Definitions

# Topological skeleton

In shape analysis, skeleton (or topological 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. Together with the distance of its points to the shape boundary, the skeleton can also serve as a representation of the shape (they contain all the information necessary to reconstruct the shape).

Skeletons have several different mathematical definitions in the technical literature, and there are many different algorithms for computing them. Various different variants of skeleton can also be found, including straight skeletons, morphological skeletons, and skeletons by influence zones (SKIZ) (also known as Voronoi diagram).

In the technical literature, the concepts of skeleton and medial axis are used interchangeably by some authors, while some other authors regard them as related, but not the same. Similarly, the concepts of skeletonization and thinning are also regarded as identical by some, and not by others.

Skeletons have been used in several applications in computer vision, image analysis, and digital image processing, including optical character recognition, fingerprint recognition, visual inspection, pattern recognition, and binary image compression.

## Mathematical Definitions

Skeletons have several different mathematical definitions in the technical literature; most of them lead to similar results in continuous spaces, but usually yield different results in discrete spaces.

### Quench points of the fire propagation model

In his seminal paper (Blum 1967), H. Blum defines a medial axis for computing a skeleton of a shape, using an intuitive model of fire propagation on a grass field, where the field has the form of the given shape. If one "sets fire" at all points on the boundary of that grass field simultaneously, then the skeleton is the set of quench points, i.e., those points where two or more wavefronts meet. This intuitive description is the starting point for a number of more precise definitions.

### Centers of maximal discs (or balls)

A disc (or ball) B is said to maximal in a set A if

• $Bsubseteq A$, and
• If another disc D contains B, then $Dnotsubseteq A$.

One way of defining the skeleton of a shape A is as the set of centers of all maximal discs in A.

### Centers of bi-tangent circles

The skeleton of a shape A can also be defined as the set of centers of the discs that touch the boundary of A in two or more locations. This assures that the skeleton points are equidistant from the shape boundary.

### Ridges of the distance function

Many definitions of skeleton make use of the concept of distance function, which is a function that returns for each point x inside a shape A its distance to the closest point on the boundary of A. Using the distance function is very attractive because its computation is relatively fast.

One of the definitions of skeleton using the distance function is as the ridges of the distance function, i.e., the points that are locally maximum.

### Other definitions

• Points with no upstream segments in the distance function. The upstream of a point x is the segment starting at x which follows the maximal gradient path.
• Points where the gradient of the distance function are different from 1 (or, equivalently, not well defined)
• Smallest possible set of lines that preserve the topology and are equidistant to the borders

## Skeletonization Algorithms

There are many different algorithms for computing skeletons for shapes in digital images, as well as continuous sets.

• Using morphological operators
• Using curve evolution
• Using level sets
• Finding ridge points on the distance function
• "Peeling" the shape, without changing the topology, until convergence

## References

• Rafael C. Gonzales and Richard E. Woods, Digital Image Processing, ISBN 0-201-18075-8 (2001)
• Ramesh Jain, Rangachar Kasturi and Brian G. Schunck, Machine Vision, ISBN 0-07-032018-7 (1995)
• Anil K. Jain, Fundamentals of Digital Image Processing, ISBN 0-13-336165-9 (1989)
• Jean Serra, Image Analysis and Mathematical Morphology, ISBN 0126372403 (1982)
• Edward R. Dougherty, An Introduction to Morphological Image Processing, ISBN 0-8194-0845-X (1992)
• J.A. Sethian, Level Set Methods and Fast Marching Methods, ISBN 0-521-64557-3 (1999)
• Maria Petrou and Pedro García SevillaImage Processing Dealing with Texture, ISBN-13: 978-0-470-02628-1, ISBN-10: 0-470-02628-6 (2006)
• R.L. Ogniewicz, "Automatic Medial Axis Pruning Based on Characteristics of the Skeleton-Space", in Shape, Structure and Pattern Recognition (D. Dori and A. Bruckstein editors), ISBN 981-02-2239-4 (1995)
• "A Transformation for Extracting New Descriptors of Shape" by H. Blum, in Models for the Perception of Speech and Visual Form, W. Whaten-Dunn (Ed.). MIT Press: Cambridge, MA, pp. 362–380