Showing papers on "Euclidean distance published in 1999"

Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming

Abstract: Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (EDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (EDM). In this paper, we follow the successful approach in [20] and solve the EDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primal-dual interior-point algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.

Scaling up Dynamic Time Warping to Massive Dataset

Abstract: There has been much recent interest in adapting data mining algorithms to time series databases. Many of these algorithms need to compare time series. Typically some variation or extension of Euclidean distance is used. However, as we demonstrate in this paper, Euclidean distance can be an extremely brittle distance measure. Dynamic time warping (DTW) has been suggested as a technique to allow more robust distance calculations, however it is computationally expensive. In this paper we introduce a modification of DTW which operates on a higher level abstraction of the data, in particular, a piecewise linear representation. We demonstrate that our approach allows us to outperform DTW by one to three orders of magnitude. We experimentally evaluate our approach on medical, astronomical and sign language data.

Large-Scale Clustering of cDNA-Fingerprinting Data

Abstract: Clustering is one of the main mathematical challenges in large-scale gene expression analysis. We describe a clustering procedure based on a sequential k-means algorithm with additional refinements that is able to handle high-throughput data in the order of hundreds of thousands of data items measured on hundreds of variables. The practical motivation for our algorithm is oligonucleotide fingerprinting—a method for simultaneous determination of expression level for every active gene of a specific tissue—although the algorithm can be applied as well to other large-scale projects like EST clustering and qualitative clustering of DNA-chip data. As a pairwise similarity measure between two p-dimensional data points, x and y, we introduce mutual information that can be interpreted as the amount of information about x in y, and vice versa. We show that for our purposes this measure is superior to commonly used metric distances, for example, Euclidean distance. We also introduce a modified version of mutual information as a novel method for validating clustering results when the true clustering is known. The performance of our algorithm with respect to experimental noise is shown by extensive simulation studies. The algorithm is tested on a subset of 2029 cDNA clones coming from 15 different genes from a cDNA library derived from human dendritic cells. Furthermore, the clustering of these 2029 cDNA clones is demonstrated when the entire set of 76,032 cDNA clones is processed.

Small distortion and volume preserving embeddings for planar and Euclidean metrics

Abstract: A finite metric space, (S,d) , contains a finite set of points and a distance function on pairs of points. A contraction is an embedding, h, of a finite metric space (S, d) into Rd where for any u, v E S, the Euclidean (&) distance between h(u) and h(v) is no more than d(u, v). The distortion of the embedding is the maximum over pairs of the ratio of d(u, w) and the Euclidean distance between h(u) and h(v). Bourgain showed that any graphical metric could be embedded with distortion O(logn). Linial, London and Rabinovich and Aumman and Rabani used such embeddings to prove an O(log k) approximate max-flow min-cut theorem for k commodity flow problems. A generalization of embeddings that preserve distances between pairs-of points are embeddings that preserve volumes of larger sets. In particular, A (k, c)-volume respecting embedding of n-points in any metric space is a contraction where every subset of k points has within an ck-’ factor of its maximal possible k l-dimensional volume. Feige invented these embeddings in devising a polylogarithmic approximation algorithm for the bandwidth problem using these embeddings. Feige’s methods have subsequently been used by Vempala for approximating versions of the VLSI layout problem. Feise showed that a (k, O(10,g~‘~ n,/m)) volume r&ecting embedding‘ eksted.” Be -recently found improved (k, 0( mdk log k + log n)) volume respecting embeddings. For metrics arising from planar graphs (planar metrics), we give (k,O(m)) volume respecting contractions. As a corollary, we give embeddings for Permission to makkr digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. XC’99 Miami Beach Florida Copyright ACM 1999 I-581 13-068-6/99/06...$5.00 planar metrics with distortion O(e). This gives rise to an O(e)-approximate max-flow min-cut theorem for multicommodity flow problems in planar graphs. We also give an improved bound for volume respecting embeddings for Euclidean metrics. In particular, we give an (k,O(dog klog D)) volume respecting embedding where D is the ratio of the largest distance to the smallest distance in the metric. Our results give improvements for Feige’s and Vempala’s approximation algorithms for planar and Euclidean metrics. For volume respecting embeddings, our embeddings do not degrade very fast when preserving the volumes of large subsets. This may be useful in the future for approximation algorithms or if volume .respecting embeddings prove to be of independent interest.

