Showing papers on "Metric (mathematics) published in 1997"

Geodesic Active Contours

Abstract: A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

M-tree: An Efficient Access Method for Similarity Search in Metric Spaces

Abstract: A new access method, called M-tree, is proposed to organize and search large data sets from a generic “metric space”, i.e. where object proximity is only defined by a distance function satisfying the positivity, symmetry, and triangle inequality postulates. We detail algorithms for insertion of objects and split management, which keep the M-tree always balanced - several heuristic split alternatives are considered and experimentally evaluated. Algorithms for similarity (range and k-nearest neighbors) queries are also described. Results from extensive experimentation with a prototype system are reported, considering as the performance criteria the number of page I/O’s and the number of distance computations. The results demonstrate that the Mtree indeed extends the domain of applicability beyond the traditional vector spaces, performs reasonably well in high-dimensional data spaces, and scales well in case of growing files.

RI-MP2: first derivatives and global consistency

Abstract: The evaluation of RI-MP2 first derivatives with respect to nuclear coordinates or with respect to an external electric field is described. The prefix RI indicates the use of an approximate resolution of identity in the Hilbert space of interacting charge distributions (Coulomb metric), i.e., the use of an auxiliary basis set to approximate charge distributions. The RI technique is applied to first derivatives of the MP2 correlation energy expression while the (restricted) Hartree-Fock reference is treated in the usual way. Computational savings by a factor of 10 over conventional approaches are demonstrated in an application to porphyrin. It is shown that the RI approximation to MP2 derivatives does not entail any significant loss in accuracy. Finally, the relative energetic stabilities of a representative sample of closed-shell molecules built from first and second row elements have been investigated by the RI-MP2 approach, and thus it is tested whether such properties that refer to potential energy hypersurfaces in a more global way can be described with similar consistency to the more locally defined derivatives.

Improved heterogeneous distance functions

Abstract: Instance-based learning techniques typically handle continuous and linear input values well, but often do not handle nominal input attributes appropriately. The Value Difference Metric (VDM) was designed to find reasonable distance values between nominal attribute values, but it largely ignores continuous attributes, requiring discretization to map continuous values into nominal values. This paper proposes three new heterogeneous distance functions, called the Heterogeneous Value Difference Metric (HVDM), the Interpolated Value Difference Metric (IVDM), and the Windowed Value Difference Metric (WVDM). These new distance functions are designed to handle applications with nominal attributes, continuous attributes, or both. In experiments on 48 applications the new distance metrics achieve higher classification accuracy on average than three previous distance functions on those datasets that have both nominal and continuous attributes.

Global Minimum for Active Contour Models: A Minimal Path Approach

Abstract: A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model‘s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the internal regularization term in the external potential term. Our method is based on finding a path of minimal length in a Riemannian metric. We then make use of a new efficient numerical method to find this shortest path. It is shown that the proposed energy, though based only on a potential integrated along the curve, imposes a regularization effect like snakes. We explore the relation between the maximum curvature along the resulting contour and the potential generated from the image. The method is capable to close contours, given only one point on the objects‘ boundary by using a topology-based saddle search routine. We show examples of our method applied to real aerial and medical images.

Introduction to Functional Analysis

Abstract: Preliminaries 1. Banach spaces and Metric Linear Spaces 2. Spectral of Theory Linear Operators 3. Frechet Spaces and their Dual Spaces

Integral Probability Metrics and Their Generating Classes of Functions

Abstract: We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

Distance-based indexing for high-dimensional metric spaces

Abstract: In many database applications, one of the common queries is to find approximate matches to a given query item from a collection of data items. For example, given an image database, one may want to retrieve all images that are similar to a given query image. Distance based index structures are proposed for applications where the data domain is high dimensional, or the distance function used to compute distances between data objects is non-Euclidean. In this paper, we introduce a distance based index structure called multi-vantage point (mvp) tree for similarity queries on high-dimensional metric spaces. The mvp-tree uses more than one vantage point to partition the space into spherical cuts at each level. It also utilizes the pre-computed (at construction time) distances between the data points and the vantage points. We have done experiments to compare mvp-trees with vp-trees which have a similar partitioning strategy, but use only one vantage point at each level, and do not make use of the pre-computed distances. Empirical studies show that mvp-tree outperforms the vp-tree 20% to 80% for varying query ranges and different distance distributions.

