Showing papers on "Metric (mathematics) published in 2006"

Region covariance: a fast descriptor for detection and classification

Abstract: We describe a new region descriptor and apply it to two problems, object detection and texture classification. The covariance of d-features, e.g., the three-dimensional color vector, the norm of first and second derivatives of intensity with respect to x and y, etc., characterizes a region of interest. We describe a fast method for computation of covariances based on integral images. The idea presented here is more general than the image sums or histograms, which were already published before, and with a series of integral images the covariances are obtained by a few arithmetic operations. Covariance matrices do not lie on Euclidean space, therefore we use a distance metric involving generalized eigenvalues which also follows from the Lie group structure of positive definite matrices. Feature matching is a simple nearest neighbor search under the distance metric and performed extremely rapidly using the integral images. The performance of the covariance features is superior to other methods, as it is shown, and large rotations and illumination changes are also absorbed by the covariance matrix.

Region Covariance : A Fast Descriptor for Detection and Classification

Abstract: We describe a new region descriptor and apply it to two problems, object detection and texture classification. The covariance of d-features, e.g., the three-dimensional color vector, the norm of first and second derivatives of intensity with respect to x and y, etc., characterizes a region of interest. We describe a fast method for computation of covariances based on integral images. The idea presented here is more general than the image sums or histograms, which were already published before, and with a series of integral images the covariances are obtained by a few arithmetic operations. Covariance matrices do not lie on Euclidean space, therefore we use a distance metric involving generalized eigenvalues which also follows from the Lie group structure of positive definite matrices. Feature matching is a simple nearest neighbor search under the distance metric and performed extremely rapidly using the integral images. The performance of the covariance features is superior to other methods, as it is shown, and large rotations and illumination changes are also absorbed by the covariance matrix.

Isolation by resistance.

Abstract: Despite growing interest in the effects of landscape heterogeneity on genetic structuring, few tools are available to incorporate data on landscape composition into population genetic studies. Analyses of isolation by distance have typically either assumed spatial homogeneity for convenience or applied theoretically unjustified distance metrics to compensate for heterogeneity. Here I propose the isolation-by-resistance (IBR) model as an alternative for predicting equilibrium genetic structuring in complex landscapes. The model predicts a positive relationship between genetic differentiation and the resistance distance, a distance metric that exploits precise relationships between random walk times and effective resistances in electronic networks. As a predictor of genetic differentiation, the resistance distance is both more theoretically justified and more robust to spatial heterogeneity than Euclidean or least cost path-based distance measures. Moreover, the metric can be applied with a wide range of data inputs, including coarse-scale range maps, simple maps of habitat and nonhabitat within a species' range, or complex spatial datasets with habitats and barriers of differing qualities. The IBR model thus provides a flexible and efficient tool to account for habitat heterogeneity in studies of isolation by distance, improve understanding of how landscape characteristics affect genetic structuring, and predict genetic and evolutionary consequences of landscape change.

Hierarchical mesh segmentation based on fitting primitives

Abstract: In this paper, we describe a hierarchical face clustering algorithm for triangle meshes based on fitting primitives belonging to an arbitrary set. The method proposed is completely automatic, and generates a binary tree of clusters, each of which is fitted by one of the primitives employed. Initially, each triangle represents a single cluster; at every iteration, all the pairs of adjacent clusters are considered, and the one that can be better approximated by one of the primitives forms a new single cluster. The approximation error is evaluated using the same metric for all the primitives, so that it makes sense to choose which is the most suitable primitive to approximate the set of triangles in a cluster.Based on this approach, we have implemented a prototype that uses planes, spheres and cylinders, and have experimented that for meshes made of 100 K faces, the whole binary tree of clusters can be built in about 8 s on a standard PC.The framework described here has natural application in reverse engineering processes, but it has also been tested for surface denoising, feature recovery and character skinning.

Ambiguous chance constrained problems and robust optimization

Abstract: In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We focus primarily on the special case where the uncertainty set ** of the distributions is of the form ** where ρp denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure **. The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. We also show that the robust sampled problem can be solved efficiently both in theory and in practice.

