A distance matrix D of order n is symmetric with elements idfj, where d,, = 0. D is Euclidean when the in(n 1) quantities dij can be generated as the distances between a set of n points, X (n X p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p = rank(X) of any generating X; in general p + 1 and p +2 are also acceptable but may include imaginary ...

For a convex body $K\subset\mathbb{R}^d$ the mean distance $\Delta(K)=\mathbb{E}|X_1-X_2|$ is expected Euclidean of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower upper bounds for ratio between $\Delta(K)$ first intrinsic volume $V_1(K)$ $K$ (normalized width) are derived degenerate extremal cases discussed. The argument relies on Riesz's rearrangement ine...

Based on a two-component descriptor, a distance label for each point, it is shown that Euclidean distance maps can be generated by effective sequential algorithms. The map indicates, for each pixel in the objects (or the background) of the originally binary picture, the shortest distance to the nearest pixel in the background (or the objects). A map with negligible errors can be produced in two...

In this article, we propose to investigate the extension of the SEDT (Squared Euclidean Distance Transformation) on irregular isothetic grids. We give two algorithms to handle different structurations of grids. We first describe a simple approach based on the complete Voronoi diagram of the background irregular pixels, very fast on sparse grids. Then, we extend the separable algorithm defined o...

Thinning is one of the most frequently used methods to know the geometrical feature of objects. It also provides the topological feature and length measurements about an object. For example, the tree structure of the bronchus is determined by using the thinned result of it. This paper presents a three dimensional thinning method which can control the quality of result concerning appearance of s...

Fermat-Weber problem in its simple form (unconstrained, single facility, Euclidean metric) is well investigated. Lot of algorithms are also developed for more complicated cases. But the generalized multi-facility problem with barriers, restricted zones and an arbitrary metric has no well-known algorithm for its solving. In this paper, we consider the planar multi-facility Weber problem with res...

1. SUMMARY Time-series, or time-sequence, data show the value of a parameter over time. A common query with time-series data is to find all sequences which are similar to a given sequence. The most common technique for evaluating similarity between two sequences involves calculating the Euclidean distance between them. However, many examples can be given where two similar sequences are separate...

Let x1, · · · ,xn be points randomly chosen from a set G ⊂ R and f(x) be a function. The Euclidean random matrix is given by Mn = (f(∥xi − xj∥))n×n where ∥ · ∥ is the Euclidean distance. When N is fixed and n → ∞ we prove that μ̂(Mn), the empirical distribution of the eigenvalues of Mn, converges to δ0 for a big class of functions of f(x). Assuming both N and n go to infinity proportionally, we ...

The Euclidean distance matrix (EDM) completion problem and the positive semidefinite (PSD) matrix completion problem are considered in this paper. Approaches to determine the location of a point in a linear manifold are studied, which are based on a referential coordinate set and a distance vector whose components indicate the distances from the point to other points in the set. For a given ref...

Distance/correlation metrics have emerged as a robust and simplified tool for assessing the spectral characteristics of hyperspectral image pixels effectively categorizing vegetation within specific study area. Correlation methods provide readily deployable computationally efficient approach, rendering them particularly advantageous applications in developing nations or regions with limited res...