Reduced-rank adaptive filtering

Abstract: A novel rank reduction scheme is introduced for adaptive filtering problems. This rank reduction method uses a cross-spectral metric to select the optimal lower dimensional subspace for reduced-rank adaptive filtering as a function of the basis vectors of the full-rank space.

Distance measures for point sets and their computation

Abstract: We consider the problem of measuring the similarity or distance between two finite sets of points in a metric space, and computing the measure. This problem has applications in, e.g., computational geometry, philosophy of science, updating or changing theories, and machine learning. We review some of the distance functions proposed in the literature, among them the minimum distance link measure, the surjection measure, and the fair surjection measure, and supply polynomial time algorithms for the computation of these measures. Furthermore, we introduce the minimum link measure, a new distance function which is more appealing than the other distance functions mentioned. We also present a polynomial time algorithm for computing this new measure. We further address the issue of defining a metric on point sets. We present the metric infimum method that constructs a metric from any distance functions on point sets. In particular, the metric infimum of the minimum link measure is a quite intuitive. The computation of this measure is shown to be in NP for a broad class of instances; it is NP-hard for a natural problem class.

Color Snakes

Abstract: A framework for object segmentation in vector-valued images is presented in this paper. The first scheme proposed is based on geometric active contours moving toward the objects to be detected in the vector-valued image. Object boundaries are obtained as geodesics or minimal weighted-distance curves, where the metric is given by a definition of edges in vector-valued data. The curve flow corresponding to the proposed active contours holds formal existence, uniqueness, stability, and correctness results. The scheme automatically handles changes in the deforming curve topology. The technique is applicable, for example, to color and texture images as well as multiscale representations. We then present an extension of these vector active contours, proposing a possible image flow for vector-valued image segmentation. The algorithm is based on moving each one of the image level sets according to the proposed vector active contours. This extension also shows the relation between active contours and a number of partial-differential-equations-based image processing algorithms as anisotropic diffusion and shock filters.

Learning topological maps with weak local odometric information

Abstract: Topological maps provide a useful abstraction for robotic navigation and planning. Although stochastic maps can theoretically be learned using the Baum-Welch algorithm, without strong prior constraint on the structure of the model it is slow to converge, requires a great deal of data, and is often stuck in local minima. In this paper, we consider a special case of hidden Markov models for robot-navigation environments, in which states are associated with points in a metric configuration space. We assume that the robot has some odometric ability to measure relative transformations between its configurations. Such odometry is typically not precise enough to suffice for building a global map, but it does give valuable local information about relations between adjacent states. We present an extension of the Baum-Welch algorithm that takes advantage of this local odometric information, yielding faster convergence to better solutions with less data.

Parsing in a Dynamical System: An Attractor-based Account of the Interaction of Lexical and Structural Constraints in Sentence Processing

Abstract: A dynamical systems approach to parsing is proposed in which syntactic hypotheses are associated with attractors in a metric space. These attractors have many of the properties of traditional syntactic categories, while at the same time encoding context-dependent, lexically specie c distinctions. Hypotheses motivated by the dynamical system theory were tested in four reading time experiments examining the interaction of simple lexica l frequencies, frequencies that are contingent on an environment deened by syntactic categories, and frequencies contingent on verb argument structure. The experiments documented a variety of contingent frequency effects that cut across traditional linguistic grains, each of which was predicted by the dynamical systems model. These effects were simulated in an implementation of the theory, employing a recurrent network trained from a corpus to construct metric representations and an algorithm implementing a gravitational dynamical system to model reading time as time to gravitate to an attractor.

The Elmore delay as a bound for RC trees with generalized input signals

Abstract: The Elmore delay is an extremely popular timing-performance metric which is used at all levels of electronic circuit design automation, particularly for resistor-capacitor (RC) tree analysis. The widespread usage of this metric is mainly attributable to it being a delay measure that is a simple analytical function of the circuit parameters. The only drawback to this delay metric is the uncertainty of its accuracy and the restriction to it being an estimate only for the step response delay. In this paper, we prove that the Elmore delay measure is an absolute upper bound on the actual 50% delay of an RC tree response. Moreover, we prove that this bound holds for input signals other than steps and that the actual delay asymptotically approaches the Elmore delay as the input signal rise time increases. A lower bound on the delay is also developed using the Elmore delay and the second moment of the impulse response. The utility of this bound is for understanding the accuracy and the limitations of the Elmore metric as we use it as a performance metric for design automation.

