1. Introduction to digital topology
The presentation has been prepared based on the following sources:
Kong T. Y., A digital fundamental group, Computer Graphics, vol. 13, pp , 1989.
Michel Couprie’s presentation on hole closing
Couprie M., Coeurjolly D., Zrour R., Discrete bisector function and Euclidean skeleton in 2D and 3D, Image Vision Comput., vol. 25, pp , 2007.
George Malandain’s webpage on digital topology:
Michel Couprie webpage on simple points:

2. Outline What is topology and digital topology
Notion of connectivity in 2D and 3D
Notion of a digital image from discreet topology point of view
Def. of connected components
2D thinning algorithm
Def. of simple points and thinning based on simple points deletion
Theorems on 2D sequential thinning algorithms
2D parallel thinning algorithms
Topological characteristic of points in 2D and 3D
Holes, cavities, concavities in 3D
Hole closing algorithm and its modification
Filtered Euclidean skeleton algorithm

3. Elementary introduction to topology
Topology grew out of geometry and set theory, and is the study of both the fine structure and global structure of space. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together or the way they are act on a space.
Paper on the problem of seven bridges of Kaliningrad by Leonhard Euler
For example, the square and the circle have some properties in common: they are both consist of one connected component and both separate the plane into two parts, the part inside and the part outside.
Euler proved that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. If we represent islands and riverbanks with vertex and bridges with edges then we can represent the situation with the graph on the left.
The proof of the Euler solution is very simply. Imagine that we start from any vertex of the graph and then if we pass an edge we delete it. When we finish all edges should be deleted and we should stay in the starting vertex. Such a closed path in a graph is called Euler cycle. So the Kaliningrad problem can be reformulated in the following way. If the graph on the left has an Euler cycle. Always If we go through a vertex we delete two edges one which we use to reach the vertex and another one which we use to leave the vertex. The same rule concerns starting vertex when we start we delete one edge which connects staring voxel and when we finish our trip we delete second edge which connects starting vertex. So finally the consideration leads to the conclusion that there is Euler path in a graph provided that even numbers of edges goes to each vertex. The condition is not fulfilled by the graph on the left so the graph does not have Euler cycle.
This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks.
Czas 1:10

4. Elementary introduction to topology part 2
homotopy - intuitively two objects are homotopic if one can be deformed into the other without cutting or gluing (without changing its topology).
The impossibility of crossing each bridge just once applies to any arrangement of bridges homotopic to those in Kaliningrad.
A cup is homotopic with a doughnut
In order to deal with problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on.
From this need arises the notion of homotopy. Przeczytac definicje homotopy where Topology of an object can be described by the number of holes in the object and number of connected components the object consits of.
In other words two objects are homotopic if thay have the same number of holes and they consist of the same number of connected components.
The homtopy relation is equivalence relation so it divides a space into classes of equivalence.
The right image presents a division of the set of English letters alphabet into sets of homotopic letters.
Czas 2 min

5. Digital topology
Digital topology is the study of the topological properties of image arrays. Its results provide a sound mathematical basis for image processing operations such as image thinning, border following, contour filling and object counting.
The general topology is inappropriate because it considers spaces in which even the smallest neighbourhood of a point contains infinitely many other points.
General topology
Digital topology
#N(p) = ∞
ε → ∞
It is obvious that such a space (and even the smallest part of it) cannot be explicitly represented in a computer. Therefore we need the topology of the so called locally finite spaces whose elements have neighbourhoods containing finite number of elements.
#N(p) = 4 p p

6. Pixel and voxel
The presentation is about 2D and 3D binary image arrays.
The elements of a 2D image array are called pixels.
The elements of a 3D image array are called voxels.
To avoid having to consider the border of the image array we assume that the array is unbounded in all directions.
Each pixel or voxel is associated with a lattice point (i.e., a point with integer coordinates) in the plane or in 3D-space.
p024
p124 …
p424
p014
p004
p40
p41
p42
p43
p44
Nw i would like to present about 20 spides on introductory notions which are necessary to understand details on some algorithms of image processing based on digital topology.
Although some authors consider higher dimanetional arrays. . . .
p421
p10
p11
p12
p13
p14
p120
p420
p110
p410
p00
p01
p02
p03
p04
p000
p100 …
p400

