Morphological skeleton

In (Lantuéjoul 1977),[1] Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image ${\displaystyle X\subset \mathbb {R} ^{2}}$:

${\displaystyle S(X)=\bigcup _{\rho >0}\bigcap _{\mu >0}\left[(X\ominus \rho B)-(X\ominus \rho B)\circ \mu {\overline {B}}\right]}$,

where ${\displaystyle \ominus }$ and ${\displaystyle \circ }$ are the morphological erosion and opening, respectively, ${\displaystyle \rho B}$ is an open ball of radius ${\displaystyle \rho }$, and ${\displaystyle {\overline {B}}}$ is the closure of ${\displaystyle B}$.

Let ${\displaystyle \{nB\}}$, ${\displaystyle n=0,1,\ldots }$, be a family of shapes, where B is a structuring element,

${\displaystyle nB=\underbrace {B\oplus \cdots \oplus B} _{n{\mbox{ times}}}}$, and

${\displaystyle 0B=\{o\}}$, 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 ${\displaystyle X\subset \mathbb {Z} ^{2}}$, the skeleton S(X) is the union of the skeleton subsets ${\displaystyle \{S_{n}(X)\}}$, ${\displaystyle n=0,1,\ldots ,N}$, where:

${\displaystyle S_{n}(X)=(X\ominus nB)-(X\ominus nB)\circ B}$.

The original shape X can be reconstructed from the set of skeleton subsets ${\displaystyle \{S_{n}(X)\}}$ as follows:

${\displaystyle X=\bigcup _{n}(S_{n}(X)\oplus nB)}$.

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

${\displaystyle \bigcup _{n\geq m}(S_{n}(X)\oplus nB)=X\circ mB}$.

Let ${\displaystyle nB_{z}}$ be the translated version of ${\displaystyle nB}$ to the point z, that is, ${\displaystyle nB_{z}=\{x\in E|x-z\in nB\}}$.

A shape ${\displaystyle nB_{z}}$ centered at z is called a maximal disk in a set A when:

• ${\displaystyle nB_{z}\in A}$, and
• if, for some integer m and some point y, ${\displaystyle nB_{z}\subseteq mB_{y}}$, then ${\displaystyle mB_{y}\not \subseteq A}$.

Each skeleton subset ${\displaystyle S_{n}(X)}$ consists of the centers of all maximal disks of size n.