A Computational Introduction to Digital Image Processing, 2nd Edition

Mathematical Morphology 291

In the right-hand grid, we have

X

0

= {p}, X

1

= {p, 1}, X

2

= {p, 1, 2}, . . .

Note that the u se of the cross-shaped structuring element means that we never cross the

boundary.

Connected Components

We use a very similar algorithm to fill a connected comp onent; we use the cross-shaped

structuring element for 4-connected components, and the square structuring element for

8-connected components. Starting with a pixel p, we fill up the rest of the component by

creating a sequence of sets

X

0

= {p}, X

1

, X

2

, . . .

such that

X

n

= (X

n−1

⊕ B) ∩A

until X

k

= X

k−1

. Figure 10.25 shows an example.

2 1 2

1

p

1

2 1 2

Using the cross

p

1 1 1

1 1

1 1 1

2

3

3

34

4 4 4