- Topological skeleton
shape analysis, skeleton (or topological skeleton) of a shapeis a thin version of that shape that is equidistantto 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 [(R. Jain 1995), Section 2.5.10, pg. 55.] (Gonzales and Woods 2001), Section 11.1.5, pg. 650] [http://people.csail.mit.edu/polina/papers/skeletons_cvpr00.pdf] [(Dougherty 1992)] [(Ogniewicz 1995)] , while some other authors(A.K. Jain 1989), Section 9.9, pg. 382] [(Serra 1982)] (Sethian 1999), Section 17.5.2, pg. 234] regard them as related, but not the same. Similarly, the concepts of "skeletonization" and
thinningare 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 imagecompression.
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
Quench points of the fire propagation model
In his seminal paper (Blum 1967), H. Blum defines a
medial axisfor 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 [http://en.wiktionary.org/wiki/quench 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)
* , and
* If another disc "D" contains "B", then .
One way of defining the skeleton of a shape "A" is as the set of centers of all maximal discs in "A" [(ajain1989|A.K. Jain 1989), Section 9.9, pg. 387] .
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.
* 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
There are many different algorithms for computing skeletons for shapes in
digital images, as well as continuoussets.
* Using morphological operators(Gonzales and Woods 2001), Section 9.5.7, pg. 543]
* Using curve evolution
* Using level sets
* Finding ridge points on the distance function
* "Peeling" the shape, without changing the topology, until convergence [(A.K. Jain 1989), Section 9.9, pg. 389]
* 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 Sevilla"Image 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
* [http://www.cee.hw.ac.uk/hipr/html/skeleton.html Skeletonization/Medial Axis Transform]
* [http://www.cs.ru.nl/~ths/rt2/col/h9/9gebiedENG.html#9.2.4 Skeletons of a region]
* [http://www.citr.auckland.ac.nz/techreports/2002/CITR-TR-112.pdf Skeletons in Digital image processing (pdf)]
* [http://www-igm.univ-mlv.fr/LabInfo/rapportsInternes/2006/01.pdf Comparision of 15 line thinning algorithms]
* [http://mecca.louisville.edu/~msabry/projects/cskel.htm Skeletonization using Level Set Methods]
* [http://www.cvip.uofl.edu/~msabry/home/Publications/Hassouna_Farag_ICCV_2007.pdf Curve Skeletons]
* [http://mecca.louisville.edu/~msabry/projects/vcomparative.htm Comparative Study of Curve Skeleton Extraction Techniques.]
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