Topological skeleton

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.

In his seminal paper, Harry Blum^[8] of the Air Force Cambridge Research Laboratories at Hanscom Air Force Base, in Bedford, Massachusetts, defined 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.

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

• ${\displaystyle B\subseteq A}$ , and
• If another disc D contains B, then ${\displaystyle D\not \subseteq A}$ .

One way of defining the skeleton of a shape A is as the set of centers of all maximal disks in A.^[9]

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.^[10] This definition assures that the skeleton points are equidistant from the shape boundary and is mathematically equivalent to Blum's medial axis transform.

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.^[3] There is a common mis-statement in the literature that the skeleton consists of points which are "locally maximum" in the distance transform. This is simply not the case, as even cursory comparison of a distance transform and the resulting skeleton will show. Ridges may have varying height, so a point on the ridge may be lower than its immediate neighbor on the ridge. It is thus not a local maximum, even though it belongs to the ridge. It is, however, less far away vertically than its ground distance would warrant. Otherwise it would be part of the slope.

• 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 continuous sets.

• Using morphological operators (See Morphological skeleton^[10])
• Supplementing morphological operators with shape based pruning^[11]
• Using intersections of distances from boundary sections^[12]
• Using curve evolution ^[13][14]
• Using level sets^[5]
• Finding ridge points on the distance function^[3]
• "Peeling" the shape, without changing the topology, until convergence^[15]
• Zhang-Suen Thinning Algorithm^[16]

Skeletonization algorithms can sometimes create unwanted branches on the output skeletons. Pruning algorithms are often used to remove these branches.

• Medial axis
• Straight skeleton
• β-skeleton
• Grassfire Transform
• Stroke-based fonts

• Abeysinghe, Sasakthi; Baker, Matthew; Chiu, Wah; Ju, Tao (2008), "Segmentation-free skeletonization of grayscale volumes for shape understanding", IEEE Int. Conf. Shape Modeling and Applications (SMI 2008) (PDF), pp. 63–71, doi:10.1109/SMI.2008.4547951, ISBN 978-1-4244-2260-9, S2CID 15148296.
• Abeysinghe, Sasakthi; Ju, Tao; Baker, Matthew; Chiu, Wah (2008), "Shape modeling and matching in identifying 3D protein structures" (PDF), Computer-Aided Design, 40 (6), Elsevier: 708–720, doi:10.1016/j.cad.2008.01.013
• Bai, Xiang; Longin, Latecki; Wenyu, Liu (2007), "Skeleton pruning by contour partitioning with discrete curve evolution" (PDF), IEEE Transactions on Pattern Analysis and Machine Intelligence, 29 (3): 449–462, doi:10.1109/TPAMI.2007.59, PMID 17224615, S2CID 14965041.
• Blum, Harry (1967), "A Transformation for Extracting New Descriptors of Shape", in Wathen-Dunn, W. (ed.), Models for the Perception of Speech and Visual Form (PDF), Cambridge, Massachusetts: MIT Press, pp. 362–380.
• Bucksch, Alexander (2014), "A practical introduction to skeletons for the plant sciences", Applications in Plant Sciences, 2 (8): 1400005, doi:10.3732/apps.1400005, PMC 4141713, PMID 25202647.
• Cychosz, Joseph (1994), Graphics gems IV, San Diego, CA, USA: Academic Press Professional, Inc., pp. 465–473, ISBN 0-12-336155-9.
• Dougherty, Edward R. (1992), An Introduction to Morphological Image Processing, ISBN 0-8194-0845-X.
• Golland, Polina; Grimson, W. Eric L. (2000), "Fixed topology skeletons" (PDF), 2000 Conference on Computer Vision and Pattern Recognition (CVPR 2000), 13-15 June 2000, Hilton Head, SC, USA, vol. 1, IEEE Computer Society, pp. 1010–1017, doi:10.1109/CVPR.2000.855792, ISBN 0-7695-0662-3, S2CID 9916140
• Gonzales, Rafael C.; Woods, Richard E. (2001), Digital Image Processing, Prentice Hall, ISBN 0-201-18075-8.
• Jain, Anil K. (1989), Fundamentals of Digital Image Processing, Bibcode:1989fdip.book.....J, ISBN 0-13-336165-9.
• Jain, Ramesh; Kasturi, Rangachar; Schunck, Brian G. (1995), Machine Vision, ISBN 0-07-032018-7.
• Kimmel, Ron; Shaked, Doron; Kiryati, Nahum; Bruckstein, Alfred M. (1995), "Skeletonization via distance maps and level sets" (PDF), Computer Vision and Image Understanding, 62 (3): 382–391, doi:10.1006/cviu.1995.1062
• Ogniewicz, R. L. (1995), "Automatic Medial Axis Pruning Based on Characteristics of the Skeleton-Space", in Dori, D.; Bruckstein, A. (eds.), Shape, Structure and Pattern Recognition, World Scientific, ISBN 981-02-2239-4.
• Petrou, Maria; García Sevilla, Pedro (2006), Image Processing Dealing with Texture, Wiley, ISBN 978-0-470-02628-1.
• Serra, Jean (1982), Image Analysis and Mathematical Morphology, Academic Press, ISBN 0-12-637240-3.
• Sethian, J. A. (1999), Level Set Methods and Fast Marching Methods, Cambridge University Press, ISBN 0-521-64557-3.
• Tannenbaum, Allen (1996), "Three snippets of curve evolution theory in computer vision", Mathematical and Computer Modelling, 24 (5): 103–118, doi:10.1016/0895-7177(96)00117-3.

• ITK (C++)
• Skeletonize3D (Java)
• Graphics gems IV (C)
• EVG-Thin (C++)
• NeuronStudio

• Skeletonization/Medial Axis Transform
• Skeletons of a region
• Skeletons in Digital image processing (pdf)
• Comparison of 15 line thinning algorithms
• Skeletonization using Level Set Methods
• Curve Skeletons
• Skeletons from laser scanned point clouds (Homepage)