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.

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:

• $nB_{z}\in A$, 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.

Notes

1. See also (Serra's 1982 book)

References

• Image Analysis and Mathematical Morphology by Jean Serra, ISBN 0-12-637240-3 (1982)
• Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances by Jean Serra, ISBN 0-12-637241-1 (1988)
• An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
• Ch. Lantuéjoul, "Sur le modèle de Johnson-Mehl généralisé", Internal report of the Centre de Morph. Math., Fontainebleau, France, 1977.