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

[math]\displaystyle{ C_1=\{(0,0),(-1,-1),(0,-1),(1,-1)\} }[/math] and [math]\displaystyle{ D_1=\{(-1,1),(0,1),(1,1)\} }[/math],

[math]\displaystyle{ C_2=\{(-1,0),(0,0),(-1,-1),(0,-1)\} }[/math] and [math]\displaystyle{ D_2=\{(0,1),(1,1),(1,0)\} }[/math]

and the three rotations of each by [math]\displaystyle{ 90^o }[/math], [math]\displaystyle{ 180^o }[/math], and [math]\displaystyle{ 270^o }[/math]. The corresponding composite structuring elements are denoted [math]\displaystyle{ B_1,\ldots,B_8 }[/math].

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

[math]\displaystyle{ X\otimes B_i=X\setminus (X\odot B_i) }[/math],

where [math]\displaystyle{ \setminus }[/math] denotes the set-theoretical difference and [math]\displaystyle{ \odot }[/math] denotes the hit-or-miss transform.

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

[math]\displaystyle{ A\otimes B_1\otimes B_2\otimes\ldots\otimes B_8\otimes B_1\otimes B_2\otimes\ldots }[/math].

Thickening

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 [math]\displaystyle{ \text{thicken}(X, B_i) = X\cup (X\odot B_i) }[/math]

where [math]\displaystyle{ \cup }[/math] denotes the set-theoretical difference and [math]\displaystyle{ \odot }[/math] denotes the hit-or-miss transform, and [math]\displaystyle{ B_i }[/math] is the structural element and [math]\displaystyle{ X }[/math] is the image being operated on.