Distance transformations: fast algorithms and applications to medical image processing

Abstract: Medical image processing is a demanding domain, both in terms of CPU and memory requirements. The volume of data to be processed is often large (a typical MRI dataset requires 10 MBytes) and many processing tools are only useful to the physician if they are available as real-time applications, i.e. if they run in a few seconds at most. Of course, a large part of these demands are - and will be - handled by the development of more powerful hardware. On the other hand, when faced with non-linear computational complexity, the development of improved algorithms is obviously the best solution. Distance transformations, a powerful image analysis tool used in a number of problems such as image registration, requires such improvements. A distance map is an image where the value of each pixel is the distance from this pixel to the nearest pixel belonging to a given set or object. A distance transformation (DT) is an algorithm that computes a distance map from a binary image representing this set of pixels. This definition is global in the sense that it requires finding the minimum on a set of distances computed between all image pixels and all object pixels. Therefore, a direct application of the definition usually leads to an unacceptable computational complexity. Numerous algorithms have been proposed to localize this definition of distance to the nearest pixel and allow a faster DT computation, but up to now, none of them combines both exactness and linear complexity. Numerous applications of distance transformations to image analysis and pattern recognition have been reported and those related to medical image processing are explored in what follows. Chapter 1 introduces a few basic concepts, a typical application of distance transformations in pattern recognition and the key challenges in producing a DT algorithm. Chapter 2 contains an exhaustive critical review of published algorithms. The strong and weak points of the most popular ones are discussed and the core principles for our original algorithms are derived. Chapters 3, 5, 6, 8 and 10 present original distance transformation algorithms. Each of those chapters is organized in a somewhat similar fashion. First we describe the algorithm. Then we evaluate its computational complexity and compare it to the state of the art. Chapter 4, 7, 9 and 11 each present an application to a particular problem in medical image processing, using the algorithm developed in the previous chapter. Ideally, the description of any medical image processing problem should include a medical justification of the need for an automated processing, a complete review of the state of the art in the field, a detailed description of the proposed processing method, and an evaluation of the accuracy of the results and their medical significance. Because of both time and space constraints in this thesis, such an exhaustive work will only be presented for the application in chapter 4, while the other applications will be described more briefly. Chapter 3 describes a new exact Euclidean distance transformation using ordered propagation. It is based on a variation of Ragnelmam's approximate Euclidean DT. We analyze the error patterns for approximate Euclidean DT using finite masks, and we derive a rule defining, for any pixel location, the size of the neighborhood that guarantees the exactness of the DT. This algorithm is particularly well-suited to implement mathematical morphology operations, which are examined in details. In Chapter 4, we apply the algorithm of chapter 3 to the segmentation of neuronal fibers from microscopic images of the sciatic nerve. In particular, it is used to determine the thickness of the myelin sheath surrounding the center of the fiber. This study was carried out in collaboration with the Neural Rehabilitation Engineering Laboratory, UCL. Chapter 5 proposes another exact Euclidean distance transformation, based on the explicit computation of the Voronoi division of the image. Possible error locations are detected at the corners of the Voronoi polygons and corrected if needed. This algorithm is shown to be the fastest exact EDT to date. It approaches the theoretical optimal complexity, a CPU time proportional to the number of pixels on which the distance is computed. Chapter 6 investigates how the algorithms of chapters 3 and 5 can be extended to 3 dimensional images. It shows the limitations of both approaches and proposes an hybrid algorithm mixing the method of chapter 5 and Saito's. In Chapter 7, the 3D Euclidean DT is applied to the registration of MR images of the brain where the matching criterion is the distance between the surfaces of similar objects (skin, cortex, ventricular system, ...) in both images. Examples are shown, from projects with the Neuro-physiology Laboratory, UCL, and with the Positron Tomography Laboratory, UCL. Chapter 8 discusses an extension of the distance transformation concept: geodesic distances on non-convex domains. Because geodesic distances are based on the notion of paths, a trade-off has to be introduced between the accuracy with which straight lines are represented and the way curves of the domain are followed. It is shown that, whatever the trade-off chosen, there is an efficient implementation of the geodesic DT by propagation. By back-tracking the geodesic distance propagation, one can find the shortest path between a target and a starting point. In chapter 9, this is used to plan the optimal path for the camera movements in virtual endoscopy, a work done in collaboration with the Surgical Planning Laboratory, Harvard Medical School, Boston. Chapter 10 extends the Euclidean distance transformation from finding the nearest object pixel to finding the k nearest object pixels. It is shown that this can be done with a complexity increasing linearly with k. In Chapter 11, the k-DT is used as a fast implementation of the k Nearest Neighbors (k-NN) classification between different tissue types in multi-modal MR imaging. This is illustrated through the classification of multiple sclerosis lesions from T1-T2 images, provided by the Radiology unit, St-Luc Hospital, UCL, via the Positron Tomography Laboratory, UCL. Finally, a general conclusion is drawn. It reviews the main contributions of the thesis, its applications and explores some new domains in which their applications could also be useful. Ultimately, the publications related to this thesis are briefly reviewed.

