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131
Wavelet Thresholding via a Bayesian Approach
 J. R. STATIST. SOC. B
, 1996
"... We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. ..."
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Cited by 262 (33 self)
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We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might b...
WaveletBased Histograms for Selectivity Estimation
"... Query optimization is an integral part of relational database management systems. One important task in query optimization is selectivity estimation, that is, given a query P, we need to estimate the fraction of records in the database that satisfy P. Many commercial database systems maintain histog ..."
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Cited by 245 (16 self)
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Query optimization is an integral part of relational database management systems. One important task in query optimization is selectivity estimation, that is, given a query P, we need to estimate the fraction of records in the database that satisfy P. Many commercial database systems maintain histograms to approximate the frequency distribution of values in the attributes of relations. In this paper, we present a technique based upon a multiresolution wavelet decomposition for building histograms on the underlying data distributions, with applications to databases, statistics, and simulation. Histograms built on the cumulative data values give very good approximations with limited space usage. We give fast algorithms for constructing histograms and using
Approximate Query Processing Using Wavelets
, 2000
"... Abstract. Approximate query processing has emerged as a costeffective approach for dealing with the huge data volumes and stringent responsetime requirements of today’s decision support systems (DSS). Most work in this area, however, has so far been limited in its query processing scope, typically ..."
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Cited by 216 (12 self)
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Abstract. Approximate query processing has emerged as a costeffective approach for dealing with the huge data volumes and stringent responsetime requirements of today’s decision support systems (DSS). Most work in this area, however, has so far been limited in its query processing scope, typically focusing on specific forms of aggregate queries. Furthermore, conventional approaches based on sampling or histograms appear to be inherently limited when it comes to approximating the results of complex queries over highdimensional DSS data sets. In this paper, we propose the use of multidimensional wavelets as an effective tool for generalpurpose approximate query processing in modern, highdimensional applications. Our approach is based on building waveletcoefficient synopses of the data and using these synopses to provide approximate answers to queries. We develop novel query processing
Approximate Computation of Multidimensional Aggregates of Sparse Data Using Wavelets
"... Computing multidimensional aggregates in high dimensions is a performance bottleneck for many OLAP applications. Obtaining the exact answer to an aggregation query can be prohibitively expensive in terms of time and/or storage space in a data warehouse environment. It is advantageous to have fast, a ..."
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Cited by 198 (3 self)
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Computing multidimensional aggregates in high dimensions is a performance bottleneck for many OLAP applications. Obtaining the exact answer to an aggregation query can be prohibitively expensive in terms of time and/or storage space in a data warehouse environment. It is advantageous to have fast, approximate answers to OLAP aggregation queries. In this paper, we present anovel method that provides approximate answers to highdimensional OLAP aggregation queries in massive sparse data sets in a timeefficient and spaceefficient manner. We construct a compact data cube, which is an approximate and spaceefficient representation of the underlying multidimensional array, based upon a multiresolution wavelet decomposition. In the online phase, each aggregation query can generally be answered using the compact data cube in one I/O or a small number of I/Os, depending upon the desired accuracy. We present two I/Oefficient algorithms to construct the compact data cube for the important case of sparse highdimensional arrays, which often arise in practice. The traditional histogram methods are infeasible for the massive highdimensional data sets in OLAP applications. Previously developed wavelet techniques are efficient only for dense data. Our online query processing algorithm is very fast and capable of refining answers as the user demands more accuracy. Experiments on real data show that our method provides significantly more accurate results for typical OLAP aggregation queries than other efficient approximation techniques such as random sampling.
Data Cube Approximation and Histograms via Wavelets (Extended Abstract)
 In CIKM
, 1998
"... ) Jeffrey Scott Vitter Center for Geometric Computing and Department of Computer Science Duke University Durham, NC 277080129 USA jsv@cs.duke.edu Min Wang y Center for Geometric Computing and Department of Computer Science Duke University Durham, NC 277080129 USA minw@cs.duke.edu Bala Iyer ..."
