Showing papers on "Metric (mathematics) published in 2000"

The Earth Mover's Distance as a Metric for Image Retrieval

Abstract: We investigate the properties of a metric between two distributions, the Earth Mover's Distance (EMD), for content-based image retrieval. The EMD is based on the minimal cost that must be paid to transform one distribution into the other, in a precise sense, and was first proposed for certain vision problems by Peleg, Werman, and Rom. For image retrieval, we combine this idea with a representation scheme for distributions that is based on vector quantization. This combination leads to an image comparison framework that often accounts for perceptual similarity better than other previously proposed methods. The EMD is based on a solution to the transportation problem from linear optimization, for which efficient algorithms are available, and also allows naturally for partial matching. It is more robust than histogram matching techniques, in that it can operate on variable-length representations of the distributions that avoid quantization and other binning problems typical of histograms. When used to compare distributions with the same overall mass, the EMD is a true metric. In this paper we focus on applications to color and texture, and we compare the retrieval performance of the EMD with that of other distances.

Real-time tracking of non-rigid objects using mean shift

Abstract: A new method for real time tracking of non-rigid objects seen from a moving camera is proposed. The central computational module is based on the mean shift iterations and finds the most probable target position in the current frame. The dissimilarity between the target model (its color distribution) and the target candidates is expressed by a metric derived from the Bhattacharyya coefficient. The theoretical analysis of the approach shows that it relates to the Bayesian framework while providing a practical, fast and efficient solution. The capability of the tracker to handle in real time partial occlusions, significant clutter, and target scale variations, is demonstrated for several image sequences.

Fast Time Sequence Indexing for Arbitrary Lp Norms

Abstract: Fast indexing in time sequence databases for similarity searching has attracted a lot of research recently. Most of the proposals, however, typically centered around the Euclidean distance and its derivatives. We examine the problem of multimodal similarity search in which users can choose the best one from multiple similarity models for their needs. In this paper, we present a novel and fast indexing scheme for time sequences, when the distance function is any of arbitrary Lp norms (p = 1; 2; : : : ;1). One feature of the proposed method is that only one index structure is needed for all Lp norms including the popular Euclidean distance (L2 norm). Our scheme achieves significant speedups over the state of the art: extensive experiments on real and synthetic time sequences show that the proposed method is comparable to the best competitor forL2 andL1 norms, but significantly (up to 10 times) faster for L1 norm.

Semi-qualitative Reasoning about Distances: A Preliminary Report

Abstract: We introduce a family of languages intended for representing knowledge and reasoning about metric (and more general distance) spaces. While the simplest language can speak only about distances between individual objects and Boolean relations between sets, the more expressive ones are capable of capturing notions such as 'somewhere in (or somewhere out of) the sphere of a certain radius', 'everywhere in a certain ring', etc. The computational complexity of the satisfiability problem for formulas in our languages ranges from NP-completeness to undecidability and depends on the class of distance spaces in which they are interpreted. Besides the class of all metric spaces, we consider, for example, the spaces R × R and N × N with their natural metrics.

Geometry of the space of phylogenetic trees

Abstract: We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.

The Space of Kähler Metrics

Abstract: This paper, the second of a series, deals with the function space \mathcal{H} of all smooth Kahler metrics in any given n-dimensional, closed complex manifold V, these metrics being restricted to a given, fixed, real cohomology class, called a polarization of V. This function space is equipped with a pre-Hilbert metric structure introduced by T. Mabuchi [10], who also showed that, formally, this metric has nonpositive curvature. In the first paper of this series [4], the second author showed that the same space is a path length space. He also proved that \mathcal{H} is geodesically convex in the sense that, for any two points of \mathcal{H}, there is a unique geodesic path joining them, which is always length minimizing and of class C1,1. This partially verifies two conjectures of Donaldson [8] on the subject. In the present paper, we show first of all, that the space is, as expected, a path length space of nonpositive curvature in the sense of A. D. Aleksandrov. A second result is related to the theory of extremal Kahler metrics, namely that the gradient flow in \mathcal{H} of the "K energy" of V has the property that it strictly decreases the length of all paths in \mathcal{H}, except those induced by one parameter families of holomorphic automorphisms of M.

