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. == Performing Morphological Skeletonization on Images ==