The internet AS-level topology: three data sources and one definitive metric

Abstract: We calculate an extensive set of characteristics for Internet AS topologies extracted from the three data sources most frequently used by the research community: traceroutes, BGP, and WHOIS. We discover that traceroute and BGP topologies are similar to one another but differ substantially from the WHOIS topology. Among the widely considered metrics, we find that the joint degree distribution appears to fundamentally characterize Internet AS topologies as well as narrowly define values for other important metrics. We discuss the interplay between the specifics of the three data collection mechanisms and the resulting topology views. In particular, we how how the data collection peculiarities explain differences in the resulting joint degree distributions of the respective topologies. Finally, we release to the community the input topology datasets, along with the scripts and output of our calculations. This supplement hould enable researchers to validate their models against real data and to make more informed election of topology data sources for their specific needs

Optimal Hand-Eye Calibration

Abstract: This paper presents a calibration method for eye-in-hand systems in order to estimate the hand-eye and the robot-world transformations. The estimation takes place in terms of a parametrization of a stochastic model. In order to perform optimally, a metric on the group of the rigid transformations SE(3) and the corresponding error model are proposed for nonlinear optimization. This novel metric works well with both common formulations AX=XB and AX=ZB, and makes use of them in accordance with the nature of the problem. The metric also adapts itself to the system precision characteristics. The method is compared in performance to earlier approaches.

General Kerr–NUT–AdS metrics in all dimensions

Abstract: The Kerr–AdS metric in dimension D has cohomogeneity [D/2]; the metric components depend on the radial coordinate r and [D/2] latitude variables μi that are subject to the constraint ∑iμ2i = 1. We find a coordinate reparametrization in which the μi variables are replaced by [D/2] − 1 unconstrained coordinates yα, and having the remarkable property that the Kerr–AdS metric becomes diagonal in the coordinate differentials dyα. The coordinates r and yα now appear in a very symmetrical way in the metric, leading to an immediate generalization in which we can introduce [D/2] − 1 NUT parameters. We find that (D − 5)/2 are non-trivial in odd dimensions whilst (D − 2)/2 are non-trivial in even dimensions. This gives the most general Kerr–NUT–AdS metric in D dimensions. We find that in all dimensions D ≥ 4, there exist discrete symmetries that involve inverting a rotation parameter through the AdS radius. These symmetries imply that Kerr–NUT–AdS metrics with over-rotating parameters are equivalent to under-rotating metrics. We also consider the BPS limit of the Kerr–NUT–AdS metrics, and thereby obtain, in odd dimensions and after Euclideanization, new families of Einstein–Sasaki metrics.

Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing

Abstract: This paper is dedicated to the statistical analysis of the space of multivariate normal distributions with an application to the processing of Diffusion Tensor Images (DTI). It relies on the differential geometrical properties of the underlying parameters space, endowed with a Riemannian metric, as well as on recent works that led to the generalization of the normal law on Riemannian manifolds. We review the geometrical properties of the space of multivariate normal distributions with zero mean vector and focus on an original characterization of the mean, covariance matrix and generalized normal law on that manifold. We extensively address the derivation of accurate and efficient numerical schemes to estimate these statistical parameters. A major application of the present work is related to the analysis and processing of DTI datasets and we show promising results on synthetic and real examples.

Large Margin Component Analysis

Abstract: Metric learning has been shown to significantly improve the accuracy of k-nearest neighbor (kNN) classification In problems involving thousands of features, distance learning algorithms cannot be used due to overfitting and high computational complexity In such cases, previous work has relied on a two-step solution: first apply dimensionality reduction methods to the data, and then learn a metric in the resulting low-dimensional subspace In this paper we show that better classification performance can be achieved by unifying the objectives of dimensionality reduction and metric learning We propose a method that solves for the low-dimensional projection of the inputs, which minimizes a metric objective aimed at separating points in different classes by a large margin This projection is defined by a significantly smaller number of parameters than metrics learned in input space, and thus our optimization reduces the risks of overfitting Theory and results are presented for both a linear as well as a kernelized version of the algorithm Overall, we achieve classification rates similar, and in several cases superior, to those of support vector machines

Another look at forecast-accuracy metrics for intermittent demand

Abstract: Some traditional measurements of forecast accuracy are unsuitable for intermittent demand data because they can give infinite or undefined values. Rob Hyndman summarizes these forecast accuracy metrics and explains their potential failings. He also introduces a new metric-the mean absolute scaled error (MASE)-which is more appropriate for intermittent-demand data. More generally, he believes that the MASE should become the standard metric for comparing forecast accuracy across multiple time series. Copyright International Institute of Forecasters, 2006

