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 openingss, 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), Lantuéjoul:

1. ,

are the morphological erosion and opening is an open ball of radius.

Discrete images

, be a family of shapes, where B is a structuring element,

1. , and

2. , where o denotes the origin.

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

, the skeleton S(X) is the union of the skeleton subsets, where:

1. .

Reconstruction from the skeleton

The original shape X as follows:

1. .

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

1. .

The skeleton as the centers of the maximal disks

to the point z.

centered at z is called a maximal disk in a set A when:

• , and

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

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.

• P. Maragos and R. W. Schafer, Morphological Skeleton Representation and Coding of Binary Images, IEEE Trans. on Acoustics, Speech, and Signal Processing, vol.34, no.5, pp.1228-1244, Oct. 1986.

• Scott E. Umbaugh (2018). Digital Image Processing and Analysis, pp 93-96. CRC Press.