Gauss–Laguerre quadrature

Library Technician Anton from Strathroy, has many passions that include r/c helicopters, property developers in condo new launch singapore and coin collecting. Finds the beauty in planing a trip to spots around the globe, recently only returning from Old Town of Corfu. 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 and 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 $X\subset {\mathbb{ℝ}}^2$:

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

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

Discrete images

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

$nB=\underset{n\text{ times}}{\underbrace{B\oplus \dots \oplus B}}$, 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 $X\subset {\mathbb{\mathbb{Z}}}^2$, the skeleton S(X) is the union of the skeleton subsets $\{S_n(X)\}$, $n=0,1,\dots ,N$, 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:

$\bigcup _{n\ge m}(S_n(X)\oplus nB)=X\circ mB$.

The skeleton as the centers of the maximal disks

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

A shape $n{B}_z$ centered at z is called a maximal disk in a set A when:

• $n{B}_z\in A$, and
• if, for some integer m and some point y, $n{B}_z\subseteq m{B}_y$, then $m{B}_y\subseteq ̸A$.

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

Notes

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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.