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.

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), Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image X\subset \mathbb{R}^2 : S(X)=\bigcup_{\rho >0}\bigcap_{\mu >0}\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\oplus\cdots\oplus B}_{n\mbox{ times}} , and 0B=\{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 \mathbb{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=\bigcup_n (S_n(X)\oplus nB) . Partial reconstructions can also be performed, leading to opened versions of the original shape: \bigcup_{n\geq m} (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=\{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. 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.

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