Thinning (morphology)

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 $E=Z^{2}$ , 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 $B_{1},\ldots ,B_{8}$ .

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

X\otimes B_{i}=X\setminus (X\odot B_{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\otimes B_{1}\otimes B_{2}\otimes \ldots \otimes B_{8}\otimes B_{1}\otimes B_{2}\otimes \ldots

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