Introduction to digital topology The presentation has been

Introduction to digital topology The presentation has been prepared based on the following sources: 1. Kong T. Y. , A digital fundamental group, Computer Graphics, vol. 13, pp. 159 -166, 1989. 2. Michel Couprie’s presentation on hole closing 3. Couprie M. , Coeurjolly D. , Zrour R. , Discrete bisector function and Euclidean skeleton in 2 D and 3 D, Image Vision Comput. , vol. 25, pp. 1543 -1556, 2007. 4. George Malandain’s webpage on digital topology: http: //wwwsop. inria. fr/epidaure/personnel/malandain/topology/ 5. www. wikipedia. org 6. Michel Couprie webpage on simple points: http: //www. esiee. fr/~info/ck/CK_simple. html 1

Outline 1. What is topology and digital topology 2. Notion of connectivity in 2 D and 3 D 3. Notion of a digital image from discreet topology point of view 4. Def. of connected components 5. 2 D thinning algorithm 6. Def. of simple points and thinning based on simple points deletion 7. Theorems on 2 D sequential thinning algorithms 8. 2 D parallel thinning algorithms 9. Topological characteristic of points in 2 D and 3 D 10. Holes, cavities, concavities in 3 D 11. Hole closing algorithm and its modification 12. Filtered Euclidean skeleton algorithm 2

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 www. wikipedia. org 3

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 www. wikipedia. org 4

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→ ∞ p p #N(p) = 4 5

Pixel and voxel • The presentation is about 2 D and 3 D binary image arrays. • The elements of a 2 D image array are called pixels. • The elements of a 3 D 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 3 D-space. p 024 p 124 … p 424 p 014 p 40 p 41 p 42 p 43 p 44 . . . p 10 p 004. . . p 11 p 01 p 12 p 02 p 13 p 03 p 14 p 04 p 421 p 120 p 420 p 110 p 000 p 410 p 100 … p 400 6

Connectivity in 2 D 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 p 4 -neighborhood p 7

Connectivity in 3 D 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 3 D space then Nm(p), for m = 6, 18, 26 denotes the set consisting of p and its m-neighbors. 26 -neighborhood p 18 -neighborhood p 6 -neighborhood p 8

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 9

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. 10

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 2 D. In 3 D the following configurations are allowed: (6, 26); (26, 6); (6, 18); (18, 6). An example of topology paradox 11

Digital picture Let use Z 2 to denote the set of lattice points in the plane and Z 3 to denote the set of lattice points in 3 D-space. A digital picture is a quadruple (V, m, n, B), where V = Z 2 or V = Z 3, B V and where (m, n) = (4, 8) or (8, 4) if V = Z 2, and (m, n) = (6, 26), (26, 6), (6, 18), or (18, 6) if V = Z 3. 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. 12

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. 13

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. 14

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? 15

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

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

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 p 18

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 p 0 to pn and is said to be a closed path if pn = p 0. A degenerate one-point path � p 0�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 {p 0, . . . , p 3}is a A simple closed black curve in (4, 8) in an (4, 8) digital picture in an (8, 4) digital picture dig. pic. but it is not in (8, 4) dig. pic. p 3 p 2 p 0 p 1 19

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 20

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. 21

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. q p 22

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 23