Showing papers on "Euclidean distance published in 1982"

Multiple stage vector quantization for speech coding

Abstract: In this paper, we present a multiple stage vector quantization technique which allows easy expansion of the original vector quantizer design to operate at higher bit rates for lower distortion. The computation and storage reduction is achieved by the fact that the overall requirements are the sum of the requirements of each stage instead of an exponentially increasing function of the bit rate as in the original one stage design. In the case of Euclidean distance measures such as the log area ratio measure, experimental results show that the quantizer performance is very close to a theoretically predicted asymptotically optimal rate distortion relationship.

Finitely-additive invariant measures on Euclidean spaces

Abstract: It is shown that for n ≥ 3 the Lebesgue measure is the unique finitely-additive isometry-invariant measure on the ring of bounded Lebesgue measurable subsets of the n-dimensional Euclidean space.

Multi-grid convergence theory

Abstract: 3. Convergence of the Two-Grid Iteration 3.1 Smoothing Property and Approximation Property 3.2 Discussion of the Approximation Property 3.2.1 Finite Element Equation (Simple Case) 3.2.2 Finite Element Equation (More General Case) 3.3 Discussion of the Smoothing Property 3.3.1 Jacobi-like Iteration for Positive Definite Matrix 3.3.2 Modified Jacobi Iteration for General Matrix 3.3.3 Smoothing Property for GauB-Seidel Iteration 3.4 Two-Grid for Finite Element Equations 3.4.1 Case of H -Regular Problems 3.4.2 Less Regular Problems 3.5 Quantitative Estimates for Symmetric Problems

On the minimum Euclidean distance for a class of signal space codes

Abstract: The minimum Euclidean distance for a class of constant envelope phase modulation codes is studied. Bandwidth and power efficient signals with continuous phase are considered. The information carrying phase varies piecewise linearly and the slopes are cyclically changed for successive symbol time intervals, yielding the so-called multi- h signals. It has previously been shown that this class of signals contains bandwidth and power efficient signals when coherent maximum likelihood sequence detection is used. Bounds on the achievable Euclidean distance for signals in the above class are given. Upper bounds are calculated as well as minimum distance results for specific multilevel multi- h signals. It is concluded that quaternary and octal muiti- h schemes considerably outperform the binary schemes. Furthermore in the important small modulation index region, 2-h codes gain the maximum 3 dB. Larger gains are not available by increasing the number of h values.

Some Comments on Non-Euclidean Mental Maps

Abstract: The Euclidean metric is perhaps the most commonly used and most convenient one for representing mapped phenomena. In this paper we examine the suitability of representing cognitive phenomena via the Euclidean metric. Some general properties of spaces are examined with particular emphasis on the properties of isotropy, incompleteness, and curvature, and a more detailed discussion is undertaken of the suitability of using curved spaces (particularly Reimannian spaces) for the representation of cognitive information. A final discussion is presented on the problems of handling manifolds with folds, warps, and tears; and speculations are made concerning the appropriateness of non-Euclidean metrics for the spatial representation of mental maps.

Partially Coherent Detection of Digital Full Response Continuous Phase Modulated Signals

Abstract: Recently bandwidth efficient constant envelope digital modulation schemes have been shown also to he power efficient, if detected coherently. In this paper the performance of such systems is analyzed for an optimum noncoherent or partially coherent detector, at large signal-to-noise ratios. The considered schemes are M -ary with arbitrary pulse shaping over one symbol interval (full response). The performance is analyzed by means of a parameter called equivalent minimum Euclidean distance, mathematically playing the same role as the minimum Euclidean distance used for coherent detectors. It is shown that noncoherent detectors perform as well as coherent, although the observation interval must be increased.

Geometric and Probabilistic Aspects of Statistical Distance Functions

Abstract: commonly used statistical distance functions. It is argued that the distance between centroids can be viewed as a probabilistic function reflecting both the geometric position of each taxon and/or the variance and correlation between the variables. A Monte Carlo simulation study is carried out to assess the effects of sample size, different variances and intercorrelations on the variability or "precision" of the distance estimates using the Euclidean distance, Pearson's Coefficient of Racial Likeness, and the Mahalanobis distance. All three distance estimators are shown to be biased and consistently overestimate the theoretical distance. The effects of these results on studies in systematic biology are discussed. [Distance statistics; Euclidean distance; Pearson's Coefficient of Racial Likeness; Mahalanobis distance; numerical taxonomy; multivariate geometry.]

A second-order method for solving the continuous multifacility location problem

Abstract: A unified and numerically stable second-order approach to the continuous multifacility location problem is presented. Although details are initially given for only the unconstrained Euclidean norm problem, we show how the framework can be readily extended to l p norm and mixed norm problem as well as to constrained problems.

On Non-Normal Invariance Principles for Multi-Response Permutation Procedures

Abstract: Summary A non-normal invariance principle is established for a restricted class of univariate multi-response permutation procedures whose distance measure is the square of Euclidean distance. For observations from a distribution with finite second moment, the test statistic is found asymptotically to have a centered chi-squared distribution. Spectral expansions are used to determine the asymptotic distribution for more general distance measures d, and it is shown that if d(x, y) = |x — y|u, u† 2, the asymptotic distribution is not invariant (i.e. it is dependent on the distribution of the observations).

Minimum permanents of doubly stochastic matrices with at least one zero entry

Abstract: It is shown that the minimum value of the permanent on the n× ndoubly stochastic matrices which contain at least one zero entry is achieved at those matrices nearest to Jn in Euclidean norm, where Jn is the n× nmatrix each of whose entries is n-1 . In case n ≠ 3 the minimum permanent is achieved only at those matrices nearest Jn ; for n= 3 it is achieved at other matrices containing one or more zero entries as well.

