Morphological Skeleton

In digital image processing, morphological skeleton is a skeleton (or medial axis) representation of a shape or binary image, computed by means of morphological operators.

Morphological skeletons are of two kinds:
* Those defined by means of morphological openings, from which the original shape can be reconstructed,
* Those computed by means of the hit-or-miss transform, which preserve the shape's topology.

Skeleton by openings

Lantuéjoul's formula

Continuous images

In ( Lantuéjoul 1977),See also ( Serra's 1982 book) Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image $X\subset \mathbb^2$:

:S(X)=\bigcup_\bigcap_\left X\ominus \rho B)-(X\ominus \rho B)\circ \mu \overline B\right/math>,

where $\ominus$ and $\circ$ are the morphological erosion and opening, respectively, $\rho B$ is an open ball of radius $\rho$, and $\overline B$ is the closure of $B$.

Discrete images

Let $\$, $n=0,1,\ldots$, be a family of shapes, where ''B'' is a structuring element,
:$nB=\underbrace_$, and
:$0B=\$, where ''o'' denotes the origin.

The variable ''n'' is called the ''size'' of the structuring element.

Lantuéjoul's formula has been discretized as follows. For a discrete binary image $X\subset \mathbb^2$, the skeleton ''S(X)'' is the union of the skeleton subsets $\$, $n=0,1,\ldots,N$, where:

:$S_n(X)=(X\ominus nB)-(X\ominus nB)\circ B$.

Reconstruction from the skeleton

The original shape ''X'' can be reconstructed from the set of skeleton subsets $\$ as follows:

:$X=\bigcup_n (S_n(X)\oplus nB)$.

Partial reconstructions can also be performed, leading to opened versions of the original shape:

:$\bigcup_ (S_n(X)\oplus nB)=X\circ mB$.

The skeleton as the centers of the maximal disks

Let $nB_z$ be the translated version of $nB$ to the point ''z'', that is, $nB_z=\$.

A shape $nB_z$ centered at ''z'' is called a maximal disk in a set ''A'' when:
* $nB_z\in A$, and
* if, for some integer ''m'' and some point ''y'', $nB_z\subseteq mB_y$, then $mB_y\not\subseteq A$.

Each skeleton subset $S_n(X)$ consists of the centers of all maximal disks of size ''n''.

Performing Morphological Skeletonization on Images

Morphological Skeletonization can be considered as a controlled erosion process. This involves shrinking the image until the area of interest is 1 pixel wide. This can allow quick and accurate image processing on an otherwise large and memory intensive operation. A great example of using skeletonization on an image is processing fingerprints. This can be quickly accomplished using bwmorph; a built-in Matlab function which will implement the Skeletonization Morphology technique to the image.

The image to the right shows the extent of what skeleton morphology can accomplish. Given a partial image, it is possible to extract a much fuller picture. Properly pre-processing the image with a simple Auto Threshold grayscale to binary converter will give the skeletonization function an easier time thinning. The higher contrast ratio will allow the lines to joined in a more accurate manner. Allowing to properly reconstruct the fingerprint.

skelIm = bwmorph(orIm,'skel',Inf); %Function used to generate Skeletonization Images

Notes

References

* ''Image Analysis and Mathematical Morphology'' by Jean Serra, (1982)
* ''Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances'' by Jean Serra, (1988)
* ''An Introduction to Morphological Image Processing'' by Edward R. Dougherty, (1992)
* Ch. Lantuéjoul, "Sur le modèle de Johnson-Mehl généralisé", ''Internal report of the Centre de Morph. Math.'', Fontainebleau, France, 1977.
* Scott E. Umbaugh (2018). Digital Image Processing and Analysis, pp 93-96. CRC Press. {{ISBN, 978-1-4987-6602-9

Mathematical morphology
Digital geometry