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1,382
Waveletbased statistical signal processing using hidden Markov models
 IEEE Transactions on Signal Processing
, 1998
"... Abstract — Waveletbased statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly Gaussian. These models are unrealistic for many realworld signals. In this paper, we develop a new framework for statistical signal pr ..."
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Cited by 325 (52 self)
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Abstract — Waveletbased statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly Gaussian. These models are unrealistic for many realworld signals. In this paper, we develop a new framework for statistical signal processing based on waveletdomain hidden Markov models (HMM’s) that concisely models the statistical dependencies and nonGaussian statistics encountered in realworld signals. Waveletdomain HMM’s are designed with the intrinsic properties of the wavelet transform in mind and provide powerful, yet tractable, probabilistic signal models. Efficient expectation maximization algorithms are developed for fitting the HMM’s to observational signal data. The new framework is suitable for a wide range of applications, including signal estimation, detection, classification, prediction, and even synthesis. To demonstrate the utility of waveletdomain HMM’s, we develop novel algorithms for signal denoising, classification, and detection. Index Terms — Hidden Markov model, probabilistic graph, wavelets.
Multiple Description Coding: Compression Meets the Network
, 2001
"... This article focuses on the compressed representations of the pictures ..."
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Cited by 295 (8 self)
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This article focuses on the compressed representations of the pictures
Regularization networks and support vector machines
 Advances in Computational Mathematics
, 2000
"... Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization a ..."
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Cited by 269 (33 self)
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Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization and Support Vector Machines. We review both formulations in the context of Vapnik’s theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics. The emphasis is on regression: classification is treated as a special case.
New tight frames of curvelets and optimal representations of objects with piecewise C² singularities
 COMM. ON PURE AND APPL. MATH
, 2002
"... This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needleshap ..."
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Cited by 253 (17 self)
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This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needleshaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2−j, each element has an envelope which is aligned along a ‘ridge ’ of length 2−j/2 and width 2−j. We prove that curvelets provide an essentially optimal representation of typical objects f which are C2 except for discontinuities along C2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the nterm partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ‖f − f C n ‖ 2 L2 ≤ C · n−2 · (log n) 3, n → ∞. This rate of convergence holds uniformly over a class of functions which are C 2 except for discontinuities along C 2 curves and is essentially optimal. In comparison, the squared error of nterm wavelet approximations only converges as n −1 as n → ∞, which is considerably worst than the optimal behavior.
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
"... ..."
Adaptive Wavelet Thresholding for Image Denoising and Compression
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2000
"... The first part of this paper proposes an adaptive, datadriven threshold for image denoising via wavelet softthresholding. The threshold is derived in a Bayesian framework, and the prior used on the wavelet coefficients is the generalized Gaussian distribution (GGD) widely used in image processing ..."
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Cited by 236 (4 self)
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The first part of this paper proposes an adaptive, datadriven threshold for image denoising via wavelet softthresholding. The threshold is derived in a Bayesian framework, and the prior used on the wavelet coefficients is the generalized Gaussian distribution (GGD) widely used in image processing applications. The proposed threshold is simple and closedform, and it is adaptive to each subband because it depends on datadriven estimates of the parameters. Experimental results show that the proposed method, called BayesShrink, is typically within 5% of the MSE of the best softthresholding benchmark with the image assumed known. It also outperforms Donoho and Johnstone's SureShrink most of the time. The second part
The steerable pyramid: A flexible architecture for multiscale derivative computation
, 1995
"... We describe an architecture for efficient and accurate linear decomposition of an image into scale and orientation subbands. The basis functions of this decomposition are directional derivative operators of any desired order. We describe the construction and implementation of the transform. 1 Differ ..."
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Cited by 232 (27 self)
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We describe an architecture for efficient and accurate linear decomposition of an image into scale and orientation subbands. The basis functions of this decomposition are directional derivative operators of any desired order. We describe the construction and implementation of the transform. 1 Differential algorithms are used in a wide variety of image processing problems. For example, gradient measurements are used as a first stage of many edge detection, depthfromstereo, and optical flow algorithms. Higherorder derivatives have also been found useful in these applications. Extraction of these derivative quantities may be viewed as a decomposition of a signal via terms of a local Taylor series expansions [1]. Another widespread tool in signal and image processing is multiscale decomposition. Apart from the advantages of decomposing signals into information at different scales, the typical recursive form of these algorithms leads to large improvements in computational efficiency. Many authors have combined multiscale decompositions with differential measurements (eg., [2, 3]). In these cases, a multiscale pyramid is constructed, and then differential operators (typically, differences of neighboring pixels) are applied to the subbands of the pyramid. Since both the pyramid decomposition and the derivative operation are linear and shiftinvariant, we may combine them into a single operation. The advantages of doing so are that the resulting derivatives may be more accurate (see [4]). In this paper, we propose a simple, efficient decomposition architecture for combining these two operations. The decomposition is the latest incarnation of 1 Source code and filter kernels for implementation of the steerable pyramid are available via anonymous ftp from ftp.cis.upenn.edu:pub/eero/steerpyr.tar.Z
A Practical Guide to Wavelet Analysis
, 1998
"... A practical stepbystep guide to wavelet analysis is given, with examples taken from time series of the El Nio Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finitelength t ..."
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Cited by 230 (2 self)
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A practical stepbystep guide to wavelet analysis is given, with examples taken from time series of the El Nio Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finitelength time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmller, crosswavelet spectra, and coherence are described. The statistical significance tests are used to give a qu...
Spherical Wavelets: Efficiently Representing Functions on the Sphere
, 1995
"... Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classic ..."
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Cited by 229 (14 self)
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Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.