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Cited by 106 (3 self)
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) Jeffrey Scott Vitter Center for Geometric Computing and Department of Computer Science Duke University Durham, NC 277080129 USA jsv@cs.duke.edu Min Wang y Center for Geometric Computing and Department of Computer Science Duke University Durham, NC 277080129 USA minw@cs.duke.edu Bala Iyer Database Technology Institute IBM Santa Teresa Laboratory P.O. Box 49023 San Jose, CA 95161 USA balaiyer@vnet.ibm.com Abstract There has recently been an explosion of interest in the analysis of data in data warehouses in the field of OnLine Analytical Processing (OLAP). Data warehouses can be extremely large, yet obtaining quick answers to queries is important. In many situations, obtaining the exact answer to an OLAP query is prohibitively expensive in terms of time and/or storage space. It can be advantageous to have fast, approximate answers to queries. In this paper, we present an I/Oefficient technique based upon a multiresolution wavelet decomposition that yields an approximate a...
Dynamic Maintenance of WaveletBased Histograms
, 2000
"... In this paper, we introduce an efficient method for the dynamic maintenance of waveletbased histograms (and other transformbased histograms). Previous work has shown that waveletbased histograms provide more accurate selectivity estimation than traditional histograms, such as equidepth histog ..."
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Cited by 92 (7 self)
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In this paper, we introduce an efficient method for the dynamic maintenance of waveletbased histograms (and other transformbased histograms). Previous work has shown that waveletbased histograms provide more accurate selectivity estimation than traditional histograms, such as equidepth histograms. But since waveletbased histograms are built by a nontrivial mathematical procedure, namely, wavelet transform decomposition, it is hard to maintain the accuracy of the histogram when the underlying data distribution changes over time. In particular, simple techniques, such as split and merge, which works well for equidepth histograms, and updating a fixed set of wavelet coefficients, are not suitable here.
Wavelet synopses with error guarantees
 In Proceedings of the 2002 ACM SIGMOD international conference on Management of data
, 2002
"... ABSTRACT Recent work has demonstrated the effectiveness of the wavelet decomposition in reducing large amounts of data to compact sets of ..."
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Cited by 82 (1 self)
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ABSTRACT Recent work has demonstrated the effectiveness of the wavelet decomposition in reducing large amounts of data to compact sets of
Wavelet Analysis and Its Statistical Applications
, 1999
"... In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this ..."
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Cited by 64 (14 self)
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In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this article is intended to give a relatively accessible introduction to standard wavelet analysis and to provide an up to date review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be e ective, rather than to researchers already familiar with the eld. Given that objective, we do not emphasise mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in...
Compressing Still and Moving Images with Wavelets
 Multimedia Systems
"... The wavelet transform has become a cuttingedge technology in image compression research. This article explains what wavelets are and provides a practical, nutsandbolts tutorial on waveletbased compression that will help readers to understand and experiment with this important new technology. Keyw ..."
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Cited by 58 (3 self)
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The wavelet transform has become a cuttingedge technology in image compression research. This article explains what wavelets are and provides a practical, nutsandbolts tutorial on waveletbased compression that will help readers to understand and experiment with this important new technology. Keywords: image coding, signal compression, wavelet transform, image transforms 1 Introduction The advent of multimedia computing has lead to an increased demand for digital images. The storage and manipulation of these images in their raw form is very expensive; for example, a standard 35mm photograph digitized at 12 ¯m per pixel requires about 18 MBytes of storage and one second of NTSCquality color video requires almost 23 MBytes of storage. To make widespread use of digital imagery practical, some form of data compression must be used. Digital images can be compressed by eliminating redundant information. There are three types of redundancy that can be exploited by image compression system...
Generalized Cross Validation for wavelet thresholding
 Signal Processing
, 1995
"... Noisy data are often fitted using a smoothing parameter, controling the importance of two objectives that are opposite to a certain extent. One of these two is smoothness and the other is closeness to the input data. The optimal value of this paramater minimizes the error of the result (as compared ..."
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Cited by 51 (18 self)
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Noisy data are often fitted using a smoothing parameter, controling the importance of two objectives that are opposite to a certain extent. One of these two is smoothness and the other is closeness to the input data. The optimal value of this paramater minimizes the error of the result (as compared to the unknown, exact data), usually expressed in the L 2 norm. This optimum cannot be found exactly, simply because the exact data are unknown. In spline theory, the Generalized Cross Validation (GCV) technique has proven to be an effective (though rather slow) statistical way for estimating this optimum. On the other hand, wavelet theory is well suited for signal and image processing. This paper investigates the possibility of using GCV in a noise reduction algorithm, based on waveletthresholding, where the threshold can be seen as a kind of smoothing parameter. The GCV method thus allows choosing the (nearly) optimal threshold, without knowing the noise variance. Both an original theore...