Clustering Categorical Data: An Approach Based on Dynamical Systems

Abstract: We describe a novel approach for clustering collections of sets, and its application to the analysis and mining of categorical data. By “categorical data,” we mean tables with fields that cannot be naturally ordered by a metric – e.g., the names of producers of automobiles, or the names of products offered by a manufacturer. Our approach is based on an iterative method for assigning and propagating weights on the categorical values in a table; this facilitates a type of similarity measure arising from the co-occurrence of values in the dataset. Our techniques can be studied analytically in terms of certain types of non-linear dynamical systems.

Metric regularity and subdifferential calculus

Abstract: The theory of metric regularity is an extension of two classical results: the Lyusternik tangent space theorem and the Graves surjection theorem. Developments in non-smooth analysis in the 1980s and 1990s paved the way for a number of far-reaching extensions of these results. It was also well understood that the phenomena behind the results are of metric origin, not connected with any linear structure. At the same time it became clear that some basic hypotheses of the subdifferential calculus are closely connected with the metric regularity of certain set-valued maps. The survey is devoted to the metric theory of metric regularity and its connection with subdifferential calculus in Banach spaces.

Efficient Minimization of New Quadric Metric for Simplifying Meshes with Appearance Attributes (Addendum to IEEE Visualization 1999 paper)

Abstract: In an earlier paper we introduced a new quadric metric for simplifying triangle meshes using the edge collapse operation. The quadric measures both the geometric accuracy of the simplified mesh surface and the fidelity of appearance fields defined on the surface (such as normals or colors). The minimization of this quadric metric involves solving a linear system of size (3+m)(3+m) ,w herem is the number of distinct appearance attributes. The system has only O(m) nonzero entries, so it can be solved in O(m 2 ) time using traditional sparse solvers such as the method of conjugate gradients. In this short addendum, we show that the special structure of the sparsity permits the system to be solved in O(m) time.

Applications of Convex Analysis to Multidimensional Scaling

Abstract: In this paper we discuss the convergence of an algorithm for metric and nonmetric multidimensional scaling that is very similar to the C-matrix algorithm of Guttman. The paper improves some earlier results in two respects. In the first place the analysis is extended to cover general Minkovski metrics, in the second place a more elementary proof of convergence based on results of Robert is presented.

Automated Derivation of Primitives for Movement Classification

Abstract: We describe a new method for representing human movement compactly, in terms of a linear super-imposition of simpler movements termed i>primitives. This method is a part of a larger research project aimed at modeling motor control and imitation using the notion of perceptuo-motor primitives, a basis set of coupled perceptual and motor routines. In our model, the perceptual system is biased by the set of motor behaviors the agent can execute. Thus, an agent can automatically classify observed movements into its executable repertoire. In this paper, we describe a method for automatically deriving a set of primitives directly from human movement data. We used movement data gathered from a psychophysical experiment on human imitation to derive the primitives. The data were first filtered, then segmented, and principal component analysis was applied to the segments. The eigenvectors corresponding to a few of the highest eigenvalues provide us with a basis set of primitives. These are used, through superposition and sequencing, to reconstruct the training movements as well as novel ones. The validation of the method was performed on a humanoid simulation with physical dynamics. The effectiveness of the motion reconstruction was measured through an error metric. We also explored and evaluated a technique of clustering in the space of primitives for generating controllers for executing frequently used movements.

On the metric structure of non-Kähler complex surfaces

Abstract: We give a characterization of a locally conformally Kahler (l.c.K.) metric with parallel Lee form on a compact complex surface. Using the Kodaira classification of surfaces, we classify the compact complex surfaces admitting such structures. This gives a classification of Sasakian structures on compact three-manifolds. A weak version of the above mentioned characterization leads to an explicit construction of l.c.K. metrics on all Hopf surfaces. We characterize the locally homogeneous l.c.K. metrics on geometric complex surfaces, and we prove that some Inoue surfaces do not admit any l.c.K. metric.

Large Complex Structure Limits of K3 Surfaces

Abstract: Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we make a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kahler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibrations associated with the large complex structure limit, namely an n-sphere, and that the metric on this Sn is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the limit discriminant locus of the fibrations. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkahler rotation trick, we reduce the problem to that of studying Kahler degenerations of elliptic K3 surfaces, with the Kahler class approaching the wall of the Kahler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricci-flat metric: if the elliptic fibres are of area ∊ > 0, then the error is O(e−C/∊) for some constant C > 0. This metric is obtained by gluing together a semi-flat metric on the smooth part of the fibration with suitable Ooguri-Vafa metrics near the singular fibres. For small ∊, this is a sufficiently good approximation that the above conjecture is then an easy consequence.