Theory of partially adaptive radar

Abstract: This work extends the recently introduced cross-spectral metric for subspace selection and dimensionality reduction to partially adaptive space-time sensor array processing. A general methodology is developed for the analysis of reduced-dimension detection tests with known and unknown covariance. It is demonstrated that the cross-spectral metric results in a low-dimensional detector which provides nearly optimal performance when the noise covariance is known. It is also shown that this metric allows the dimensionality of the detector to be reduced below the dimension of the noise subspace eigenstructure without significant loss. This attribute provides robustness in the subspace selection process to achieve reduced-dimensional target detection. Finally, it is demonstrated that the cross-spectral subspace reduced-dimension detector can outperform the full-dimension detector when the noise covariance is unknown, closely approximating the performance of the matched filter.

A plug-in approach to support estimation

Abstract: We suggest a new approach, based on the use of density estimators, for the problem of estimating the (compact) support of a multivariate density. This subject (motivated in terms of pattern analysis by Grenander) has interesting connections with detection and clustering. A natural class of density-based estimators is defined. Universal consistency results and convergence rates are established for these estimators, with respect to the usual measure-based metric $d_{\mu}$ between sets. Further convergence rates (with respect to both $d_{\mu}$ and the Hausdorff metric $d_H$) are also obtained under some, fairly intuitive, shape restrictions.

The moduli space of special lagrangian submanifolds

Abstract: This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is a smooth manifold with a natural $L^2$ metric. It is shown that the metric is induced from a local Lagrangian immersion into the product of cohomology groups $H^1(L)\times H^{n-1}(L)$. Using this approach, an interpretation of the mirror symmetry discussed by Strominger, Yau and Zaslow is given in terms of the classical Legendre transform.

Configuration Controllability of Simple Mechanical Control Systems

Abstract: In this paper we present a definition of "configuration controllability" for mechanical systems whose Lagrangian is kinetic energy with respect to a Riemannian metric minus potential energy. A computable test for this new version of controllability is derived. This condition involves an object which we call the symmetric product. Of particular interest is a definition of "equilibrium controllability" for which we are able to derive computable sufficient conditions. Examples illustrate the theory.

Anisotropic mesh optimization and its application in adaptivity

Abstract: The construction of solution-adapted meshes is addressed within an optimization framework. An approximation of the second spatial derivative of the solution is used to get a suitable metric in the computational domain. A mesh quality is proposed and optimized under this metric, accounting for both the shape and the size of the elements. For this purpose, a topological and geometrical mesh improvement method of high generality is introduced. It is shown that the adaptive algorithm that results recovers optimal convergence rates in singular problems, and that it captures boundary and internal layers in convection-dominated problems. Several important implementation issues are discussed. © 1997 John Wiley & Sons, Ltd.

Subspace selection for partially adaptive sensor array processing

Abstract: This paper introduces a cross-spectral metric for subspace selection and rank reduction in partially adaptive minimum variance array processing. The counter-intuitive result that it is suboptimal to perform rank reduction via the selection of the subspace formed by the principal eigenvectors of the array covariance matrix is demonstrated. A cross-spectral metric is shown to be the optimal criterion for reduced-rank Wiener filtering.

Text Segmentation Using Exponential Models

Abstract: This paper introduces a new statistical approach to partitioning text automatically into coherent segments. Our approach enlists both short-range and long-range language models to help it sniff out likely sites of topic changes in text. To aid its search, the system consults a set of simple lexical hints it has learned to associate with the presence of boundaries through inspection of a large corpus of annotated data. We also propose a new probabilistically motivated error metric for use by the natural language processing and information retrieval communities, intended to supersede precision and recall for appraising segmentation algorithms. Qualitative assessment of our algorithm as well as evaluation using this new metric demonstrate the effectiveness of our approach in two very different domains, Wall Street Journal articles and the TDT Corpus, a collection of newswire articles and broadcast news transcripts.

