Thinning (morphology)

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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.


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.