Showing papers on "Euclidean distance published in 2004"

Robust image segmentation using FCM with spatial constraints based on new kernel-induced distance measure

Abstract: Fuzzy c-means clustering (FCM) with spatial constraints (FCM/spl I.bar/S) is an effective algorithm suitable for image segmentation. Its effectiveness contributes not only to the introduction of fuzziness for belongingness of each pixel but also to exploitation of spatial contextual information. Although the contextual information can raise its insensitivity to noise to some extent, FCM/spl I.bar/S still lacks enough robustness to noise and outliers and is not suitable for revealing non-Euclidean structure of the input data due to the use of Euclidean distance (L/sub 2/ norm). In this paper, to overcome the above problems, we first propose two variants, FCM/spl I.bar/S/sub 1/ and FCM/spl I.bar/S/sub 2/, of FCM/spl I.bar/S to aim at simplifying its computation and then extend them, including FCM/spl I.bar/S, to corresponding robust kernelized versions KFCM/spl I.bar/S, KFCM/spl I.bar/S/sub 1/ and KFCM/spl I.bar/S/sub 2/ by the kernel methods. Our main motives of using the kernel methods consist in: inducing a class of robust non-Euclidean distance measures for the original data space to derive new objective functions and thus clustering the non-Euclidean structures in data; enhancing robustness of the original clustering algorithms to noise and outliers, and still retaining computational simplicity. The experiments on the artificial and real-world datasets show that our proposed algorithms, especially with spatial constraints, are more effective.

Improved MDS-based localization

Abstract: It is often useful to know the geographic positions of nodes in a communications network, but adding GPS receivers or other sophisticated sensors to every node can be expensive. MDS-MAP is a recent localization method based on multidimensional scaling (MDS). It uses connectivity information - who is within communications range of whom - to derive the locations of the nodes in the network, and can take advantage of additional data, such as estimated distances between neighbors or known positions for certain anchor nodes, if they are available. However, MDS-MAP is an inherently centralized algorithm and is therefore of limited utility in many applications. In this paper, we present a new variant of the MDS-MAP method, which we call MDS-MAP(P) standing for MDS-MAP using patches of relative maps, that can be executed in a distributed fashion. Using extensive simulations, we show that the new algorithm not only preserves the good performance of the original method on relatively uniform layouts, but also performs much better than the original on irregularly-shaped networks. The main idea is to build a local map at each node of the immediate vicinity and then merge these maps together to form a global map. This approach works much better for topologies in which the shortest path distance between two nodes does not correspond well to their Euclidean distance. We also discuss an optional refinement step that improves solution quality even further at the expense of additional computation.

Making Time-Series Classification More Accurate Using Learned Constraints.

Abstract: It has long been known that Dynamic Time Warping (DTW) is superior to Euclidean distance for classification and clustering of time series. However, until lately, most research has utilized Euclidean distance because it is more efficiently calculated. A recently introduced technique that greatly mitigates DTWs demanding CPU time has sparked a flurry of research activity. However, the technique and its many extensions still only allow DTW to be applied to moderately large datasets. In addition, almost all of the research on DTW has focused exclusively on speeding up its calculation; there has been little work done on improving its accuracy. In this work, we target the accuracy aspect of DTW performance and introduce a new framework that learns arbitrary constraints on the warping path of the DTW calculation. Apart from improving the accuracy of classification, our technique as a side effect speeds up DTW by a wide margin as well. We show the utility of our approach on datasets from diverse domains and demonstrate significant gains in accuracy and efficiency. E u clid ean D istan c e D yn am ic T im e W arp in g D is tan ce Figure 1: Note that while the two time series have an overall similar shape, they are not aligned in the time axis. Euclidean distance, which assumes the i point in one sequence is aligned with the i point in the other, will produce a pessimistic dissimilarity measure. The non-linear Dynamic Time Warped alignment allows a more intuitive distance measure to be calculated.

