Digital Topology

CIS 601 Fall 2004Longin Jan Latecki

Relationship between pixels

Neighbors of a pixel

– 4-neighbors (N,S,W,E pixels) == N4(p).

A pixel p at coordinates (x,y) has

four horizontal and vertical neighbors:

• (x+1,y), (x-1, y), (x,y+1), (x, y-1)

– You can add the four diagonal neighbors to give the 8-neighbor set. Diagonal neighbors == ND(p).

– 8-neighbors: include diagonal pixels == N8(p).

Pixel Connectivity

Connectivity is used to trace contours, to define object boundaries, and for segmentation.

In order for two pixels to be connected, they must be

“neighbors” sharing a common property—satisfy some

similarity criterion.

In a binary image with pixel values “0” and “1”,

two neighboring pixels are said to be connected if they have the same value.

Let V: Set of binary or gray level values used to define

connectivity; e.g., V={1}.

Connectivity

4-adjacency: Two pixels p and q with values

in V are 4-adjacent if q is in the set N4(p).

8-adjacency: q is in the set N8(p).

A 4 (8)-path from p to q is a sequence of points

p=p0, p1, …, pn=q such that each pn+1 is 4 (8)-adjacent to pn

Connected components

Let S represent a subset of pixels in an image.

Set S is connected if for each pair of points p and q in S, there is a path from p to q contained entirely S.

A connected component is a maximal set of pixels in S that is connected.

There can be more than one such set within a given S.

We assume that points r and t have been visited and labeled.

Now we visit point p: p=0: no action; p=1: check r and t.

– both r and t = 0; assign new label to p;

– only one of r and t is a 1. assign its label to p;

– both r and t are 1:

• same label => assign it to p;

• different label=> assign one of them to p and

establish equivalence between labels (they are the same.)

Second pass over the image to merge equivalent labels.

rpt

4-connected component labeling for points

Homework 9

Develop a similar algorithm for 8-connectivity.

Matlab example: BW = logical([1 1 1 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0]);L = bwlabel(BW,4); [r,c] = find(L == 2);

Problems with 4- and 8-connectivity

Neither method is satisfactory.

– Why? A simple closed curve divides a plane into

two simply connected regions.

– However, neither 4-connectivity nor 8-connectivity

can achieve this for discrete labeled components.

– Give some examples.

Border following

Let our object b be an 8-connected set of 1’s that is not a single isolated point.

A pixel P is a border pixel of b if P is 4-adjacent to at least one 0.

Let the 8 neighbors of P be ordered counterclockwise as follows, which implies that the outer border will be followed counterclockwise.

 

N4 N3 N2

N5 P N1N6 N7 N8

Border following algorithm:

Let (P_i,Q_i) be such that P_i is in b (P_i is 1), Q_i is 0,Q_i = N_1, and P_i and Q_i are 4-adjacent.

Then P_i+1 = N_k, where N_k is the first of the 8 neighbors of P_i that is 1(it exists since b is not a single isolated point)and Q_i = N_k-1.

The algorithm stops when we come back to the initial point pair (P_1, Q_1).

This stop condition is necessary, since we can pass through a point twice (give an example).

In Matlab use function

B = boundaries(BW);

Q3

Q4 1 P3 1 P2 Q2

1 P4 1 1 P1 Q1

Q5 1 P5 1 P6 1 P7

Q6 Q7

Example of the border following algorithm: