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). 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, etc. 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 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 and plant morphology on various biological scales. 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 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 One way of defining the skeleton of a shape A is as the set of centers of all maximal disks in A. 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.