Morphological skeleton

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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[edit]

Lantuéjoul's formula[edit]

Continuous images[edit]

In (Lantuéjoul 1977),[1] Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image :


where and are the morphological erosion and opening, respectively, is an open ball of radius , and is the closure of .

Discrete images[edit]

Let , , be a family of shapes, where B is a structuring element,

, and
, 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 , the skeleton S(X) is the union of the skeleton subsets , , where:


Reconstruction from the skeleton[edit]

The original shape X can be reconstructed from the set of skeleton subsets as follows:


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


The skeleton as the centers of the maximal disks[edit]

Let be the translated version of to the point z, that is, .

A shape centered at z is called a maximal disk in a set A when:

  • , and
  • if, for some integer m and some point y, , then .

Each skeleton subset consists of the centers of all maximal disks of size n.


  1. ^ See also (Serra's 1982 book)


  • 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.