Clustering large datasets in arbitrary metric spaces

Abstract: Clustering partitions a collection of objects into groups called clusters, such that similar objects fall into the same group. Similarity between objects is defined by a distance function satisfying the triangle inequality; this distance function along with the collection of objects describes a distance space. In a distance space, the only operation possible on data objects is the computation of distance between them. All scalable algorithms in the literature assume a special type of distance space, namely a k-dimensional vector space, which allows vector operations on objects. We present two scalable algorithms designed for clustering very large datasets in distance spaces. Our first algorithm BUBBLE is, to our knowledge, the first scalable clustering algorithm for data in a distance space. Our second algorithm BUBBLE-FM improves upon BUBBLE by reducing the number of calls to the distance function, which may be computationally very expensive. Both algorithms make only a single scan over the database while producing high clustering quality. In a detailed experimental evaluation, we study both algorithms in terms of scalability and quality of clustering. We also show results of applying the algorithms to a real life dataset.

Segmentation of measured point data using a parametric quadric surface approximation

Abstract: In reverse engineering, a shape containing multi-patched surfaces is digitized, the boundaries of these surfaces should be detected. The objective of this paper is to introduce a new and computationally efficient segmentation technique for extracting edges, and partitioning the 3D measured point data based on the location of the boundaries. The procedure begins with the identification of edge points. An automatic edge-based approach is developed on the basis of local geometry. A parametric quadric surface approximation method is used to estimate the local surface curvature properties. The least-square approximation scheme minimizes the sum of the squares of the actual Euclidean distance between the neighborhood data points and the parametric quadric surface. The surface curvatures and the principal directions are computed from the locally approximated surfaces. Edge points are identified as the curvature extremes, and zero crossings, which are found from the estimated surface curvatures. After edge points are identified, edge-neighborhood chain-coding algorithm is used for forming boundary curves. The original point set is then broken down into subsets, which meet along the boundaries, by scan line algorithm. All point data are applied to each boundary loops to partition the points to different regions. Experimental results are presented to verify the developed methods.

Relevance feedback retrieval of time series data

Abstract: There has been much recent interest in retrieval of time series data. Earlier work has used a fixed similarity metric (e.g., Euclidean distance) to determine the similarity between a userspecified query and items in the database. Here, we describe a novel approach to retrieval of time series data by using relevance feedback from the user to adjust the similarity metric. This is important because the Euclidean distance metric does not capture many notions of similarity between time series. In particular, Euclidean distance is sensitive to various “distortions” such as offset translation, amplitude scaling, etc. Depending on the domain and the user, one may wish a query to be sensitive or insensitive to these distortions to varying degrees. This paper addresses this problem by introducing a profile that encodes the user's subjective notion of similarity in a domain. These profiles can be learned continuously from interaction with the user. We further show how the user profile may be embedded in a system that uses relevance feedback to modify the query in a manner analogous to the familiar text retrieval algorithms.

