Showing papers on "Euclidean distance published in 1987"

Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis

Abstract: The main problem investigated in this paper is that of restricted invertibility of linear operators acting on finite dimensionallp-spaces. Our initial motivation to study such questions lies in their applications. The results obtained below enable us to complete earlier work on the structure of complemented subspaces ofLp-spaces which have extremal euclidean distance.

A weighted cepstral distance measure for speech recognition

Abstract: A weighted cepstral distance measure is proposed and is tested in a speaker-independent isolated word recognition system using standard DTW (dynamic time warping) techniques. The measure is a statistically weighted distance measure with weights equal to the inverse variance of the cepstral coefficients. The experimental results show that the weighted cepstral distance measure works substantially better than both the Euclidean cepstral distance and the log likelihood ratio distance measures across two different databases. The recognition error rate obtained using the weighted cepstral distance measure was about 1 percent for digit recognition. This result was less than one-fourth of that obtained using the simple Euclidean cepstral distance measure and about one-third of the results using the log likelihood ratio distance measure. The most significant performance characteristic of the weighted cepstral distance was that it tended to equalize the performance of the recognizer across different talkers.

Conformal rotation in perturbative gravity

Abstract: The classical Euclidean action for general relativity is unbounded below; therefore Euclidean functional integrals weighted by this action are manifestly divergent. However, as a consequence of the positive-energy theorem, physical amplitudes for asymptotically flat spacetimes can indeed be expressed as manifestly convergent Euclidean functional integrals formed in terms of the physical degrees of freedom. From these integrals, we derive expressions for these same physical quantities as Euclidean integrals over the full set of variables for gravity computed as metric perturbations off a flat background. These parametrized Euclidean functional integrals are weighted by manifestly positive actions with rotated conformal factors. They are similar in form to Euclidean functional integrals obtained by the Gibbons-Hawking-Perry prescription of contour rotation.

State-space measures for stability robustness

Abstract: Stability robustness measures for a perturbed linear feedback system are derived based on state-space models of the system. The system may be a continuous-time or discrete-time system. The perturbations are modeled as additive perturbation matrices. Necessary and sufficient conditions for the stability of the perturbed closed-loop system for all perturbations of norm bounded by some positive number are obtained. The destabilizing perturbations of minimal norm are characterized. It is shown by an example that there are cases when the destabilizing perturbations of minimal norm are all complex. The results are expressed in terms of induced operator norms. These are later specialized to the Euclidean norm and expressed in terms of singular values. A simple example is also included to illustrate an application of the results of this note.

A fast procedure for computing the distance between complex objects in three space

Abstract: An efficient and reliable algorithm for computing the Euclidean distance between a pair of convex sets in Rmdescribed. Extensive numerical experience with a broad family of polytopes in Rsshows that the computational cost is approximately linear in the total number of vertices specifying the two polytopes. The algorithm has special features which make its application in a variety of robotics problems attractive. These are discussed and an example of collision detection is given.

Minimum link paths in polygons and related problems

Abstract: Consider the motion of a robot in a two-dimensional Euclidean space that is bounded by a simple polygon. The robot can move only along straight-line segments and the boundary of the polygon represents an impenetrable obstacle. Suppose that it is easier for the robot to move in a straight line rather than to turn and rotate. In this thesis, we investigate problems related to paths of fewest number of turns (or minimum link paths), under the simplifying assumption that robot is a point. In particular, we present efficient algorithms for the following problems in a polygon P: (1) Find a minimum link path between two given points of P. (2) Find minimum link paths between a fixed point and all the vertices of P. (3) Preprocess P with respect to a fixed point x such that the number of edges in a minimum link path between x and a query point can be determined in logarithmic time. (4) Compute the link diameter of P (which is the maximum number of edges in any minimum link path of P). We also consider problems where the complexity of a path is measured by its Euclidean length rather than by its number of turns. In this geodesic metric, the distance between two points of the polygon is the length of the shortest Euclidean path between them. We present fast algorithms for the following problems in P: (1) Compute the geodesic diameter of P (which is the maximum distance between any two points of P). (2) Compute geodesic furthest neighbors for all the vertices of P (furthest neighbor of a point x is another point in P whose distance from x is maximum). Several open problems are discussed at the conclusion of the work.

On the efficiency of Newton's method in approximating all zeros of a system of complex polynomials

Abstract: This paper studies the efficiency of an algorithm based on Newton's method is approximating all zeros of a system of polynomials f = (f1, f2, …, fn): ℂn → ℂn. The criteria for a successful approximation y of a zero w of f include the following: given ϵ > 0, y is within distance ϵ of w; Newton's method applied to f and initiated at y results in quadratic convergence to w; given ϵ > 0, |fi(y)| < ϵ for all i = 1, 2, …, n, where | | is the Euclidean norm on ℂ. It is shown that, probabilistically, each zero of f is successfully approximated within a determined number of steps.