Merging Occupancy Grid Maps From Multiple Robots

Abstract: Mapping can potentially be speeded up in a significant way by using multiple robots exploring different parts of the environment. But the core question of multirobot mapping is how to integrate the data of the different robots into a single global map. A significant amount of research exists in the area of multirobot mapping that deals with techniques to estimate the relative robots poses at the start or during the mapping process. With map merging, the robots in contrast individually build local maps without any knowledge about their relative positions. The goal is then to identify regions of overlap at which the local maps can be joined together. A concrete approach to this idea is presented in form of a special similarity metric and a stochastic search algorithm. Given two maps m and m', the search algorithm transforms m' by rotations and translations to find a maximum overlap between m and m'. In doing so, the heuristic similarity metric guides the search algorithm toward optimal solutions. Results from experiments with up to six robots are presented based on simulated as well as real-world map data

Image Retrieval and Classification Using Local Distance Functions

Abstract: In this paper we introduce and experiment with a framework for learning local perceptual distance functions for visual recognition. We learn a distance function for each training image as a combination of elementary distances between patch-based visual features. We apply these combined local distance functions to the tasks of image retrieval and classification of novel images. On the Caltech 101 object recognition benchmark, we achieve 60.3% mean recognition across classes using 15 training images per class, which is better than the best published performance by Zhang, et al.

Continuous nearest neighbor monitoring in road networks

Abstract: Recent research has focused on continuous monitoring of nearest neighbors (NN) in highly dynamic scenarios, where the queries and the data objects move frequently and arbitrarily. All existing methods, however, assume the Euclidean distance metric. In this paper we study k-NN monitoring in road networks, where the distance between a query and a data object is determined by the length of the shortest path connecting them. We propose two methods that can handle arbitrary object and query moving patterns, as well as fluctuations of edge weights. The first one maintains the query results by processing only updates that may invalidate the current NN sets. The second method follows the shared execution paradigm to reduce the processing time. In particular, it groups together the queries that fall in the path between two consecutive intersections in the network, and produces their results by monitoring the NN sets of these intersections. We experimentally verify the applicability of the proposed techniques to continuous monitoring of large data and query sets.

Approximation Algorithms for Metric Facility Location Problems

Abstract: In this paper we present a 1.52-approximation algorithm for the metric uncapacitated facility location problem, and a 2-approximation algorithm for the metric capacitated facility location problem with soft capacities. Both these algorithms improve the best previously known approximation factor for the corresponding problem, and our soft-capacitated facility location algorithm achieves the integrality gap of the standard linear programming relaxation of the problem. Furthermore, we will show, using a result of Thorup, that our algorithms can be implemented in quasi-linear time.

A New look at the Plebanski-Demianski family of solutions

Abstract: The Plebanski–Demianski metric, and those that can be obtained from it by taking coordinate transformations in certain limits, include the complete family of space–times of type D with an aligned electromagnetic field and a possibly non-zero cosmological constant. Starting with a new form of the line element which is better suited both for physical interpretation and for identifying different subfamilies, we review this entire family of solutions. Our metric for the expanding case explicitly includes two parameters which represent the acceleration of the sources and the twist of the repeated principal null congruences, the twist being directly related to both the angular velocity of the sources and their NUT-like properties. The non-expanding type D solutions are also identified. All special cases are derived in a simple and transparent way.

Spatio-temporal data reduction with deterministic error bounds

Abstract: A common way of storing spatio-temporal information about mobile devices is in the form of a 3D (2D geography + time) trajectory. We argue that when cellular phones and Personal Digital Assistants become location-aware, the size of the spatio-temporal information generated may prohibit efficient processing. We propose to adopt a technique studied in computer graphics, namely line-simplification, as an approximation technique to solve this problem. Line simplification will reduce the size of the trajectories. Line simplification uses a distance function in producing the trajectory approximation. We postulate the desiderata for such a distance-function: it should be sound, namely the error of the answers to spatio-temporal queries must be bounded. We analyze several distance functions, and prove that some are sound in this sense for some types of queries, while others are not. A distance function that is sound for all common spatio-temporal query types is introduced and analyzed. Then we propose an aging mechanism which gradually shrinks the size of the trajectories as time progresses. We also propose to adopt existing linguistic constructs to manage the uncertainty introduced by the trajectory approximation. Finally, we analyze experimentally the effectiveness of line-simplification in reducing the size of a trajectories database.