Indexing spatio-temporal trajectories with Chebyshev polynomials

Abstract: In this paper, we attempt to approximate and index a d- dimensional (d ≥ 1) spatio-temporal trajectory with a low order continuous polynomial. There are many possible ways to choose the polynomial, including (continuous)Fourier transforms, splines, non-linear regressino, etc. Some of these possiblities have indeed been studied beofre. We hypothesize that one of the best possibilities is the polynomial that minimizes the maximum deviation from the true value, which is called the minimax polynomial. Minimax approximation is particularly meaningful for indexing because in a branch-and-bound search (i.e., for finding nearest neighbours), the smaller the maximum deviation, the more pruning opportunities there exist. However, in general, among all the polynomials of the same degree, the optimal minimax polynomial is very hard to compute. However, it has been shown thta the Chebyshev approximation is almost identical to the optimal minimax polynomial, and is easy to compute [16]. Thus, in this paper, we explore how to use the Chebyshev polynomials as a basis for approximating and indexing d-dimenstional trajectories.The key analytic result of this paper is the Lower Bounding Lemma. that is, we show that the Euclidean distance between two d-dimensional trajectories is lower bounded by the weighted Euclidean distance between the two vectors of Chebyshev coefficients. this lemma is not trivial to show, and it ensures that indexing with Chebyshev cofficients aedmits no false negatives. To complement that analystic result, we conducted comprehensive experimental evaluation with real and generated 1-dimensional to 4-dimensional data sets. We compared the proposed schem with the Adaptive Piecewise Constant Approximation (APCA) scheme. Our preliminary results indicate that in all situations we tested, Chebyshev indexing dominates APCA in pruning power, I/O and CPU costs.

Similarity between Euclidean and cosine angle distance for nearest neighbor queries

Abstract: Understanding the relationship among different distance measures is helpful in choosing a proper one for a particular application. In this paper, we compare two commonly used distance measures in vector models, namely, Euclidean distance (EUD) and cosine angle distance (CAD), for nearest neighbor (NN) queries in high dimensional data spaces. Using theoretical analysis and experimental results, we show that the retrieval results based on EUD are similar to those based on CAD when dimension is high. We have applied CAD for content based image retrieval (CBIR). Retrieval results show that CAD works no worse than EUD, which is a commonly used distance measure for CBIR, while providing other advantages, such as naturally normalized distance.

A PCA-based similarity measure for multivariate time series

Abstract: Multivariate time series (MTS) datasets are common in various multimedia, medical and financial applications. We propose a similarity measure for MTS datasets, Eros Extended Frobenius norm), which is based on Principal Component Analysis (PCA). Eros applies PCA to MTS datasets represented as matrices to generate principal components and associated eigenvalues. These principal components and eigenvalues are then used to compare the similarity between MTS matrices. Though Eros in itself does not satisfy the triangle inequality, without which existing multidimensional indexing structures may not be utilized, the lower and upper bounds to satisfy the triangle inequality are obtained. In order to show the validity of Eros for similarity search on MTS datasets, we performed several experiments on three datasets (2 real-world and 1 synthetic). The results show the superiority of our approaches as compared to the traditional similarity measures for MTS datasets, such as Euclidean Distance (ED), Dynamic Time Warping (DTW), Weighted Sum SVD (WSSVD) and PCA similarity factor (SPCA) in precision/recall.

