# General connectivity

The notion of neighbor connectivity is discussed in most image processing textbooks. Specifically, what is the set of neighbors of a pixel? For example, a commonly-used neighborhood connectivity is 4-connected, where each pixel has four neighbors. The neighborhood looks like this:

   1
1 1 1
1

Another common connectivity is 8-connected, where each pixel has eight neighbors. The neighborhood looks like this:

 1 1 1
1 1 1
1 1 1

Many Image Processing Toolbox functions depend on a definition of connectivity. Most of these functions allow you to specify the desired connectivity in a very general, flexible way.

Consider bwperim. This function computes perimeter pixels by finding the foreground pixels that are connected to background pixels. Some definition of connectivity is therefore required. Let's look at an example.

bw = [ 0     0     0     0     0     0     0
0     0     1     1     1     0     0
0     1     1     1     1     1     0
0     1     1     1     1     1     0
0     1     1     1     1     1     0
0     0     1     1     1     0     0
0     0     0     0     0     0     0 ];

Find the foreground pixels that are 4-connected to the background.

perim4 = bwperim(bw, 4)
perim4 =

0     0     0     0     0     0     0
0     0     1     1     1     0     0
0     1     0     0     0     1     0
0     1     0     0     0     1     0
0     1     0     0     0     1     0
0     0     1     1     1     0     0
0     0     0     0     0     0     0



Now find the foreground pixels that are 8-connected to the background.

perim8 = bwperim(bw, 8)
perim8 =

0     0     0     0     0     0     0
0     0     1     1     1     0     0
0     1     1     0     1     1     0
0     1     0     0     0     1     0
0     1     1     0     1     1     0
0     0     1     1     1     0     0
0     0     0     0     0     0     0



In addition to specifying connectivity as "4" or "8", you can also use a 3-by-3 matrix of 0s and 1s to specify other kinds of connectivity. The 3-by-3 connectivity matrix below says that a pixel has only two neighbors: The pixel above and the pixel below.

conn2 = [ 0 1 0
0 1 0
0 1 0 ];

perim2 = bwperim(bw, conn2)
perim2 =

0     0     0     0     0     0     0
0     0     1     1     1     0     0
0     1     0     0     0     1     0
0     0     0     0     0     0     0
0     1     0     0     0     1     0
0     0     1     1     1     0     0
0     0     0     0     0     0     0



Note that the (4,2) and (4,6) pixels are not considered to be perimeter pixels with this connectivity. That's because their upper and lower neighbors are both part of the foreground, not the background.

Sometimes it's useful to define 6-connectivity. There are two different flavors, both of which can be expressed using the 3-by-3 matrix form:

conn6a = [ 1 1 0
1 1 1
0 1 1 ];

conn6b = [ 0 1 1
1 1 1
1 1 0 ];

For processing three-dimensional arrays, you specify 6-connectivity (face-adjacent pixels), 18-connectivity (face- or edge-adjacent pixels), or 26-connectivity (face-, edge-, or vertex-adjacent pixels). But you can specify your own flavor of connectivity by using a 3-by-3-by-3 matrix. Here's an example.

bw_3d = false(5,5,3);
bw_3d(2:4,2:4,:) = true;

% Label three-dimensional connected components.
L1 = bwlabeln(bw_3d)
L1(:,:,1) =

0     0     0     0     0
0     1     1     1     0
0     1     1     1     0
0     1     1     1     0
0     0     0     0     0

L1(:,:,2) =

0     0     0     0     0
0     1     1     1     0
0     1     1     1     0
0     1     1     1     0
0     0     0     0     0

L1(:,:,3) =

0     0     0     0     0
0     1     1     1     0
0     1     1     1     0
0     1     1     1     0
0     0     0     0     0



There's only one three-dimensional connected component. But that's using the default connectivity. Let's define a three-dimensional connectivity such that each pixel has neighbors only in the same plane.

conn_3d = cat(3, zeros(3,3), ones(3,3), zeros(3,3))
conn_3d(:,:,1) =

0     0     0
0     0     0
0     0     0

conn_3d(:,:,2) =

1     1     1
1     1     1
1     1     1

conn_3d(:,:,3) =

0     0     0
0     0     0
0     0     0


L2 = bwlabeln(bw_3d, conn_3d)
L2(:,:,1) =

0     0     0     0     0
0     1     1     1     0
0     1     1     1     0
0     1     1     1     0
0     0     0     0     0

L2(:,:,2) =

0     0     0     0     0
0     2     2     2     0
0     2     2     2     0
0     2     2     2     0
0     0     0     0     0

L2(:,:,3) =

0     0     0     0     0
0     3     3     3     0
0     3     3     3     0
0     3     3     3     0
0     0     0     0     0



Now there are three connected components, one in each plane of the array.

The Image Processing Toolbox functions bwlabel, bweuler, bwboundaries, and bwtraceboundary can accept either 4 or 8 connectivity.

The toolbox functions bwareaopen, bwlabeln, bwperim, bwulterode, imclearborder, imextendedmax, imextendedmin, imhmax, imhmin, imimposemin, imreconstruct, imregionalmax, imregionalmin, and watershed can all accept general 3-by-3 (or higher dimensional) connectivity matrices.

Have you used this general connectivity capability before? If so, I'd like to hear about it.

Published with MATLAB® 7.5

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