Graph-based text classification: learn from your neighbors

Abstract: Automatic classification of data items, based on training samples, can be boosted by considering the neighborhood of data items in a graph structure (e.g., neighboring documents in a hyperlink environment or co-authors and their publications for bibliographic data entries). This paper presents a new method for graph-based classification, with particular emphasis on hyperlinked text documents but broader applicability. Our approach is based on iterative relaxation labeling and can be combined with either Bayesian or SVM classifiers on the feature spaces of the given data items. The graph neighborhood is taken into consideration to exploit locality patterns while at the same time avoiding overfitting. In contrast to prior work along these lines, our approach employs a number of novel techniques: dynamically inferring the link/class pattern in the graph in the run of the iterative relaxation labeling, judicious pruning of edges from the neighborhood graph based on node dissimilarities and node degrees, weighting the influence of edges based on a distance metric between the classification labels of interest and weighting edges by content similarity measures. Our techniques considerably improve the robustness and accuracy of the classification outcome, as shown in systematic experimental comparisons with previously published methods on three different real-world datasets.

Minimum Risk Annealing for Training Log-Linear Models

Abstract: When training the parameters for a natural language system, one would prefer to minimize 1-best loss (error) on an evaluation set. Since the error surface for many natural language problems is piecewise constant and riddled with local minima, many systems instead optimize log-likelihood, which is conveniently differentiable and convex. We propose training instead to minimize the expected loss, or risk. We define this expectation using a probability distribution over hypotheses that we gradually sharpen (anneal) to focus on the 1-best hypothesis. Besides the linear loss functions used in previous work, we also describe techniques for optimizing nonlinear functions such as precision or the BLEU metric. We present experiments training log-linear combinations of models for dependency parsing and for machine translation. In machine translation, annealed minimum risk training achieves significant improvements in BLEU over standard minimum error training. We also show improvements in labeled dependency parsing.

Comparing Partial Rankings

Abstract: We provide a comprehensive picture of how to compare partial rankings, that is, rankings that allow ties. We propose several metrics to compare partial rankings and prove that they are within constant multiples of each other.

Perceptually driven 3D distance metrics with application to watermarking

Abstract: This paper presents an objective structural distortion measure which reflects the visual similarity between 3D meshes and thus can be used for quality assessment. The proposed tool is not linked to any specific application and thus can be used to evaluate any kinds of 3D mesh processing algorithms (simplification, compression, watermarking etc.). This measure follows the concept of structural similarity recently introduced for 2D image quality assessment by Wang et al.1 and is based on curvature analysis (mean, standard deviation, covariance) on local windows of the meshes. Evaluation and comparison with geometric metrics are done through a subjective experiment based on human evaluation of a set of distorted objects. A quantitative perceptual metric is also derived from the proposed structural distortion measure, for the specific case of watermarking quality assessment, and is compared with recent state of the art algorithms. Both visual and quantitative results demonstrate the robustness of our approach and its strong correlation with subjective ratings.

Interference temperature: a metric for dynamic spectrum utilization

Abstract: Interference temperature has been proposed by the FCC as a metric for interference analysis. The purpose of the metric is to demystify and remove the subjective context that has been the basis of interference analysis within the regulatory agencies. The development of an interference metric is critical if more intensive, dynamic use of the spectrum is desired. There has been very little specified as to the origins, use, and challenges of an interference metric. This paper provides an introductory analysis of the interference temperature metric.

An efficient algorithm for local distance metric learning

Abstract: Learning application-specific distance metrics from labeled data is critical for both statistical classification and information retrieval. Most of the earlier work in this area has focused on finding metrics that simultaneously optimize compactness and separability in a global sense. Specifically, such distance metrics attempt to keep all of the data points in each class close together while ensuring that data points from different classes are separated. However, particularly when classes exhibit multimodal data distributions, these goals conflict and thus cannot be simultaneously satisfied. This paper proposes a Local Distance Metric (LDM) that aims to optimize local compactness and local separability. We present an efficient algorithm that employs eigenvector analysis, and bound optimization to learn the LDM from training data in a probabilistic framework. We demonstrate that LDM achieves significant improvements in both classification and retrieval accuracy compared to global distance learning and kernel-based KNN.

Metric-based iterative closest point scan matching for sensor displacement estimation

Abstract: This paper addresses the scan matching problem for mobile robot displacement estimation. The contribution is a new metric distance and all the tools necessary to be used within the iterative closest point framework. The metric distance is defined in the configuration space of the sensor, and takes into account both translation and rotation error of the sensor. The new scan matching technique ameliorates previous methods in terms of robustness, precision, convergence, and computational load. Furthermore, it has been extensively tested to validate and compare this technique with existing methods