Slim-Trees: High Performance Metric Trees Minimizing Overlap Between Nodes

Abstract: In this paper we present the Slim-tree, a dynamic tree for organizing metric datasets in pages of fixed size. The Slim-tree uses the "fat-factor" which provides a simple way to quantify the degree of overlap between the nodes in a metric tree. It is well-known that the degree of overlap directly affects the query performance of index structures. There are many suggestions to reduce overlap in multidimensional index structures, but the Slim-tree is the first metric structure explicitly designed to reduce the degree of overlap. Moreover, we present new algorithms for inserting objects and splitting nodes. The new insertion algorithm leads to a tree with high storage utilization and improved query performance, whereas the new split algorithm runs considerably faster than previous ones, generally without sacrificing search performance. Results obtained from experiments with real-world data sets show that the new algorithms of the Slim-tree consistently lead to performance improvements. After performing the Slim-down algorithm, we observed improvements up to a factor of 35% for range queries.

Image-driven simplification

Abstract: We introduce the notion of image-driven simplification, a framework that uses images to decide which portions of a model to simplify. This is a departure from approaches that make polygonal simplification decisions based on geometry. As with many methods, we use the edge collapse operator to make incremental changes to a model. Unique to our approach, however, is the use at comparisons between images of the original model against those of a simplified model to determine the cost of an ease collapse. We use common graphics rendering hardware to accelerate the creation of the required images. As expected, this method produces models that are close to the original model according to image differences. Perhaps more surprising, however, is that the method yields models that have high geometric fidelity as well. Our approach also solves the quandary of how to weight the geometric distance versus appearance properties such as normals, color, and texture. All of these trade-offs are balanced by the image metric. Benefits of this approach include high fidelity silhouettes, extreme simplification of hidden portions of a model, attention to shading interpolation effects, and simplification that is sensitive to the content of a texture. In order to better preserve the appearance of textured models, we introduce a novel technique for assigning texture coordinates to the new vertices of the mesh. This method is based on a geometric heuristic that can be integrated with any edge collapse algorithm to produce high quality textured surfaces.

On a Connection between Kernel PCA and Metric Multidimensional Scaling

Abstract: In this note we show that the kernel PCA algorithm of Scholkopf, Smola, and Muller (Neural Computation, 10, 1299–1319.) can be interpreted as a form of metric multidimensional scaling (MDS) when the kernel function k(x, y) is isotropic, i.e. it depends only on Vx − yV. This leads to a metric MDS algorithm where the desired configuration of points is found via the solution of an eigenproblem rather than through the iterative optimization of the stress objective function. The question of kernel choice is also discussed.

Evaluating the scalability of distributed systems

Abstract: Many distributed systems must be scalable, meaning that they must be economically deployable in a wide range of sizes and configurations. This paper presents a scalability metric based on cost-effectiveness, where the effectiveness is a function of the system's throughput and its quality of service. It is part of a framework which also includes a sealing strategy for introducing changes as a function of a scale factor, and an automated virtual design optimization at each scale factor. This is an adaptation of concepts for scalability measures in parallel computing. Scalability is measured by the range of scale factors that give a satisfactory value of the metric, and good scalability is a joint property of the initial design and the scaling strategy. The results give insight into the scaling capacity of the designs, and into how to improve the design. A rapid simple bound on the metric is also described. The metric is demonstrated in this work by applying it to some well-known idealized systems, and to real prototypes of communications software.

GPS GDOP metric

Abstract: The authors present a review of the GDOP metric as used in GPS. Their goal is to review this metric and many of its known bounds as well as to report some new results. They use a formal linear algebraic framework to aid further study and insight.

