Generalized Distances in Digital Geometry

ABSTBACT An analysis of paths and distances in n dimensions is carried out using variable neigbborhood sequences. A symbolic expression for the distance function between any two points in this quantized space is derived. An algorithm for finding the shortest path is presented. The necessary and sufficient condition for such distance functions to satisfy the properties of a metric has been derived. Certain practical and efficient methods to check for metric properties are also presented.

wellbehaved, where p is the length of the sequence B. We have also reviewed the other weaker and stronger conditions of triangularity and have introduced a new weaker condition here to reformulate the strategy for deciding the metricity of a d(B).

A generalized distance measure called t-Cost-m-Neighbour (tCmN) distance in n-D grid point space is presented.

Digital distance geometry (DDG) is the study of distances in the geometry of digitized spaces. This was introduced approximately 25 years ago, when the study of digital geometry itself began, for providing a theoretical background to digital picture processing algorithms. In this survey we focus our attention on the DDG of arbitrary dimensions and other related issues and compile an up-to-date list of references on the topic.

A new class of distance functions has been defined in n-D, where the distance between neighboring points may be more than unity. A necessary and sufficient condition for such distance functions to satisfy the properties of a metric has been derived. These metrics, called t-cost distance, give the length of the shortest t-path between two points in n-D digital space. Some properties of their hyperspheres are also studied. Suitability of these distances as viable alternative to Euclidean distance in n-D has been explored using absolute and relative error criteria. It is shown that lower dimension (2-D and 3-D) distance measures presently used in digital geometry can be easily derived as special cases. Finally most of these results have been extended for the natural generalization of integral costs to real costs.

In this paper a geometric approach is suggested to ®nd the closest approximation to Euclidean metric based on geometric measures of the digital circles in 2D and the digital spheres in 3D for the generalized octagonal distances. First we show that the vertices of the digital circles (spheres) for octagonal distances can be suitably approximated as a function of the number of neighborhood types used in the sequence. Then we use these approximate vertex formulae to compute the geometric features in an approximate way. Finally we minimize the errors of these measurements with respect to respective Euclidean discs to identify the best distances. We have veri®ed our results by experimenting with analytical error measures suggested earlier. We have also compared the performances of the good octagonal distances with good weighted distances. It has been found that the best octagonal distance in 2D (f1Y 1Y 2g) performs equally good with respect to the best one for the weighted distances (h3Y 4i). In fact in 3D, the octagonal distance f1Y 1Y 3g has an edge over the other good weighted distances. Ó

The notion of rn-Neighbour Distance d, m integer, in the n-D digital geometry has been extended under the name of real rn—Neighbour Distance 6m' in this paper, to n-D real space. Complete analyses of the hyperspheres H(rn,n;r) of 6 have been carried out to show that the rnaxirna of the absolute and relative errors between this rnetric and the Euclidean norm E minimizes at certain extreme symmetric points on the hypersphere. The coherence between these results and those already available in the digital domain has been mentioned to project as a powerful tool in metric analyses in digital geometry. The paper also rnakes fundamental contributions in the study of non—Euclidean metric spaces, extending the L1 = 6 and I = 6 norms in a natural yet non-Minkowski way. Finally it is shown that real rn-neighbour distance has direct applications in scheduling problems.

Theoretical Computer Science, 2012

In image processing, the distance transform (DT), in which each object grid point is assigned the distance to the closest background grid point, is a powerful and often used tool. In this paper, distance functions defined as minimal cost-paths are used and a number of algorithms that can be used to compute the DT are presented. We give proofs of the correctness of the algorithms.

Journal of Mathematical Imaging and Vision, 2018

In this paper, we present a general framework for digital distance functions, defined as minimal cost paths, on the square grid. Each path is a sequence of pixels, where any two consecutive pixels are adjacent and associated with a weight. The allowed weights between any two adjacent pixels along a path are given by a weight sequence, which can hold an arbitrary number of weights. We build on our previous results, where only two or three unique weights are considered, and present a framework that allows any number of weights. We show that the rotational dependency can be very low when as few as three or four unique weights are used. Moreover, by using n weights, the Euclidean distance can be perfectly obtained on the perimeter of a square with side length 2n. A sufficient condition for weight sequences to provide metrics is proven.

SIAM Review, 2014

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.