Morphological skeleton

In digital image processing, morphological skeleton is a skeleton 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 derived the following morphological formula for the skeleton of a continuous binary image :
where and are the morphological erosion and opening, respectively, is an open ball of radius, and is the closure of.

Discrete images

Let,, be a family of shapes, where B is a structuring element,
The variable n is called the size of the structuring element.
Lantuéjoul's formula has been discretized as follows. For a discrete binary image, the skeleton S is the union of the skeleton subsets,, where:

Reconstruction from the skeleton

The original shape X can be reconstructed from the set of skeleton subsets as follows:
Partial reconstructions can also be performed, leading to opened versions of the original shape:

The skeleton as the centers of the maximal disks

Let be the translated version of to the point z, that is,.
A shape centered at z is called a maximal disk in a set A when:

, and
if, for some integer m and some point y,, then.

Each skeleton subset 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; %Function used to generate Skeletonization Images