A Nearly Linear-Time Approximation Scheme for the Euclidean kappa-median Problem

Abstract: In the k-median problem we are given a set S of n points in a metric space and a positive integer k. The objective is to locate k medians among the points so that the sum of the distances from each point in S to its closest median is minimized. The k-median problem is a well-studied, NP-hard, basic clustering problem which is closely related to facility location. We examine the version of the problem in Euclidean space. Obtaining approximations of good quality had long been an elusive goal and only recently Arora, Raghavan and Rao gave a randomized polynomial-time approximation scheme for the Euclidean plane by extending techniques introduced originally by Arora for Euclidean TSP. For any fixed ? > 0; their algorithm outputs a (1 + ?)-approximation in O(nknO(1/?) log n) time.In this paper we provide a randomized approximation scheme for points in d- dimensional Euclidean space, with running time O(21/?d n log n log k); which is nearly linear for any fixed ? and d. Our algorithm provides the first polynomialtime approximation scheme for k-median instances in d-dimensional Euclidean space for any fixed d > 2: To obtain the drastic running time improvement we develop a structure theorem to describe hierarchical decomposition of solutions. The theorem is based on a novel adaptive decomposition scheme, which guesses at every level of the hierarchy the structure of the optimal solution and modifies accordingly the parameters of the decomposition. We believe that our methodology is of independent interest and can find applications to further geometric problems.

Spectra of Euclidean Random Matrices

Abstract: We study the spectrum of a random matrix, whose elements depend on the Euclidean distance between points randomly distributed in space. This problem is widely studied in the context of the Instantaneous Normal Modes of fluids and is particularly relevant at the glass transition. We introduce a systematic study of this problem through its representation by a field theory. In this way we can easily construct a high density expansion, which can be resummed producing an approximation to the spectrum similar to the Coherent Potential Approximation for disordered systems.

Edge detection on color images using RGB vector angles

Abstract: This paper introduces a new edge detection approach for color images. The method is based on the calculation of the vector angle between two adjacent pixels. Unlike Euclidean distance in RGB space, the vector angle distinguishes differences in chromaticity, independent of luminance or intensity. It is particularly well suited to applications where differences in illumination are irrelevant. Both metrics were implemented as modified Roberts edge operators to determine their effectiveness on an artificial image. The Euclidean method found edges across both luminance and chromatic boundaries whereas the vector angle method detected only chromatic differences.

Fast retrieval of similar subsequences in long sequence databases

Abstract: Although the Euclidean distance has been the most popular similarity measure in sequence databases, recent techniques prefer to use high-cost distance functions such as the time warping distance and the editing distance for wider applicability However, if these distance functions are applied to the retrieval of similar subsequences, the number of subsequences to be inspected during the search is quadratic to the average length L~ of data sequences We propose a novel subsequence matching scheme, called the aligned subsequence matching, where the number of subsequences to be compared with a query sequence is reduced to linear to L~ We also present an indexing technique to speed-up the aligned subsequence matching using the similarity measure of the modified time warping distance Experiments on synthetic data sequences demonstrate the effectiveness of our proposed approach; ours consistently outperformed sequential scanning and achieved an up to 65 times speed-up

Isotropic surface area measures

Abstract: The purpose of this note is to bring into attention an apparently forgotten result of C. M. Petty: a convex body has minimal surface area among its affine transformations of the same volume if, and only if, its area measure is isotropic. We obtain sharp affine inequalities which demonstrate the fact that this "surface isotropic" position is a natural framework for the study of hyperplane projections of convex bodies. §1. Introduction. We shall work in R" equipped with a fixed Euclidean structure and write | • | for the corresponding Euclidean norm. We denote the Euclidean unit ball and the unit sphere by Dn and S"~ ) respectively, and we write a for the rotationally invariant probability measure on S"~ l . The volume of appropriate dimension will be also denoted by | • |. We shall write con for the volume of the Euclidean unit ball in R". Finally, L(R", R") is the space of all linear transformations of R". Let K be a convex body in R". The area measure aK is denned on S"~ ) and corresponds to the usual surface measure on K via the Gauss map. If A is a Borel subset of S"~\ then

