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370
A new fast and efficient image codec based on set partitioning in hierarchical trees
 IEEE Trans. on Circuits and Systems for Video Technology
, 1996
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Quantization
 IEEE TRANS. INFORM. THEORY
, 1998
"... The history of the theory and practice of quantization dates to 1948, although similar ideas had appeared in the literature as long ago as 1898. The fundamental role of quantization in modulation and analogtodigital conversion was first recognized during the early development of pulsecode modula ..."
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Cited by 877 (12 self)
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The history of the theory and practice of quantization dates to 1948, although similar ideas had appeared in the literature as long ago as 1898. The fundamental role of quantization in modulation and analogtodigital conversion was first recognized during the early development of pulsecode modulation systems, especially in the 1948 paper of Oliver, Pierce, and Shannon. Also in 1948, Bennett published the first highresolution analysis of quantization and an exact analysis of quantization noise for Gaussian processes, and Shannon published the beginnings of rate distortion theory, which would provide a theory for quantization as analogtodigital conversion and as data compression. Beginning with these three papers of fifty years ago, we trace the history of quantization from its origins through this decade, and we survey the fundamentals of the theory and many of the popular and promising techniques for quantization.
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combin ..."
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Cited by 423 (37 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Multiresolution Analysis for Surfaces Of Arbitrary . . .
, 1993
"... Multiresolution analysis provides a useful and efficient tool for representing shape and analyzing features at multiple levels of detail. Although the technique has met with considerable success when applied to univariate functions, images, and more generally to functions defined on lR , to our k ..."
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Cited by 390 (3 self)
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Multiresolution analysis provides a useful and efficient tool for representing shape and analyzing features at multiple levels of detail. Although the technique has met with considerable success when applied to univariate functions, images, and more generally to functions defined on lR , to our knowledge it has not been extended to functions defined on surfaces of arbitrary genus. In this
Fast Multiresolution Image Querying
, 1995
"... We present a method for searching in an image database using a query image that is similar to the intended target. The query image may be a handdrawn sketch or a (potentially lowquality) scan of the image to be retrieved. Our searching algorithm makes use of multiresolution wavelet decompositions ..."
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Cited by 313 (4 self)
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We present a method for searching in an image database using a query image that is similar to the intended target. The query image may be a handdrawn sketch or a (potentially lowquality) scan of the image to be retrieved. Our searching algorithm makes use of multiresolution wavelet decompositions of the query and database images. The coefficients of these decompositions are distilled into small "signatures" for each image. We introduce an "image querying metric" that operates on these signatures. This metric essentially compares how many significant wavelet coefficients the query has in common with potential targets. The metric includes parameters that can be tuned, using a statistical analysis, to accommodate the kinds of image distortions found in different types of image queries. The resulting algorithm is simple, requires very little storage overhead for the database of signatures, and is fast enough to be performed on a database of 20,000 images at interactive rates (on standard...
Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage
 IEEE Trans. Image Processing
, 1996
"... This paper examines the relationship between waveletbased image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following pro ..."
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Cited by 257 (12 self)
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This paper examines the relationship between waveletbased image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem: given an image F defined on a square I, minimize over all g in the Besov space B 1 1 (L1 (I)) the functional #F  g# 2 L 2 (I) + ##g# B 1 1 (L 1 (I)) .Weusethetheoryof nonlinear wavelet image compression in L2 (I) to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i.i.d., mean zero, Gaussian noise. A new signaltonoise ratio, which we claim more accurately reflects the visual perception of noise in images, arises in this derivation. We present extensive computations that support the hypothesis that nearoptimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F:thelarge...
Nonuniform Sampling and Reconstruction in ShiftInvariant Spaces
, 2001
"... This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. It is a survey as well as a research paper and provides a unied framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by bringing together ..."
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Cited by 219 (13 self)
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This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. It is a survey as well as a research paper and provides a unied framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well dened within the setting of shiftinvariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any suciently dense nonuniform sampling set is obtained by ecient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted Lpspaces.
Analysis Of Multiresolution Image Denoising Schemes Using GeneralizedGaussian Priors
 IEEE TRANS. INFO. THEORY
, 1998
"... In this paper, we investigate various connections between wavelet shrinkage methods in image processing and Bayesian estimation using Generalized Gaussian priors. We present fundamental properties of the shrinkage rules implied by Generalized Gaussian and other heavytailed priors. This allows us to ..."
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Cited by 215 (9 self)
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In this paper, we investigate various connections between wavelet shrinkage methods in image processing and Bayesian estimation using Generalized Gaussian priors. We present fundamental properties of the shrinkage rules implied by Generalized Gaussian and other heavytailed priors. This allows us to show a simple relationship between differentiability of the logprior at zero and the sparsity of the estimates, as well as an equivalence between universal thresholding schemes and Bayesian estimation using a certain Generalized Gaussian prior.
Adaptive Greedy Approximations
"... The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the ..."
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Cited by 187 (0 self)
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The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function's structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model. We derive a notion of the coherence of a signal with respect to a dict...