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:

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 \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:

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

Notes

References