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146
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
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Efficient time series matching by wavelets
 Proc. of 15th Int'l Conf. on Data Engineering
, 1999
"... Time series stored as feature vectors can be indexed by multidimensional index trees like RTrees for fast retrieval. Due to the dimensionality curse problem, transformations are applied to time series to reduce the number of dimensions of the feature vectors. Different transformations like Discrete ..."
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Cited by 203 (1 self)
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Time series stored as feature vectors can be indexed by multidimensional index trees like RTrees for fast retrieval. Due to the dimensionality curse problem, transformations are applied to time series to reduce the number of dimensions of the feature vectors. Different transformations like Discrete Fourier Transform (DFT), Discrete Wavelet Transform (DWT), KarhunenLoeve (KL) transform or Singular Value Decomposition (SVD) can be applied. While the use of DFT and KL transform or SVD have been studied in the literature, to our knowledge, there is no indepth study on the application of DWT. In this paper, we propose to use Haar Wavelet Transform for time series indexing. The major contributions are: (1) we show that Euclidean distance is preserved in the Haar transformed domain and no false dismissal will occur, (2) we show that Haar transform can outperform DFT through experiments, (3) a new similarity model is suggested to accommodate vertical shift of time series, and (4) a twophase method is proposed for efficientnearest neighbor query in time series databases. 1.
Scalable compression and transmission of Internet multicast video
, 1996
"... In just a few years the "Internet Multicast Backbone", or MBone, has risen from a small, research curiosity to a large scale and widely used communications infrastructure. A driving force behind this growth was our development of multipoint audio, video, and shared whiteboard conferencing applicatio ..."
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Cited by 104 (5 self)
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In just a few years the "Internet Multicast Backbone", or MBone, has risen from a small, research curiosity to a large scale and widely used communications infrastructure. A driving force behind this growth was our development of multipoint audio, video, and shared whiteboard conferencing applications that are now used daily by the large and growing MBone community. Because these realtime media are transmitted at a uniform rate to all the receivers in the network, the source must either run below the bottleneck rate or overload portions of the multicast distribution tree. In this dissertation, we propose a solution to this problem by moving the burden of rateadaptation from the source to the receivers with a scheme we call Receiverdriven Layered Multicast, or RLM. In RLM, a source distr...
Wavelet transforms versus Fourier transforms
 Department of Mathematics, MIT, Cambridge MA
, 213
"... Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transfo ..."
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Cited by 69 (2 self)
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Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higherorder wavelets are constructed, and it is surprisingly quick to compute with them — always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including highdefinition television). So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine Transform — is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory. 1. The Haar wavelet To explain wavelets we start with an example. It has every property we hope for, except one. If that one defect is accepted, the construction is simple and the computations are fast. By trying to remove the defect, we are led to dilation equations and recursively defined functions and a small world of fascinating new problems — many still unsolved. A sensible person would stop after the first wavelet, but fortunately mathematics goes on. The basic example is easier to draw than to describe: W(x)
Matrix Refinement Equations: Existence and Uniqueness
 J. Fourier Anal. Appl
, 1996
"... . Matrix refinement equations are functional equations of the form f(x) = P N k=0 c k f(2x \Gamma k), where the coefficients c k are matrices and f is a vectorvalued function. Refinement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of sca ..."
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Cited by 51 (3 self)
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. Matrix refinement equations are functional equations of the form f(x) = P N k=0 c k f(2x \Gamma k), where the coefficients c k are matrices and f is a vectorvalued function. Refinement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of scalar refinement equations (where the coefficients c k are scalars) are known. This paper considers analogous questions for matrix refinement equations. Conditions for existence and uniqueness of compactly supported distributional solutions are given in terms of the convergence properties of an infinite product of the matrix \Delta = 1 2 P c k with itself. Fundamental differences between solutions of matrix equations and scalar refinement equations are examined. In particular, it is shown that "constrained" solutions of the matrix refinement equation can exist even when the infinite product diverges. The existence of constrained solutions is related to the eigenvalue structure of \Delta; so...
TSAtree: A waveletbased approach to improve the efficiency of multilevel surprise and trend queries on timeseries data
 IN SSDBM
, 2000
"... We introduce a novel waveletbased tree structure, termed TSAtree, which improves the efficiency of multilevel trend and surprise queries on time sequence data. With the explosion of scientific observation data (some conceptualized as timesequences), we are facing the challenge of efficiently stor ..."
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Cited by 43 (0 self)
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We introduce a novel waveletbased tree structure, termed TSAtree, which improves the efficiency of multilevel trend and surprise queries on time sequence data. With the explosion of scientific observation data (some conceptualized as timesequences), we are facing the challenge of efficiently storing, retrieving and analyzing this data. Frequent queries on this data set is to find trends (e.g., global warming) or surprises (e.g., undersea volcano eruption) within the original timeseries. The challenge, however, is that these trend and surprise queries are needed at different levels of abstractions (e.g., within the last week, last month, last year or last decade). To support these multilevel trend and surprise queries, sometimes huge subset of raw data needs to be retrieved and processed. To
Multiwavelets: Theory and Applications
, 1996
"... A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\Gamma1 V j = L 2 (R), T 1 j=\Gam ..."
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Cited by 35 (4 self)
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A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\Gamma1 V j = L 2 (R), T 1 j=\Gamma1 V j = f0g, and translates OE(t \Gamma k) constitute a basis of V 0 . Then a basis fw jk : w jk = w(2 j t \Gamma k) j; k 2 Zg of L 2 (R) is generated by a wavelet w(t), whose translates w(t \Gamma k) form a basis of W 0 , V 1 = V 0 \Phi W 0 . A standard (scalar) MRA assumes that there is only one scaling function. We make a step forward and allow several scaling functions OE 0 (t); : : : ; OE r\Gamma1 (t) to generate a basis of V 0 . The vector OE(t) = [OE 0 (t) : : : OE r\Gamma1 (t)] T satisfies a dilation equation with matrix coefficients C k . Associated with OE(t) is a multiwavelet w(t) = [w 0 (t) : : : w r\Gamma1 (t)] T . Unlike the scalar case, construction of a multiwave...
A survey on wavelet applications in data mining
 SIGKDD Explor. Newsl
"... Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a highlevel datamining framework tha ..."
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Cited by 30 (3 self)
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Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a highlevel datamining framework that reduces the overall process into smaller components. Then applications of wavelets for each component are reviewd. The paper concludes by discussing the impact of wavelets on data mining research and outlining potential future research directions and applications. 1.
GalerkinWavelet Methods For TwoPoint Boundary Value Problems
 NUMER. MATH
, 1992
"... Antiderivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to twopoint boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the ..."
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Cited by 29 (6 self)
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Antiderivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to twopoint boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.