In digital image processing

Digital image processing

Digital image processing is the use of computer algorithms to perform image processing on digital images. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing...

, morphological skeleton is a skeleton

Topological skeleton

In shape analysis, skeleton of a shape is a thin version of that shape that is equidistant to its boundaries. The skeleton usually emphasizes geometrical and topological properties of the shape, such as its connectivity, topology, length, direction, and width...

 (or medial axis

Medial axis

The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced by Blum as a tool for biological shape recognition....

) representation of a shape

Shape

The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...

 or binary image

Binary image

A binary image is a digital image that has only two possible values for each pixel. Typically the two colors used for a binary image are black and white though any two colors can be used. The color used for the object in the image is the foreground color while the rest of the image is the...

, computed by means of morphological operators

Mathematical morphology

Mathematical morphology is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions...

.

Morphological skeletons are of two kinds:

• Those defined and by means of morphological opening
  Opening (morphology)
  In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:A\circ B = \oplus B, \, where \ominus and \oplus denote erosion and dilation, respectively....
  
  s, from which the original shape can be reconstructed,
• Those computed by means of the hit-or-miss transform
  Hit-or-miss transform
  In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements...
  
  , which preserve the shape's topology
  Topology
  Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
  
  .

Continuous images

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

where and are the morphological erosion

Erosion (morphology)

For use of "Erosion" in dermatopathology, see Erosion Erosion is one of two fundamental operations in Morphological image processing from which all other morphological operations are based...

 and opening

Opening (morphology)

In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:A\circ B = \oplus B, \, where \ominus and \oplus denote erosion and dilation, respectively....

, respectively, and is an open ball of radius

Radius

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

 .

Discrete images

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

Structuring element

In mathematical morphology, a structuring element is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image...

,, 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

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

Let be the translated version 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.