The weighted region problem

Abstract: We present an algorithm for determining the shortest path between a source and a destination through a planar subdivision in which each region has an associated weight Distances are measured according to a weighted Euclidean metric: Each region of the subdivision has associated with it a weight, and the weighted distance between two points in a convex region is the product of the corresponding weight and the Euclidean distance between them Our algorithm runs in time O(n7L) and requires O(n3) space, where n is the number of edges of the subdivision, and L is the precision of the problem instance (including the number of bits in a user-specified tolerance ∈, which is the percentage the solution is allowed to differ from an optimal solution) The algorithm uses the fact that shortest paths obey Snell's Law of Refraction at region boundaries, a local optimality property of shortest paths that is well-known from the analogous optics model

Trellis-Coded MPSK with Reference Phase Errors

Abstract: With trellis-coded MPSK schemes, reference phase errors have a major impact on the system performance. This paper analyzes the interrelation among the coded-modulation format, the reference phase tracker, and the carrier phase noise. A tight upper bound to the error probability of a maximum likelihood decoder is introduced. This bound reveals eight new distance-type functions which, along with the wellknown minimum Euclidean distance, characterize the coded-modulation scheme. Relationships between the new distance-type functions and the various system parameters as well as the irreducible probability of error are pointed out. Numerical results for uncoded QPSK and three Ungerboeck encoded 8-PSK schemes showy that excessive smoothing of the reference phase in presence of carder phase noise can degrade the performance of coded 8-PSK modulation considerably.

Invariant planar shape recognition using dynamic alignment

Abstract: A technique for classifying closed planar shapes is described in which a shape is characterized by an ordered sequence that represents the Euclidean distance between the centroid and all contour pixels of the shape. Shapes belonging to the same class have similar sequences, hence a procedure for classifying shapes is based on the degree of similarity between these sequences. In order to determine the similarity between sequences, a non-linear alignment process is developed to find the best correspondence between the sequences. Optimum alignment is obtained by expanding segments of the sequences to minimize a dissimilarity function between the sequences. Normalization with respect to scaling and rotation is described and an example illustrating the use of dynamic alignment for the classification of noisy shapes is presented.

Partitioning Euclidean space

Abstract: R denotes the set of real numbers. For any finite n, Rn denotes n-dimensional space with the usual (Hilbert space) metric, d.

Probabilistic Multidimensional Scaling Models for Analyzing Consumer Choice

Abstract: We review the development of two new stochastic multidimensional scaling (MDS) methodologies that operate on paired comparisons choice data and render a spatial representation of subjects and stimuli. In the probabilistic vector MDS model, subjects are represented as vec­tors and stimuli as points in a T-dimensional space, where the scalar products or projections of the stimulus points onto the subject vectors provide information about the utility of the stimuli to the subjects.In the probabilistic unfolding MDS model, subjects are represented as ideal points and stimuli as points in a T-dimensional space, where the Euclidean distance between the stimulus points and the subject ideal points provides information as to the respective utility of the stimuli to the subjects. To illustrate the versatility of the two models, a market­ing application measuring consumer choice for fourteen actual brands of over-the-counter analgesics, utilizing optional reparameterizations, is described. Finally, other applications are identified.

Euclidean distance for combinations of some Hamming codes and binary CPFSK modulation (Corresp.)

Abstract: The integration of channel coding and modulation in a communication system to increase the Euclidean distance between modulated signals is analyzed. Systems using binary continuous-phase frequency-shift keying modulation and some block codes, such as Hamming codes and shortened Hamming codes, are considered. It is shown that the minimum Euclidean distance depends on the configuration of the parity-check matrix H of the code. For the examined codes the optimum configurations of H , which give the maximum values of the minimum Euclidean distance, are determined.