Breadth-first maximum likelihood sequence detection: basics

Abstract: The problem of performing breadth-first maximum likelihood sequence detection (MLSD) under given structural and complexity constraints is solved and results in a family of optimal detectors. Given a trellis with S states, these are partitioned into C classes where B paths into each class are selected recursively in each symbol interval. The derived result is to retain only those paths which are closest to the received signal in the Euclidean (Hamming) distance sense. Each member in the SA(B, C) family of sequence detectors (SA denotes search algorithm) performs complexity constrained MLSD for the additive white Gaussian noise (AWGN) (BSC) channel. The unconstrained solution is the Viterbi algorithm (VA). Analysis tools are developed for each member of the SA(B, C) class and the asymptotic (SNR) probability of losing the correct path is associated with a new Euclidean distance measure for the AWGN case, the vector Euclidean distance (VED). The traditional Euclidean distance is a scalar special case of this, termed the scalar Euclidean distance (SED). The generality of this VED is pointed out. Some general complexity reductions exemplify those associated with the VA approach.

Comprehensive analysis of edge detection in color image processing

Abstract: Various approaches to edge detection for color images, in- cluding techniques extended from monochrome edge detection as well as vector space approaches, are examined. In particular, edge detection techniques based on vector order statistic operators and difference vec- tor operators are studied in detail. Numerous edge detectors are ob- tained as special cases of these two classes of operators. The effect of distance measures on the performance of different color edge detectors is studied by employing distance measures other than the Euclidean norm. Variations are introduced to both the vector order statistic opera- tors and the difference vector operators to improve noise performance. They both demonstrate the ability to attenuate noise with added algo- rithm complexity. Among them, the difference vector operator with adap- tive filtering shows the most promising results. Other vector directional filtering techniques are also introduced and utilized for color edge detec- tion. Both quantitative and subjective tests are performed in evaluating the performance of the edge detectors, and a detailed comparison is presented. © 1999 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(99)00904-6)

Superintegrability in three-dimensional Euclidean space

Abstract: Potentials for which the corresponding Schrodinger equation is maximally superintegrable in three-dimensional Euclidean space are studied. The quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates.

Multiobjective criteria for neural network structure selection and identification of nonlinear systems using genetic algorithms

Abstract: An approach to model selection and identification of nonlinear systems via neural networks and genetic algorithms is presented based on multiobjective performance criteria. It considers three performance indices or cost functions as the objectives, which are the Euclidean distance (L/sub 2/-norm) and maximum difference (L/spl infin/-norm) measurements between the real nonlinear system and the nonlinear model, and the complexity measurement of the nonlinear model, instead of a single performance index. An algorithm based on the method of inequalities, least squares and genetic algorithms is developed for optimising over the multiobjective criteria. Genetic algorithms are also used for model selection in which the structure of the neural networks is determined. The Volterra polynomial basis function network and the Gaussian radial basis function network are applied to the identification of a liquid-level nonlinear system.

Fast and exact signed Euclidean distance transformation with linear complexity

Abstract: We propose a new signed or unsigned Euclidean distance transformation algorithm, based on the local corrections of the well-known 4SED algorithm of Danielsson (1980). Those corrections are only applied to a small neighborhood of a small subset of pixels from the image, which keeps the cost of the operation low. In contrast with all fast algorithms previously published, our algorithm produces perfect Euclidean distance maps in a time linearly proportional to the number of pixels in the image. The computational cost is close to the cost of the 4SSED approximation.

Fuzzy Clustering Based on Modified Distance Measures

Abstract: The well-known fuzzy c-means algorithm is an objective function based fuzzy clustering technique that extends the classical k-means method to fuzzy partitions. By replacing the Euclidean distance in the objective function other cluster shapes than the simple (hyper-)spheres of the fuzzy c-means algorithm can be detected, for instance ellipsoids, lines or shells of circles and ellipses. We propose a modified distance function that is based on the dot product and allows to detect a new kind of cluster shape and also lines and (hyper-)planes.