7. Connectivity in 2D Two lattice points in the plane are said to be:
8‑adjacent if they are distinct and each coordinate of one differs from the corresponding coordinate of the other by at most 1.
4-adjacent if they are 8-adjacent and differ in at most one of their coordinates.
For n = 4, 8 an n-neighbor of a lattice point p is a point that is n-adjacent to p.
8-neighborhood
4-neighborhood p p
Sometimes we say that two points are n-adjacent sometimes we say that two points are n-neighbours

8. Connectivity in 3D Two lattice points are said to be
26-adjacent if they are distinct and each coordinate of one differs from the corresponding coordinate of the other by at most 1.
18-adjacent if they are 26-adjacent and differ in at most two of their coordinates
6-adjacent if they are 26-adjacent and differ in at most one coordinate.
An n-neighbor of a lattice point p is a point that is n-adjacent to p.
If p is a lattice point in 3D space then Nm(p), for m = 6, 18, 26 denotes the set consisting of p and its m-neighbors. p
26-neighborhood p
18-neighborhood p
6-neighborhood
Similarly like for 2D we can introduce the notion of n neighbour for n = 18, or 26.

9. Adjacency between a point and a set
A point p is said to be adjacent to a set of points S if p is adjacent to some point in S. Two sets A, B are n-adjacent if there are points: a ∈ A, b ∈ B
which are n-adjacent.
A point adjacency to a set
Adjacency between sets
The red point is 4 and 8 adjacent to the set of black points but the blue point is only 8 adjacent to the set of black points
The set of black points is 4 and 8 adjacent to the set of blue points. The set of red points is not adjacent to the set of ble points

10. n-connected set, n-component
We say a set S of lattice points is n-connected if S cannot be partitioned into two subsets that are not n-adjacent to each other.
An n-component of a set of lattice points S is a non-empty n-connected subset of S that is not n-adjacent to any other point in S.
An 8-connected set with exactly three 4-components.
In the bottom of the slide we have an example. Asume that we consider a set of points represented with black dots. The set is 8-connected but it is not 4-connected because it can be partitioned to three sets which are not 4-adjacent to one and other.

11. Black and white
A lattice point associated with a pixel or voxel that has value 1 is called a black point;( ) a lattice point associated with a pixel or voxel with value 0 is called a white point ( ).
To avoid topology paradoxes [Duda 67] use different adjacency relations for black and white points in 2D. In 3D the following configurations are allowed: (6, 26); (26, 6); (6, 18); (18, 6).
An example of topology paradox
Each figure shows only a finite subset of the set of all lattice points. In all presented cases it does not matter which of the other lattice points (which are not presented) are white points and which are black points. So I use the convention that all lattice points not shown in the figure are white points.
Arguably, the starting point of research on digital topology was the simple but important idea of using different adjacency relations for black and white points, a device which as far as we know was first recommended by Duda, Hart, and Munson [32]. The reason for this at first sight rather bizarre decision was to avoid paradoxes such as those pointed out in [115]: in Figure
if 4-adjacency is used for all pairs of points then the black points are totally disconnected but still separate the set of white points into two components, while if 8-adjacency is used for all pairs of points then the black points form the discrete analog of a Jordan curve but they do not separate the white points.
The difficulty is resolved if we use 8-adjacency for the white points and 4-adjacency for the black, or vice versa. In three dimensions analogous paradoxes are similarly avoided if 6-adjacency is used for the white points and either 18- or 26-adjacency for the black, or vice versa.

12. Digital picture
Let use Z2 to denote the set of lattice points in the plane and Z3 to denote the set of lattice points in 3D-space.
A digital picture is a quadruple (V, m, n, B), where V = Z2 or V = Z3, B V and where (m, n) = (4, 8) or (8, 4) if V = Z2, and (m, n) = (6, 26), (26, 6), (6, 18), or (18, 6) if V = Z3.
The points in B are called the black points of the picture; the points in V — B are called the white points of the picture. Usually B is a finite set; so then P is said to be finite.
Two black points in a digital picture (V, m, n, B) are said to be adjacent if they are m‑adjacent, and two white points or a white point and a black point are said to be adjacent if they are n-adjacent.
The digital picture P = (V, m, n, B) is called two-dimensional or three-dimensional according as V = Z2 or V = Z3. The elements of V are called the points of the digital picture.
Very important fact is that we have m-adjacency between objects pixels and n-adjacency between background pixels and background and object pixels

13. Digital picture: example
A digital picture (V, m, n, B) will also be shortly called an (m, n) digital picture
The adjacencies in an (8, 4) digital picture.
The adjacencies in an (4, 8) digital picture.