Numerical analysis of volatile oil data in systematic studies of australian rainforest trees

Abstract: Preparatory to the use of volatile oils in systematic studies of Australian rainforest trees, a number of different techniques for handling volatile oil data are reviewed. Compared are two different schemes of character weighting (unweighted and F-weighted), three different schemes of standardisation (unstandardised, standardisation by range, standardisation by standard deviation), and three different distance measures (Squared Euclidean Distance, Euclidean Distance, Manhattan Metric distance). It is determined empirically that, for volatile oil data, where weighting is both possible and valid, a combination of standardisation and weighting proves most effective; on the other hand, if weighting is not possible, standardisation should not be undertaken. Euclidean Distance and Manhattan Metric distance prove to be the best distance

Numerical analysis of volatile oil data in systematic

Abstract: Summary Preparatory to the use of volatile oils in systematic studies of Australian rainforest trees, a number of different techniques for handling volatile oil data are reviewed. Compared are two different schemes of character weighting (unweighted and F-weighted), three different schemes of standardisation (unstandardised, standardisation by range, standardisation by standard deviation), and three different distance measures (Squared Euclidean Distance, Euclidean Distance, Manhattan Metric distance). It is determined empirically that, for volatile oil data, where weighting is both possible and valid, a combination of standardisation and weighting proves most effective; on the other hand, if weighting is not possible, standardisation should not be undertaken. Euclidean Distance and Manhattan Metric distance prove to be the best distance measures.

Technical Note—Extension of the Elzinga-Hearn Algorithm to the Weighted Case

Abstract: The problem considered in this note is that of locating a single new facility among m existing facilities with the objective of minimizing the maximum weighted Euclidean distance of the new facility from the existing facilities, without making the assumption that all the weights are equal. The new algorithm takes into consideration the structure of the problem, and it will terminate in a finite number of iterations if exact arithmetic is used.

Independence of the increments of Gaussian random fields

Abstract: Let be a mean zero Gaussian random field ( n ⋜ 2). We call X Euclidean if the probability law of the increments X(A ) − X(B ) is invariant under the Euclidean motions. For such an X , the variance of X(A ) − X(B ) can be expressed in the form r (| A − B |) with a function r(t ) on [0, ∞) and the Euclidean distance |A − B| .

A note on solving multifacility location problems involving euclidean distances

Abstract: This note considers a recently proposed solution method for a multifacility location problem. It is shown that the method does not always produce an optimal solution.

Position operators as integrals with respect to Euclidean systems of covariance

Abstract: Position operators (p.o.) for relativistic elementary quantum systems are constructed as operator-valued integrals with respect to Euclidean systems of covariance (ESC), i.e., positive operator-valued (POV) measures being covariant under the Euclidean group, and are expressed in terms of the generators of the Poincare transformations. These p.o. are partly well-known in the literature where they are found by other methods.

A new class of four-point properties which characterize euclidean spaces

Abstract: Characterizations of generalized euclidean spaces by means of euclidean four-point properties.state that every metric space which is complete, and which contains a metric line joining each two of its points is a generalized euclidean space if and only if each quadruple from a certain class of quadruples of the space is congruent with a quadruple of points in a euclidean space. It is known that it suffices to consider only quadruples containing a linear triple, or quadruples in which one of the linear points is a metric midpoint of the other two. Another class of four-point properties involves quadruples which contain a linear triple and a point equidistant from two of the linear points. The present paper presents three characterizations of euclidean spaces based on four-point properties in which the embedded quadruples contain a linear triple and some three of the distances determined by the four points are equal.

A detailed study of the duality relation for the least squares adjustment in euclidean spaces

Abstract: The Euclidean spaces with their inner products are used to describe methods of least squares adjustment as orthogonal projections on finite-dimensional subspaces. A unified Euclidean space approach to the least squares adjustment methods “observation equations” and “condition equations” is suggested. Hence not only the two adjustment solutions are treated from the view-point of Euclidean space theory in a unified frame but also the existing duality relation between the methods of “observation equations” and “condition equations” is discussed in full detail. Another purpose of this paper is to contribute to the development of some familiarity with Euclidean and Hilbert space concepts. We are convinced that Euclidean and Hilbert space techniques in least squares adjustment are elegant and powerful geodetic methods.

Digital Filtering With the Second Moment Norm

Abstract: The Backus-Gilbert inverse method relates model estimates to actual earth models by use of a resolving kernel. This inversion can in turn be related to various digital filter designs. If the second moment norm is used to define the resolving kernel, two types of filters are produced by an eigenvector decomposition of a time-weighted autocorrelation matrix. The eigenvector corresponding to the largest eigenvalue of this matrix is similar to the output energy filter, while the eigenvector for the smallest eigenvalue performs more like a deconvolution filter. Synthetic and real data examples demonstrate the characteristics of these filters and compare them to the familiar square norm filters. Our experiments suggest that second moment norm filters offer no significant advantages over their Euclidean norm relatives.

Kakutani’s theorem for real-valued maps on a compact surface

Abstract: Let M be a compact 2-manifold (without boundary) Cl-embedded in R3. Then there exists positive a such that, given any positive r < a and any continuous map f: M -+ R, there exist points p, q, r E M, satisfying llq rl = jjr pll = Ilp qjj = r in the euclidean norm, for which f(p) = f(q) = f(r).