Color Image Edge Detection and Segmentation: A Comparison of the Vector Angle and the Euclidean Distance Color Similarity Measures

Abstract: ii I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Acknowledgements I would like to thank Dr. Ed Jernigan for his continued support and help in the long years that lead to the preparation of this thesis. I am also grateful to him for allowing me to choose my own topic of study and for letting me carry out the research at my own pace. I would also like to thank Dr. Bob Dony for the rich discussions on color image processing and in particular for the insight of using the vector angle as a color similarity measure. I would like to thank him, as well as Dr. Catherine Burns for taking the time to review this thesis and for providing very helpful comments. To NCR belongs the credit of allowing me to attend the University of Waterloo on a part-time basis while I worked there full time and for the tuition reimbursement program which provided much needed financial support. I would also like to acknowledge the warm support of my dear companion, my family and my friends in encouraging me to finish this work. Abstract This work is based on Shafer's Dichromatic Reflection Model as applied to color image formation. The color spaces RGB, XYZ, CIELAB, CIELUV, rgb, l 1 l 2 l 3 , and the new h 1 h 2 h 3 color space are discussed from this perspective. Two color similarity measures are studied: the Euclidean distance and the vector angle. The work in this thesis is motivated from a practical point of view by several shortcomings of current methods. The first problem is the inability of all known methods to properly segment objects from the background without interference from object shadows and highlights. The second shortcoming is the non-examination of the vector angle as a distance measure that is capable of directly evaluating hue similarity without considering intensity especially in RGB. Finally, there is inadequate research on the combination of hue-and intensity-based similarity measures to improve color similarity calculations given the advantages of each color distance measure. These distance measures were used for two image understanding tasks: edge detection, and one strategy for color image segmentation, namely color clustering. Edge …

Computing a consistent approximation to a generalized pairwise comparisons matrix

Abstract: This paper presents an algorithm for computing a consistent approximation to a generalized pairwise comparisons matrix (that is, without the reciprocity property or even 1s on the main diagonal). The algorithm is based on a logarithmic transformation of the generalized pairwise comparisons matrix into a linear space with the Euclidean metric. It uses both the row and (reciprocals of) column geometric means and is thus a generalization of the ordinary geometric means method. The resulting approximation is not only consistent, but also closest to the original matrix, i.e., deviates least from an expert's original judgments. The computational complexity of the algorithm is O ( n 2 ).

Comparison analysis of methods for deriving priorities in the analytic hierarchy process

Abstract: Presents a comparison study of some methods for prioritisation in the analytic hierarchy process. The main objective of this analysis is to compare and evaluate a fuzzy preference programming method with the most popular prioritisation techniques: the eigenvector method, the weighted least squares method, the logarithmic least squared method and the goal programming method. The analysis is based on three evaluation criteria: the total deviation, measuring the Euclidean distance between the ratios obtained by the derived weights and the decision-maker ratios; the minimum violations criterion, measuring the rank reversal properties of the methods; and the conformity, indicating the outliers with respect to the other methods. The evaluation procedure uses randomly generated inconsistent pairwise comparison matrices of different dimensions.

Sierpinski gasket as a Martin boundary II: the intrinsic metric

Abstract: It is shown in [DS] that the Sierpinski gasket ^aR can be represented as the Martin boundary of a certain Markov chain and hence carries a canonical metric pM induced by the embedding into an associated Martin space M. It is a natural question to compare this metric pM with the Euclidean metric. We show first that the harmonic measure coincides with the normalized //=(log(Af+l)/log2)-dimensional Hausdorff measure with respect to the Euclidean metric. Secondly, we define an intrinsic metric p which is Lipschitz equivalent to pM and then show that p is not Lipschitz equivalent to the Euclidean metric, but the Hausdorff dimension remains unchanged and the Hausdorff measure in p is infinite. Finally, using the metric p, we prove that the harmonic extension of a continuous boundary function converges to the boundary value at every boundary point. §

Global color image segmentation strategies: Euclidean distance vs. vector angle

Abstract: In the past few years, researchers have been increasingly interested in color image segmentation. We analyze two different global image segmentation algorithms each using its own distance metric: k-means and a mixture of principal components (MPC) neural network. The k-means uses Euclidean distance for color comparisons while the MPC neural network uses vector angles. Two variants of the algorithms are examined. The first uses the RGB pixel itself for clustering while the second uses a 3/spl times/3 neighborhood. Preliminary results on a staged scene image are shown and discussed.

A Note on the Number of Distinct Distances

Abstract: We refine a method introduced in [1] and [2] for studying the number of distinct values taken by certain polynomials of two real variables on Cartesian products. We apply it to prove a "gap theorem", improving a recent lower bound on the number of distinct distances between two collinear point sets in the Euclidean space.