Bypassing the embedding: algorithms for low dimensional metrics

Abstract: The doubling dimension of a metric is the smallest k such that any ball of radius 2r can be covered using 2k balls of radius r This concept for abstract metrics has been proposed as a natural analog to the dimension of a Euclidean space If we could embed metrics with low doubling dimension into low dimensional Euclidean spaces, they would inherit several algorithmic and structural properties of the Euclidean spaces Unfortunately however, such a restriction on dimension does not suffice to guarantee embeddibility in a normed spaceIn this paper we explore the option of bypassing the embedding In particular we show the following for low dimensional metrics: Quasi-polynomial time (1+e)-approximation algorithm for various optimization problems such as TSP, k-median and facility location (1+e)-approximate distance labeling scheme with optimal label length (1+e)-stretch polylogarithmic storage routing scheme

Indexing large human-motion databases

Abstract: Data-driven animation has become the industry standard for computer games and many animated movies and special effects In particular, motion capture data recorded from live actors, is the most promising approach offered thus far for animating realistic human characters However, the manipulation of such data for general use and re-use is not yet a solved problem Many of the existing techniques dealing with editing motion rely on indexing for annotation, segmentation, and re-ordering of the data Euclidean distance is inappropriate for solving these indexing problems because of the inherent variability found in human motion The limitations of Euclidean distance stems from the fact that it is very sensitive to distortions in the time axis A partial solution to this problem, Dynamic Time Warping (DTW), aligns the time axis before calculating the Euclidean distance However, DTW can only address the problem of local scaling As we demonstrate in this paper, global or uniform scaling is just as important in the indexing of human motion We propose a novel technique to speed up similarity search under uniform scaling, based on bounding envelopes Our technique is intuitive and simple to implement We describe algorithms that make use of this technique, we perform an experimental analysis with real datasets, and we evaluate it in the context of a motion capture processing system The results demonstrate the utility of our approach, and show that we can achieve orders of magnitude of speedup over the brute force approach, the only alternative solution currently available

Euclidean Embedding of Co-Occurrence Data

Abstract: Embedding algorithms search for low dimensional structure in complex data, but most algorithms only handle objects of a single type for which pairwise distances are specified. This paper describes a method for embedding objects of different types, such as images and text, into a single common Euclidean space based on their co-occurrence statistics. The joint distributions are modeled as exponentials of Euclidean distances in the low-dimensional embedding space, which links the problem to convex optimization over positive semidefinite matrices. The local structure of our embedding corresponds to the statistical correlations via random walks in the Euclidean space. We quantify the performance of our method on two text datasets, and show that it consistently and significantly outperforms standard methods of statistical correspondence modeling, such as multidimensional scaling and correspondence analysis.

BoostMap: A method for efficient approximate similarity rankings

Abstract: This paper introduces BoostMap, a method that can significantly reduce retrieval time in image and video database systems that employ computationally expensive distance measures, metric or non-metric. Database and query objects are embedded into a Euclidean space, in which similarities can be rapidly measured using a weighted Manhattan distance. Embedding construction is formulated as a machine learning task, where AdaBoost is used to combine many simple, ID embeddings into a multidimensional embedding that preserves a significant amount of the proximity structure in the original space. Performance is evaluated in a hand pose estimation system, and a dynamic gesture recognition system, where the proposed method is used to retrieve approximate nearest neighbors under expensive image and video similarity measures: In both systems, in quantitative experiments, BoostMap significantly increases efficiency, with minimal losses in accuracy. Moreover, the experiments indicate that BoostMap compares favorably with existing embedding methods that have been employed in computer vision and database applications, i.e., FastMap and Bourgain embeddings.

Content-based music similarity search and emotion detection

Abstract: The paper investigates the use of acoustic based features for music information retrieval. Two specific problems are studied: similarity search (searching for music sound files similar to a given music sound file) and emotion detection (detection of emotion in music sounds). The Daubechies wavelet coefficient histograms (Li, T. et al., SIGIR'03, p.282-9, 2003), which consist of moments of the coefficients calculated by applying the Db8 wavelet filter, are combined with the timbral features extracted using the MARSYAS system of G. Tzanctakis and P. Cook (see IEEE Trans. on Speech and Audio Process., vol.10, no.5, p.293-8, 2002) to generate compact music features. For the similarity search, the distance between two sound files is defined to be the Euclidean distance of their normalized representations. Based on the distance measure, the closest sound files to an input sound file are obtained. Experiments on jazz vocal and classical sound files achieve a very high level of accuracy. Emotion detection is cast as a multiclass classification problem, decomposed as a multiple binary classification problem, and is resolved with the use of support vector machines trained on the extracted features. Our experiments on emotion detection achieved reasonably accurate performance and provided some insights on future work.

