More on path generated digital metrics

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

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

Symmetry, 2024

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

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.

Pattern Recognition Letters, 1988

Using the knight's moves in the game of chess, the knight's distance is delined for the digital planc, hs i'tmctional form is presented. An algorithm is given for tracing a minimal knight's path. The properties of some related topological entities are cx01orcd. Pinally, the knight's transform is defined.

This paper focuses on the design of an effective method that computes the measure of circularity of an open or closed digital curve. Thanks to its geometric interpretation, an algorithm that only uses classical tools of computational geometry is derived. Even if a sophisticated machinery coming from linear programming can provide a linear time algorithm, its O(n log n) time complexity is better than many quadratic methods based on Voronoi diagrams. Moreover, this bound can be improve in the case of convex digital curves to reach linear time. * Supported by a grant from the french DGA (Jacques Blanc-Talon).

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.

Discrete Applied Mathematics, 2011

A new class of distances for graph vertices is proposed. This class contains parametric families of distances which reduce to the shortest-path, weighted shortestpath, and the resistance distances at the limiting values of the family parameters. The main property of the class is that all distances it comprises are graph-geodetic: d(i, j) + d(j, k) = d(i, k) if and only if every path from i to k passes through j. The construction of the class is based on the matrix forest theorem and the transition inequality.

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.

Pattern Recognition, 2010

This paper focuses on the design of an effective method that computes the measure of circularity of an open or closed digital curve. Thanks to its geometric interpretation, an algorithm that only uses classical tools of computational geometry is derived. Even if a sophisticated machinery coming from linear programming can provide a linear time algorithm, its O(n log n) time complexity is better than many quadratic methods based on Voronoi diagrams. Moreover, this bound can be improve in the case of convex digital curves to reach linear time. * Supported by a grant from the french DGA (Jacques Blanc-Talon).

2007 5th International Symposium on Image and Signal Processing and Analysis, 2007

Distance functions defined by the minimal costpath using weights and neighbourhood sequences (n.s.) are considered for the constrained distance transform (CDT). The CDT is then used to find one minimal cost-path between two points. The behaviour of some path-based distance functions is analyzed and a new error function is introduced. It is concluded that the weighted n.s.-distance with two weights (3 × 3 neighbourhood) and the weighted distance with three weights (5 × 5 neighbourhood) have similar properties in terms of minimal cost-path computation, while the former is more efficient to compute.

The notion of gap is quite important in combinatorial image analysis and it finds several useful applications in fields as CAD and computer graphics. On the other hand, dimension is a fundamental concept in General Topology and it was recently extended to digital objects. In this paper, we show that the dimension of a 2D digital object equipped with an adjacency relation A ( 2 f0; 1g) can be determinated by the number of its gaps besides some other parameters like the number of its pixel, vertices and edges.

A geodesic is the real world analog of a straight line. Where a straight line on a flat piece of paper minimizes the distance between two points, a geodesic minimizes the distance between two points on any surface; be it flat or not.Supposing that we have a surface in spacegiven by the equation z = f (x, y).The search for a geodesic line on this surface, or more generally in the plan provided by an arbitrary metric, may be made by solving the coupled differential second orderequations of Euler-Lagrange system. More precisely, the search of the shortest path connecting two given points may be made by solving that system for a specific initial velocity. In this paper we determine the geodesic lines corresponding the metric of type g =(dx2 + dy2) for f (x, y) defines positive. Starting from a metric of this type, we determine the Euler-Lagrange system correspondence; its solutions are geodesics. We designed geodesics and the shortest path for the given metric and a specific function f (x, y). We will need to determine the appropriate initial velocity for the system's numerical resolution of two differential equations of second order. Therefore, we are providing a suitable method for this.

A subclass of general octagonal distances defined by neighbourhood sequences have been characterized here which have a strikingly simple closed functional form. These are called simple distances. Minimization of the average absolute (normalized) and average relative errors of these simple distances with regard to the euclidean norm have been carried out to identify the best approximate digital distances in 2-D digital geometry. The direct errors have also been analyzed and the effect of finite domain sizes on the approximation has been highlighted. It is shown that the neighbourhood sequences {2}, { 1, 2},

Dictionary of Distances, 2006