Source localization using recursively applied and projected (RAP) MUSIC

Abstract: A new method for source localization is described that is based on a modification of the well-known MUSIC algorithm. In classical MUSIC, the array manifold vector is projected onto an estimate of the signal subspace, but errors in the estimate can make location of multiple sources difficult. Recursively applied and projected (RAP) MUSIC uses each successively located source to form an intermediate array gain matrix, and projects both the array manifold and the signal subspace estimate into its orthogonal complement. The MUSIC projection is then performed in this reduced subspace. Using the metric of principal angles, we describe a general form of the RAP-MUSIC algorithm for the case of diversely polarized sources. Through a uniform linear array simulation, we demonstrate the improved Monte Carlo performance of RAP-MUSIC relative to MUSIC and two other sequential subspace methods, S- and IES-MUSIC.

Method and apparatus for dynamically optimizing an executable computer program using input data

Abstract: The present invention provides a method and apparatus for using input data to optimize a computer program. Initially, the computer program is divided into one or more logical units of code. Next, a CPU simulator is used to simulate execution of each logical unit using the input data. The output from the simulation is used to generate a first optimization metric value and corresponding state information for each logical unit. In one embodiment, the first optimization metric value and corresponding state information are stored in a first optimization vector. Using well known optimization techniques, the instructions within each logical unit are optimized iteratively until additional optimizations would result in very small incremental performance improvements. A second simulation is performed using the same input data except that this time the optimized logical units are used. This second simulation is used to measure how much the optimizer has improved the code. The output from the second simulation is used to generate a second optimization metric value and corresponding state information. The degree of optimization is determined by determining the difference between the first optimization metric value and the second optimization metric value for the sum of the logical units. If the difference is less than a predetermined threshold value, additional optimization iterations would provide little code improvement and thus the optimization is complete. However, if the difference is greater than or equal to the predetermined threshold value, additional optimizations would likely improve performance. In the latter case, the present invention would repeat the optimization process described above.

Non-literal textual search using fuzzy finite-state linear non-deterministic automata

Abstract: Method and system for selectively retrieving information contained in a stored document set using a metric-based or "fuzzy" finite-state non-deterministic automation. The system receives a generalized regular search expression from a user. The system then performs prematching during which it estimates a dissimilarity metric for each target string in the stored document set with respect to the search expression. The strings are then sorted by dissimilarity metric, with the best matches, i.e., the strings having the lowest dissimilarity metrics, first. The search expression is broken down into one or more segments. A linear fizzy finite-state non-deterministic automation is constructed (501) by matching each segment of the search expression with a corresponding set of states and transitions. The automation is initialized and then processes target strings read (502) from the sorted list, thereby generating a dissimilarity value for each target string. A dissimilarity value for each string is determined based upon penalties associated with one-to-one fuzzy character substitution, exchanged adjacent characters, one-to-many, many-to-one, and many-to-many character substitution, and other differences between the search expression and a target string read from storage.

Yamabe constants and the perturbed Seiberg–Witten equations

Abstract: Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-K\"ahler 4-manifolds.

Matching and indexing sequences of different lengths

Abstract: In this paper, we consider the problem of efficient matching and retrieval of sequences of different lengths. Most of the previous research is concentrated on similarity matching and retrieval of sequences of the same length using Euclidean distance metric. For similarity matching of sequences, we use a modified version of the edit distance function, and consider two sequences matching if a majority of the elements in the sequences match. In the matching process a mapping among non-matching elements is created to check if there are unacceptable deviations among them. This means that two matching sequences should have lengths that are comparable. For efficient retrieval of matching sequences, we propose an indexing scheme which is totally based on lengths and relative distances between sequences. We use vp-trees as the underlying distance-based index structures in our method.

Abadie's Constraint Qualification, Metric Regularity, and Error Bounds for Differentiable Convex Inequalities

Abstract: In this paper we study differentiable convex inequalities and prove that metric regularity and Abadie's constraint qualification (CQ) are equivalent for such inequalities. For convex quadratic inequalities, we show that metric regularity, the existence of a global error bound, and Abadie's CQ are mutually equivalent. As a consequence, we derive two new characterizations of weak sharp minima of a convex quadratic programming problem.