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=\$ and $D_1=\$,
:$C_2=\$ and $D_2=\$
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
$\text(X, B_i) = X\cup (X\odot B_i)$

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.

References

{{Reflist

Mathematical morphology
Digital geometry