Learning with distance substitution kernels

Abstract: During recent years much effort has been spent in incorporating problem specific a-priori knowledge into kernel methods for machine learning. A common example is a-priori knowledge given by a distance measure between objects. A simple but effective approach for kernel construction consists of substituting the Euclidean distance in ordinary kernel functions by the problem specific distance measure. We formalize this distance substitution procedure and investigate theoretical and empirical effects. In particular we state criteria for definiteness of the resulting kernels. We demonstrate the wide applicability by solving several classification tasks with SVMs. Regularization of the kernel matrices can additionally increase the recognition accuracy.

Euclidean distance between syntactically linked words.

Abstract: We study the Euclidean distance between syntactically linked words in sentences. The average distance is significantly small and is a very slowly growing function of sentence length. We consider two nonexcluding hypotheses: (a) the average distance is minimized and (b) the average distance is constrained. Support for (a) comes from the significantly small average distance real sentences achieve. The strength of the minimization hypothesis decreases with the length of the sentence. Support for (b) comes from the very slow growth of the average distance versus sentence length. Furthermore, (b) predicts, under ideal conditions, an exponential distribution of the distance between linked words, a trend that can be identified in real sentences.

Clustering objects on a spatial network

Abstract: Clustering is one of the most important analysis tasks in spatial databases. We study the problem of clustering objects, which lie on edges of a large weighted spatial network. The distance between two objects is defined by their shortest path distance over the network. Past algorithms are based on the Euclidean distance and cannot be applied for this setting. We propose variants of partitioning, density-based, and hierarchical methods. Their effectiveness and efficiency is evaluated for collections of objects which appear on real road networks. The results show that our methods can correctly identify clusters and they are scalable for large problems.

Approximate protein structural alignment in polynomial time.

Abstract: Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of optimization problems and develop an approximate polynomial-time algorithm to solve them. For a commonly used scoring function, the algorithm runs in O(n10/e6) time, for globular protein of length n, and it detects alignments that score within an additive error of e from all optima. Thus, we prove that this task is computationally feasible, although the method that we introduce is too slow to be a useful everyday tool. We argue that such approximate solutions are, in fact, of greater interest than exact ones because of the noisy nature of experimentally determined protein coordinates. The measurement of similarity between a pair of protein structures used by our algorithm involves the Euclidean distance between the structures (appropriately rigidly transformed). We show that an alternative approach, which relies on internal distance matrices, must incorporate sophisticated geometric ingredients if it is to guarantee optimality and run in polynomial time. We use these observations to visualize the scoring function for several real instances of the problem. Our investigations yield insights on the computational complexity of protein alignment under various scoring functions. These insights can be used in the design of scoring functions for which the optimum can be approximated efficiently and perhaps in the development of efficient algorithms for the multiple structural alignment problem.

Clustering time series from ARMA models with clipped data

Abstract: Clustering time series is a problem that has applications in a wide variety of fields, and has recently attracted a large amount of research. In this paper we focus on clustering data derived from Autoregressive Moving Average (ARMA) models using k-means and k-medoids algorithms with the Euclidean distance between estimated model parameters. We justify our choice of clustering technique and distance metric by reproducing results obtained in related research. Our research aim is to assess the affects of discretising data into binary sequences of above and below the median, a process known as clipping, on the clustering of time series. It is known that the fitted AR parameters of clipped data tend asymptotically to the parameters for unclipped data. We exploit this result to demonstrate that for long series the clustering accuracy when using clipped data from the class of ARMA models is not significantly different to that achieved with unclipped data. Next we show that if the data contains outliers then using clipped data produces significantly better clusterings. We then demonstrate that using clipped series requires much less memory and operations such as distance calculations can be much faster. Finally, we demonstrate these advantages on three real world data sets.

