Topological skeleton - AbsoluteAstronomy.com

In shape analysis, skeleton (or topological skeleton) 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...

is a thin version of that shape that is equidistant

Equidistant

A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal.In two-dimensional Euclidian geometry the locus of points equidistant from two given points is their perpendicular bisector...

to its boundaries

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

. The skeleton usually emphasizes geometrical and topological properties of the shape, such as its connectivity

Connectedness

In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...

, 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...

, length

Length

In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

, 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 skeleton

Straight skeleton

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.Straight skeletons...

s, morphological skeleton

Morphological skeleton

In digital image processing, morphological skeleton is a skeleton representation of a shape or binary image, computed by means of morphological operators.Morphological skeletons are of two kinds:...

s, and skeletons by influence zones (SKIZ) (also known as Voronoi diagram

Voronoi diagram

In mathematics, a Voronoi diagram is a special kind of decomposition of a given space, e.g., a metric space, determined by distances to a specified family of objects in the space...

).

In the technical literature, the concepts of skeleton and 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....

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

Thinning

Thinning is a term used in agricultural sciences to mean the removal of some plants, or parts of plants, to make room for the growth of others.- Forestry :...

are also regarded as identical by some, and not by others.

Skeletons have been used in several applications in computer vision

Computer vision

Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...

, image analysis

Image analysis

Image analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques...

, and 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...

, including optical character recognition

Optical character recognition

Optical character recognition, usually abbreviated to OCR, is the mechanical or electronic translation of scanned images of handwritten, typewritten or printed text into machine-encoded text. It is widely used to convert books and documents into electronic files, to computerize a record-keeping...

, fingerprint recognition, visual inspection, pattern recognition

Pattern recognition

In machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...

, 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...

compression

Image compression

The objective of image compression is to reduce irrelevance and redundancy of the image data in order to be able to store or transmit data in an efficient form.- Lossy and lossless compression :...

, and protein folding

Protein folding

Protein folding is the process by which a protein structure assumes its functional shape or conformation. It is the physical process by which a polypeptide folds into its characteristic and functional three-dimensional structure from random coil....

Mathematical definitions

Skeletons have several different mathematical definitions in the technical literature; most of them lead to similar results in continuous spaces

Continuum (topology)

In the mathematical field of point-set topology, a continuum is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff topological space...

, but usually yield different results in discrete space

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...

Quench points of the fire propagation model

In his seminal paper, of the Air Force Cambridge Research Laboratories in Cambridge, Massachusetts

Cambridge, Massachusetts

Cambridge is a city in Middlesex County, Massachusetts, United States, in the Greater Boston area. It was named in honor of the University of Cambridge in England, an important center of the Puritan theology embraced by the town's founders. Cambridge is home to two of the world's most prominent...

, defined a medial axis

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

Disk (mathematics)

In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

(or ball

Ball (mathematics)

In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....

) B is said to maximal in a set A if

, 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 disks 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 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 ridge

Ridge

A ridge is a geological feature consisting of a chain of mountains or hills that form a continuous elevated crest for some distance. Ridges are usually termed hills or mountains as well, depending on size. There are several main types of ridges:...

s of the distance function. 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.

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

Continuous function (set theory)

In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits of all values at previous stages...

Using 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...
Supplementing morphological operators with shape based pruning
Pruning (morphology)
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. In this case 'parasitic' components refer to branches of a line which are not key to...
Using curve evolution
Using level sets
Finding ridge points on the distance function
"Peeling" the shape, without changing the topology, until convergence

Skeletonization algorithms can sometimes create unwanted branches on the output skeletons. Pruning algorithms

Pruning (morphology)

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. In this case 'parasitic' components refer to branches of a line which are not key to...

are often used to remove these branches.

Open source software

External links

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