# Topological skeleton

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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).

## Contents

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, etc.

In the technical literature, the concepts of skeleton and medial axis are used interchangeably by some authors, [1] [2] [3] [4] [5] while some other authors [6] [7] [8] regard them as related, but not the same. Similarly, the concepts of skeletonization and thinning are also regarded as identical by some, [2] and not by others. [6]

Skeletons are widely used in computer vision, image analysis, pattern recognition and digital image processing for purposes such as optical character recognition, fingerprint recognition, visual inspection or compression. Within the life sciences skeletons found extensive use to characterize protein folding [9] and plant morphology on various biological scales. [10]

## 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, Harry Blum [11] 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.

### Centers of maximal disks (or balls)

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. [12]

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

### 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. [6] 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.

### 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.

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

## Notes

1. Jain, Kasturi & Schunck (1995), Section 2.5.10, p. 55.
2. Gonzales & Woods (2001), Section 11.1.5, p. 650
3. A. K.Jain ( 1989 ), Section 9.9, p. 382.
4. Sethian (1999), Section 17.5.2, p. 234.
5. HarryBlum ( 1967 )
6. A. K.Jain ( 1989 ), Section 9.9, p. 387.
7. Gonzales & Woods (2001), Section 9.5.7, p. 543.
8. R. Kimmel, D. Shaked, N. Kiryati, and A. M. Bruckstein. https://www.cs.technion.ac.il/~ron/PAPERS/skeletonization_CVIU_1995.pdf Comp. Vision and Image Understanding, 62(3):382-391, 1995.
9. A. K.Jain ( 1989 ), Section 9.9, p. 389.

## Related Research Articles

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Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it can be employed as well on graphs, surface meshes, solids, and many other spatial structures.

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A distance transform, also known as distance map or distance field, is a derived representation of a digital image. The choice of the term depends on the point of view on the object in question: whether the initial image is transformed into another representation, or it is simply endowed with an additional map or field.

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In computer vision and image processing feature detection includes methods for computing abstractions of image information and making local decisions at every image point whether there is an image feature of a given type at that point or not. The resulting features will be subsets of the image domain, often in the form of isolated points, continuous curves or connected regions.

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The pruning algorithm is a technique used in digital image processing based on mathematical morphology. It is used as a complement to the skeleton and thinning algorithms to remove unwanted parasitic components (spurs). In this case 'parasitic' components refer to branches of a line which are not key to the overall shape of the line and should be removed. These components can often be created by edge detection algorithms or digitization. Common uses for pruning include automatic recognition of hand-printed characters. Often inconsistency in letter writing creates unwanted spurs that need to be eliminated for better characterization.

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. In mathematics the closure of the medial axis is known as the cut locus.

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Ridge detection is the attempt, via software, to locate ridges in an image.

In geometry, a straight skeleton is a method of representing a polygon by a topological skeleton. It is similar in some ways to the medial axis but differs in that the skeleton is composed of straight line segments, while the medial axis of a polygon may involve parabolic curves. However, both are homotopy-equivalent to the underlying polygon.

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In image processing, the grassfire transform is the computation of the distance from a pixel to the border of a region. It can be described as "setting fire" to the borders of an image region to yield descriptors such as the region's skeleton or medial axis. Harry Blum introduced the concept in 1967.

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Discrete Skeleton Evolution (DSE) describes an iterative approach to reducing a morphological or topological skeleton. It is a form of pruning in that it removes noisy or redundant branches (spurs) generated by the skeletonization process, while preserving information-rich "trunk" segments. The value assigned to individual branches varies from algorithm to algorithm, with the general goal being to convey the features of interest of the original contour with a few carefully chosen lines. Usually, clarity for human vision is valued as well. DSE algorithms are distinguished by complex, recursive decision-making processes with high computational requirements. Pruning methods such as by structuring element (SE) convolution and the Hough transform are general purpose algorithms which quickly pass through an image and eliminate all branches shorter than a given threshold. DSE methods are most applicable when detail retention and contour reconstruction are valued.

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