Stochastic Gaussian model for low-bit rate coding of LPC area parameters

Abstract: Accurate quantization of LPC parameters with a minimum number of bits is necessary for synthesizing high quality speech at low bit rates. Earlier work by Juang and Gray has shown that vector quantization can provide a significant reduction in the bit rate needed to quantize the LPC parameters. Previous work on vector quantization of LPC parameters employed trained codebooks. In this paper, we describe a stochastic model of LPC-derived log areas that eliminates training of the codebook by constructing codebook entries from random sequences. Each vector of LPC parameters is modelled as a sample function of zero mean Gaussian stochastic process with known covariances. We generate an ensemble of Gaussian codewords with a specified distribution where the number of codewords in the ensemble is determined by the number of bits used to quantize the LPC parameter vector. The optimum codeword is selected by exhaustive search to minimize the Euclidean distance between the original and quantized parameters. Our results show that vector quantization using random codebooks can provide a SNR of 20 dB in quantizing 10 log area parameters with 28 bits/frame. An important advantage of random codebook is that they provide robust performance across different speakers and speech recording conditions.

The optimization of robot motion in the presence of obstacles

Abstract: In this thesis we develop a general algorithm for optimizing robot motion in the presence of obstacles. It can incorporate useful measures of system performance, accurate descriptions of motion, fixed or time-dependent boundary conditions, general state and control constraints, and most importantly, obstacle avoidance. The main idea is to express obstacle avoidance in terms of the Euclidean distance between potentially colliding objects. The mathematical properties of the distance as a function of system configuration are studied, and it is seen that various types of derivatives are easily characterized. The results lead to the formulation of optimal-path planning as a problem in optimal control. Our solution approach is straightforward. It relies on a spline function representation of paths in configuration space to parameterize the problem and exactly satisfy both the equations of motion and the boundary conditions. The constraints on the state and control and those corresponding to obstacle avoidance are easily evaluated in terms of the spline parameters, and are imposed using penalty methods. Local solutions are obtained using a modified BFGS optimization algorithm. As part of the optimal-path planning procedure, we develop an efficient and reliable (in the presence of round-off errors) algorithm for computing the distance between objects in three-dimensional space. For convex polytopes with known vertices, the algorithm terminates after a finite number of steps with highly accurate results and exhibits only small linear growth in computational time as the total number of vertices in a polytope pair is increased. The main advantages of the presented algorithmic approach to optimal-path planning is its generality, its comprehensive treatment of obstacles and its production of a smooth approximation to the optimal configuration and input time histories. The computations are expensive, but realistic and practical results are achieved. Examples are given which include minimum-energy and minimum-time problems involving a Cartesian manipulator and a system of two cylindrical manipulators cooperatively interacting in a three-dimensional workspace.

An iterative euclidean algorithm

Abstract: On the basis of the results by JL. DORNSTETTER [3] showing the equivalence between BERLEKAMP's and EUCLID's algorithm, we present an iterative Euclidean extended algorithm. We show how all polynomials obtained by the classical extended Euclidean algorithm are actually automatically produced by that iterative process.

Holomorphic and real Euclidean supersymmetries in three and four dimensions

Abstract: The description of allN-extendedD=3 andD=4 Euclidean SUSYs with spinorial super-charges is given. The most general form of the sector of central charges is considered. The dimensional reduction ofD=4 Minkowski and Euclidean SUSY toD=3 Euclidean SUSY is discussed.

The Schücking problem

Abstract: The embedding problem for a three‐parametric family of homogeneous three‐spaces into a higher‐dimensional Euclidean space is considered. These three‐spaces occur as space sections in cosmological models. After general consideration a certain two‐parametric family is embedded into a five‐dimensional Euclidean space, deferring the solution of the general case to later papers.

Performance Evaluation of Combined Coding and Modulation Schemes for Nonlinear Channels

Abstract: The performance of combined coding and modulation schemes in channels with intersymbol interference and nonlinearities is studied using an analytical approach. The class of possible receivers is presented, and a generalization of the minimum Euclidean distance used for the asymptotic analysis of the performance of combined coding and modulation schemes on AWGN channels is introduced. Numerical results are presented for coded 16-PSK modulation.

An evaluation of the M-statistic in human odontomorphometric distance analyses

Abstract: The usefulness of the M-statistic in odontomorphometric distance analyses was evaluated against a battery of more traditional metrics, which included Mahalanobis' D2, Penrose's shape metric, the Manhattan distance and Delta. Odontometric data used for the analyses were derived from 202 Paraguayan Lengua Indians and 125 contemporary caucasoids. Efron's Bootstrap procedure was used to evaluate the statistical accuracy of the different metrics, when each was applied to the same populations. Additionally, metric stability in the face of reduced sample size, statistical bias resulting from over- and underestimation, and the effects of standardization, were investigated. Our results indicated that Penrose's shape metric rather that the recently introduced M-statistic was the most reliable metric evaluated. Penrose's shape remained the most reliable when sample size was artificially reduced and when raw data were used. Interestingly, Mahalanobis' generalized distance emerged as the least reliable statistics, especially when used on small sample sizes.