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.

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

${\displaystyle C_{1}=\{(0,0),(-1,-1),(0,-1),(1,-1)\}}${\displaystyle C_{1}=\{(0,0),(-1,-1),(0,-1),(1,-1)\}} and ${\displaystyle D_{1}=\{(-1,1),(0,1),(1,1)\}}${\displaystyle D_{1}=\{(-1,1),(0,1),(1,1)\}},

${\displaystyle C_{2}=\{(-1,0),(0,0),(-1,-1),(0,-1)\}}${\displaystyle C_{2}=\{(-1,0),(0,0),(-1,-1),(0,-1)\}} and ${\displaystyle D_{2}=\{(0,1),(1,1),(1,0)\}}${\displaystyle D_{2}=\{(0,1),(1,1),(1,0)\}}

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

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

${\displaystyle X\otimes B_{i}=X\setminus (X\odot B_{i})}${\displaystyle X\otimes B_{i}=X\setminus (X\odot B_{i})},

where ${\displaystyle \setminus }${\displaystyle \setminus } denotes the set-theoretical difference and ${\displaystyle \odot }${\displaystyle \odot } denotes the hit-or-miss transform.

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

${\displaystyle A\otimes B_{1}\otimes B_{2}\otimes \ldots \otimes B_{8}\otimes B_{1}\otimes B_{2}\otimes \ldots }${\displaystyle A\otimes B_{1}\otimes B_{2}\otimes \ldots \otimes B_{8}\otimes B_{1}\otimes B_{2}\otimes \ldots }.

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] ${\displaystyle {\text{thicken}}(X,B_{i})=X\cup (X\odot B_{i})}${\displaystyle {\text{thicken}}(X,B_{i})=X\cup (X\odot B_{i})}

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

• Mathematical morphology
• Digital geometry

• CS1 maint: location missing publisher