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 , … , B 8 .
For any i between 1 and 8, and any binary image X, define
where ∖ denotes the set-theoretical difference and ⊙ denotes the hit-or-miss transform.
The thinning of an image A is obtained by cyclically iterating until convergence:
A ⊗ B 1 ⊗ B 2 ⊗ … ⊗ B 8 ⊗ B 1 ⊗ B 2 ⊗ … .