Existence of optimal transport maps for crystalline norms

Abstract: We show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in ℝn, assuming that the initial distribution of mass is absolutely continuous with respect to $\mathcal{L}$n. The proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the Euclidean distance as cost function. Moreover, improving a construction by Larman, we show the existence of a Nikodym set in ℝ3 having full measure in the unit cube, intersecting each element of a family of pairwise disjoint open lines only in one point. This example can be used to show that the regularity of the decomposition in transport rays plays an essential role in Sudakov-type arguments for proving the existence of optimal transport maps.

Deformable spanners and applications

Abstract: For a set S of points in R3, an s-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1+e)-spanner with O(n/ed) edges, where e is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion ofpoints, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers.

An efficient euclidean distance transform

Abstract: Within image analysis the distance transform has many applications. The distance transform measures the distance of each object point from the nearest boundary. For ease of computation, a commonly used approximate algorithm is the chamfer distance transform. This paper presents an efficient linear- time algorithm for calculating the true Euclidean distance-squared of each point from the nearest boundary. It works by performing a 1D distance transform on each row of the image, and then combines the results in each column. It is shown that the Euclidean distance squared transform requires fewer computations than the commonly used 5x5 chamfer transform.

The euclidean algorithm in algebraic number fields

Abstract: This article, which is an update of a version published 1995 in Expo. Math., intends to survey what is known about Euclidean number fields; we will do this from a number theoretical (and number geometrical) point of view. We have also tried to put some emphasis on the open problems in this field.

Robust Analysis of Linear Models

Abstract: This paper presents three lectures on a robust analysis of linear models. One of the main goals of these lectures is to show that this analysis, similar to the traditional least squares-based analysis, offers the user a unified methodology for inference procedures in general linear models. This discussion is facilitated throughout by the simple geometry underlying the analysis. The traditional analysis is based on the least squares fit which minimizes the Euclidean norm, while the robust analysis is based on a fit which minimizes another norm. Several examples involving real data sets are used in the lectures to help motivate the discussion.

Mining Navigation Patterns Using a Sequence Alignment Method

Abstract: In this article, a new method is illustrated for mining navigation patterns on a web site. Instead of clustering patterns by means of a Euclidean distance measure, in this approach users are partitioned into clusters using a non-Euclidean distance measure called the Sequence Alignment Method (SAM). This method partitions navigation patterns according to the order in which web pages are requested and handles the problem of clustering sequences of different lengths. The performance of the algorithm is compared with the results of a method based on Euclidean distance measures. SAM is validated by means of user-traffic data of two different web sites. Empirical results show that SAM identifies sequences with similar behavioral patterns not only with regard to content, but also considering the order of pages visited in a sequence.

Spatial Queries in the Presence of Obstacles

Abstract: Despite the existence of obstacles in many database applications, traditional spatial query processing utilizes the Euclidean distance metric assuming that points in space are directly reachable. In this paper, we study spatial queries in the presence of obstacles, where the obstructed distance between two points is defined as the length of the shortest path that connects them without crossing any obstacles. We propose efficient algorithms for the most important query types, namely, range search, nearest neighbors, e-distance joins and closest pairs, considering that both data objects and obstacles are indexed by R-trees. The effectiveness of the proposed solutions is verified through extensive experiments.

Approximating geometric bottleneck shortest paths

Abstract: In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S consisting of all edges whose lengths are less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on Euclidean minimum spanning trees, spanners, and the Delaunay triangulation. A result of independent interest is the following. For any two points p and q of S, there is a path between p and q in the Delaunay triangulation, whose length is less than or equal to 2π/(3 cos(π/6)) times the Euclidean distance |pq| between p and q, and all of whose edges have length at most |pq|.