Thinning (morphology) explained

Thinning is the transformation of a digital image into a simplified, but topologically equivalent image. It is a type of topological skeleton, but computed using mathematical morphology operators.

Example

Let

E=Z²

, and consider the eight composite structuring elements, composed by:

C_{1=\{(0,0),(-1,-1),(0,-1),(1,-1)\}}

and

D_{1=\{(-1,1),(0,1),(1,1)\}}

C_{2=\{(-1,0),(0,0),(-1,-1),(0,-1)\}}

and

D_{2=\{(0,1),(1,1),(1,0)\}}

and the three rotations of each by

90^o

180^o

, and

270^o

. The corresponding composite structuring elements are denoted

B_1,\ldots,B₈

For any i between 1 and 8, and any binary image X, define

X ⊗ B_i=X\setminus(X\odotB_i)

,where

\setminus

denotes the set-theoretical difference and

\odot

denotes the hit-or-miss transform.

The thinning of an image A is obtained by cyclically iterating until convergence:

A ⊗ B_{1 ⊗}B_{2 ⊗ \ldots ⊗}B_{8 ⊗}B_{1 ⊗}B_{2 ⊗ \ldots}

Thickening

Thickening is the dual of thinning that is used to grow selected regions of foreground pixels. In most cases in image processing thickening is performed by thinning the background ^[1]

thicken(X,B_i)=X\cup(X\odotB_i)

where

\cup

denotes the set-theoretical difference and

\odot

denotes the hit-or-miss transform, and

B_i

is the structural element and

is the image being operated on.

Notes and References

Book: Gonzalez, Rafael C.. Digital image processing. Woods, Richard E. (Richard Eugene), 1954-. 2002. 0-201-18075-8. 2nd. Upper Saddle River, N.J.. 48944550.