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

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Let  E = Z 2 {\displaystyle E=Z^{2}} , and consider the eight composite structuring elements, composed by:

C 1 = { ( 0 , 0 ) , ( − 1 , − 1 ) , ( 0 , − 1 ) , ( 1 , − 1 ) } {\displaystyle C_{1}=\\{(0,0),(-1,-1),(0,-1),(1,-1)\\}} and  D 1 = { ( − 1 , 1 ) , ( 0 , 1 ) , ( 1 , 1 ) } {\displaystyle D_{1}=\\{(-1,1),(0,1),(1,1)\\}} ,
C 2 = { ( − 1 , 0 ) , ( 0 , 0 ) , ( − 1 , − 1 ) , ( 0 , − 1 ) } {\displaystyle C_{2}=\\{(-1,0),(0,0),(-1,-1),(0,-1)\\}} and  D 2 = { ( 0 , 1 ) , ( 1 , 1 ) , ( 1 , 0 ) } {\displaystyle D_{2}=\\{(0,1),(1,1),(1,0)\\}}

and the three rotations of each by  90 o {\displaystyle 90^{o}} ,  180 o {\displaystyle 180^{o}} , and  270 o {\displaystyle 270^{o}} . The corresponding composite structuring elements are denoted  B 1 , … , B 8 {\displaystyle B_{1},\ldots ,B_{8}} .

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

X ⊗ B i = X ∖ ( X ⊙ B i ) {\displaystyle X\otimes B_{i}=X\setminus (X\odot B_{i})} ,

where  ∖ {\displaystyle \setminus } denotes the set-theoretical difference and  ⊙ {\displaystyle \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 ⊗ … ⊗ B 8 ⊗ B 1 ⊗ B 2 ⊗ … {\displaystyle A\otimes B_{1}\otimes B_{2}\otimes \ldots \otimes B_{8}\otimes B_{1}\otimes B_{2}\otimes \ldots } .

## Thickening

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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 ∪ ( X ⊙ B i ) {\displaystyle {\text{thicken}}(X,B_{i})=X\cup (X\odot B_{i})}

where  ∪ {\displaystyle \cup } denotes the set-theoretical difference and  ⊙ {\displaystyle \odot } denotes the hit-or-miss transform, and  B i {\displaystyle B_{i}} is the structural element and  X {\displaystyle X} is the image being operated on.

## References

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1. ^ Gonzalez, Rafael C. (2002). Digital image processing. Woods, Richard E. (Richard Eugene), 1954- (2nd ed.). Upper Saddle River, N.J. ISBN 0-201-18075-8. OCLC 48944550.`{{cite book}}`: CS1 maint: location missing publisher (link)