Multilevel k-way Hypergraph Partitioning

Abstract: In this paper, we present a new multilevel k-way hypergraph partitioning algorithm that substantially outperforms the existing state-of-the-art K-PM/LR algorithm for multi-way partitioning, both for optimizing local as well as global objectives. Experiments on the ISPD98 benchmark suite show that the partitionings produced by our scheme are on the average 15% to 23% better than those produced by the K-PM/LR algorithm, both in terms of the hyperedge cut as well as the (K – 1) metric. Furthermore, our algorithm is significantly faster, requiring 4 to 5 times less time than that required by K-PM/LR.

Transversing itemset lattices with statistical metric pruning

Abstract: We study how to efficiently compute significant association rules according to common statistical measures such as a chi-squared value or correlation coefficient. For this purpose, one might consider to use of the Apriori algorithm, but the algorithm needs major conversion, because none of these statistical metrics are anti-monotone, and the use of higher support for reducing the search space cannot guarantee solutions in its the search space. We here present a method of estimating a tight upper bound on the statistical metric associated with any superset of an itemset, as well as the novel use of the resulting information of upper bounds to prune unproductive supersets while traversing itemset lattices. Experimental tests demonstrate the efficiency of this method.

A metric for ARMA processes

Abstract: Autoregressive-moving-average (ARMA) models seek to express a system function of a discretely sampled process as a rational function in the z-domain. Treating an ARMA model as a complex rational function, we discuss a metric defined on the set of complex rational functions. We give a natural measure of the "distance" between two ARMA processes. The paper concentrates on the mathematics behind the problem and shows that the various algebraic structures endow the choice of metric with some interesting and remarkable properties, which we discuss. We suggest that the metric can be used in at least two circumstances: (i) in which we have signals arising from various models that are unknown (so we construct the distance matrix and perform cluster analysis) and (ii) where there are several possible models M/sub i/, all of which are known, and we wish to find which of these is closest to an observed data sequence modeled as M.

Multiple model adaptive control. Part 1: Finite controller coverings

Abstract: We consider the problem of determining an appropriate model set on which to design a set of controllers for a multiple model switching adaptive control scheme. We show that, given mild assumptions on the uncertainty set of linear time-invariant plant models, it is possible to determine a "nite set of controllers such that for each plant in the uncertainty set, satisfactory performance will be obtained for some controller in the "nite set. We also demonstrate how such a controller set may be found. The analysis exploits the Vinnicombe metric and the fact that the set of approximately bandand time-limited transfer functions is approximately "nite-dimensional. Copyright ( 2000 John Wiley & Sons, Ltd.

Video summarization using singular value decomposition

Abstract: The authors propose a novel technique for video summarization based on singular value decomposition (SVD). For the input video sequence, we create a feature-frame matrix A, and perform the SVD on it. From this SVD, we are able, to not only derive the refined feature space to better cluster visually similar frames, but also define a metric to measure the amount of visual content contained in each frame cluster using its degree of visual changes. Then, in the refined feature space, we find the most static frame cluster, define it as the content unit, and use the context value computed from it as the threshold to cluster the rest of the frames. Based on this clustering result, either the optimal set of keyframes, or a summarized motion video with the user specified time length can be generated to support different user requirements for video browsing and content overview. Our approach ensures that the summarized video representation contains little redundancy, and gives equal attention to the same amount of contents.

The Anchors Hierarchy: Using the Triangle Inequality to Survive High Dimensional Data

Abstract: This paper is about metric data structures in high-dimensional or non-Euclidean space that permit cached sufficient statistics accelerations of learning algorithms. It has recently been shown that for less than about 10 dimensions, decorating kd-trees with additional "cached sufficient statistics" such as first and second moments and contingency tables can provide satisfying acceleration for a very wide range of statistical learning tasks such as kernel regression, locally weighted regression, k-means clustering, mixture modeling and Bayes Net learning. In this paper, we begin by defining the anchors hierarchy--a fast data structure and algorithm for localizing data based only on a triangle-inequality-obeying distance metric. We show how this, in its own right, gives a fast and effective clustering of data. But more importantly we show how it can produce a well-balanced structure similar to a Ball-Tree (Omohundro, 1991) or a kind of metric tree (Uhlmann, 1991; Ciaccia, Patella, & Zezula, 1997) in a way that is neither "topdown" nor "bottom-up" but instead "middleout". We then show how this structure, decorated with cached sufficient statistics, allows a wide variety of statistical learning algorithms to be accelerated even in thousands of dimensions.