sampling method: increase the resolution and repair eachcritical configuration locally.

Summary

nD discrete object boundary is expected to be

a Jordan curve/surface,

a n-manifold.

For this aim, we need to choose appropriate shape representations,such as inter-pixel boundary, with

”good” adjacency pair, or

verifying if a given image is well-composed.

Summary

a n-manifold.

Summary

a n-manifold.

Summary

a n-manifold.

References

Reinhard Klette and Azriel Rosenfeld.Chapters 4 and 7 in ”Digital geometry: geometric methods fordigital picture analysis”, San Diego: Morgan Kaufmann, 2004.

David Coeurjolly, Annick Montanvert, et Jean-Marc Chassery.”Elements de base”, Chapitre 1 dans ”Geometrie discrete etimages numeriques”, Hermes Lavoisier, 2007.

Jacques-Olivier Lachaud et Remy Malgouyres.”Topologie, courves et surfaces discretes”, Chapitre 3 dans”Geometrie discrete et images numeriques”, Hermes Lavoisier,2007.

Gabor T. Herman.”Geometry of digital spaces”, Birkhauser, 1998.

Longin Jan Latecki.”Discrete representation of spatial objects in computer vision”,Kluwer Academic Publishers, 1998.

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