14. Connected set and component set in a digital picture
Let consider (m, n) digital picture and a set S of white and black pixels. We can not consider S as an n-connected or m-connected according to the definition from slide 10.
We say a set S of black and/or white points in a digital picture is connected if S cannot be partitioned into two subsets that are not adjacent to each other.
A component of a set of black and/or white points S is a non-empty connected subset of S which is not adjacent to any other point in S.
In an (m, n) digital picture a component of a set of black points is an m-component, whereas a component of a set of white points is an n-component.
Ad 1) because there is different connectivity between points of the set. From the same reason a subset of a set S can not be considered as n-component or m-component. So if we want to consider components and connected sets in digital picture we have to introduce new definitions. Which will take into account two different types of connectivity which occur in the picture.

15. Components in a digital picture: example
Let consider (8, 4) digital picture. How many 8-components in a set of all black points and how many 4-components in a set of all white points are there in the picture?

16. Components in a digital picture: example
As an (8, 4) digital picture this has 3 8-components and 3 4-components.

17. Component of a set
As a (4, 8) digital picture this has 5 4-components and 2 8-components.

18. Black and white component
A component of the set of all black points of a digital picture is called a black component, and a component of the set of all white points is called a white component.
In a finite digital picture there is a unique infinite white component, which is called the background.
(8, 4) digital picture. Pixels from a set S are marked with a square. {p, q} is 8-component of the set S but it is not a black component q
There is only one black component in the picture. p

19. Path of points and a closed curve
For any set of points S a path in S is a sequence 〈pi | 0 ≤ i ≤ n 〉 of points in S such that pi is adjacent to pi+1 for all 0 ≤ i < n.
A path 〈pi | 0 ≤ i ≤ n〉 is said to be a path from p0 to pn and is said to be a closed path if pn = p0. A degenerate one-point path 〈p0〉 is a special case of a closed path.
A simple closed black curve in a digital picture is a connected set of black points each of which is adjacent to exactly two other points in the set.
Closed path of points {p0,...,p3}is a simple closed black curve in (4, 8) dig. pic. but it is not in (8, 4) dig. pic.
A simple closed black curve in an (8, 4) digital picture
A simple closed black curve in an (4, 8) digital picture
Each closed curve can be transferred to a closed path. The transformation consist in putting points from a simple closed black curve into a sequence. p3 p2 p0 p1

20. Surround
If X and Y are sets of points in a digital picture (V, m, n, B) and X is connected then we say X surrounds Y if each point in Y is contained in a finite component of V - X.
In a finite digital picture the background surrounds the set of all black points.
X - set of rounded points; Y - set of squared points.
X surrounds Y in (8, 4) pic.
V - X

21. Hole, cavity
In a digital picture P, a white component that is adjacent to and surrounded by a black component C is called a hole in C if P is a two-dimensional digital picture and a cavity in C if P is a three-dimensional digital picture. By a hole of P (cavity of P) we mean a hole (cavity) in some black component of P.
The black component shown here has two holes if it is regarded as part of an (8, 4) digital picture. The black component has just one hole if it is regarded as part of a (4, 8) digital picture.

22. Isolated, Border, Interior Point
A black point is said to be isolated if it is not adjacent to any other black point. A black point is called a border point if it is adjacent to one or more white points; otherwise it is called an interior point. The border (interior) of a black component C of P is the set of all border points (resp., all interior points) in C.
The border of a black component C with respect to a white component D is the set of points in C that are adjacent to D. Recall that in this context adjacent means n-adjacent if P is an (m, n) digital picture.
In a (4, 8) digital picture both p and q would be isolated points. In an (8, 4) digital picture p would be an isolated point but q would not.
In an (8, 4) digital picture the boxed points SB would be border points; in a (4, 8) digital picture both the boxed points [] and the ringed points  would be border points.
Same notation as in previous image except that only those border points which belong to the border of the black component with respect to its (unique) hole are marked. q p

23. Simple black arc
A simple black arc is a connected set of black points each of which is adjacent to just two other points in the set, with the exception of two points— the end points of the arc — that are each adjacent to just one other point in the set.
If we remove any point from a simple closed black curve then the remaining points will form a simple black arc.
A simple black arc in the (8, 4) digital picture
A simple black arc in the (4, 8) digital picture