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 $E=Z^{2}$, 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:

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

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

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