Morphological skeleton explained

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),^[1] Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image

X\subset R ^{2} :S(X) = c up _{\rho} c ap _{\mu} \left[(X\ominus \rho B) - (X\ominus \rho B)\circ \mu \overline B\right] ,

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

\{nB\} , n =0 , 1 ,\ldots , be a family of shapes, where B is a structuring element,nB = \underbrace{B ⊕ … ⊕ B} _{ntimes} , and0 B = \{o\} , 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 Z ^{2} , the skeleton S(X) is the union of the skeleton subsets \{S _{n(X)\}} , 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

\{S _{n(X)\}} as follows:X = c up _{n} (S _{n(X) ⊕} nB) .

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

c up _{n\geq} (S _{n(X) ⊕} 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=\{x\in} E| x - z\in nB\} .

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

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.

Notes and References

1. See also (Serra's 1982 book)