Results 11  20
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1,584
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
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An EM Algorithm for WaveletBased Image Restoration
, 2002
"... This paper introduces an expectationmaximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with lowcomplexity, expressed in terms of the wavelet coecients, taking a ..."
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Cited by 233 (21 self)
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This paper introduces an expectationmaximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with lowcomplexity, expressed in terms of the wavelet coecients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated waveletbased restoration but, except for certain special cases, the resulting criteria are solved approximately or require very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation oered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. The algorithm alternates between an Estep based on the fast Fourier transform (FFT) and a DWTbased Mstep, resulting in an ecient iterative process requiring O(N log N) operations per iteration. Thus, it is the rst image restoration algorithm that optimizes a waveletbased penalized likelihood criterion and has computational complexity comparable to that of standard wavelet denoising or frequency domain deconvolution methods. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.
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 232 (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.
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 207 (22 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
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 combinato ..."
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Cited by 202 (31 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
Blind Source Separation by Sparse Decomposition in a Signal Dictionary
, 2000
"... Introduction In blind source separation an Nchannel sensor signal x(t) arises from M unknown scalar source signals s i (t), linearly mixed together by an unknown N M matrix A, and possibly corrupted by additive noise (t) x(t) = As(t) + (t) (1.1) We wish to estimate the mixing matrix A and the M ..."
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Cited by 193 (32 self)
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Introduction In blind source separation an Nchannel sensor signal x(t) arises from M unknown scalar source signals s i (t), linearly mixed together by an unknown N M matrix A, and possibly corrupted by additive noise (t) x(t) = As(t) + (t) (1.1) We wish to estimate the mixing matrix A and the Mdimensional source signal s(t). Many natural signals can be sparsely represented in a proper signal dictionary s i (t) = K X k=1 C ik ' k (t) (1.2) The scalar functions ' k
A generalized uncertainty principle and sparse representation in pairs of bases
 IEEE Trans. Inform. Theory
, 2002
"... An elementary proof of a basic uncertainty principle concerning pairs of representations of ℜ N vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pair ..."
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Cited by 173 (12 self)
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An elementary proof of a basic uncertainty principle concerning pairs of representations of ℜ N vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pairs of orthonormal bases as overcomplete dictionaries. The main contribution in this paper is the improvement of an important result due to Donoho and Huo concerning the replacement of the l0 optimization problem by a linear programming minimization when searching for the unique sparse representation. 1
Saliency, Scale and Image Description
, 2001
"... Many computer vision problems can be considered to consist of two main tasks: the extraction of image content descriptions and their subsequent matching. The appropriate choice of type and level of description is of course task dependent, yet it is generally accepted that the lowlevel or so called ..."
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Cited by 156 (0 self)
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Many computer vision problems can be considered to consist of two main tasks: the extraction of image content descriptions and their subsequent matching. The appropriate choice of type and level of description is of course task dependent, yet it is generally accepted that the lowlevel or so called early vision layers in the Human Visual System are context independent. This paper concentrates on the use of lowlevel approaches for solving computer vision problems and discusses three interrelated aspects of this: saliency; scale selection and content description. In contrast to many previous approaches which separate these tasks, we argue that these three aspects are intrinsically related. Based on this observation, a multiscale algorithm for the selection of salient regions of an image is introduced and its application to matching type problems such as tracking, object recognition and image retrieval is demonstrated.
Empirical properties of asset returns: stylized facts and statistical issues
 Quantitative Finance
, 2001
"... We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then des ..."
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Cited by 149 (2 self)
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We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
New EdgeDirected Interpolation
 IEEE Transactions on Image Processing
, 2001
"... This paper proposes an edgedirected interpolation algorithm for natural images. The basic idea is to first estimate local covariance coefficients from a lowresolution image and then use these covariance estimates to adapt the interpolation at a higher resolution based on the geometric duality betw ..."
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Cited by 142 (1 self)
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This paper proposes an edgedirected interpolation algorithm for natural images. The basic idea is to first estimate local covariance coefficients from a lowresolution image and then use these covariance estimates to adapt the interpolation at a higher resolution based on the geometric duality between the lowresolution covariance and the highresolution covariance. The edgedirected property of covariancebased adaptation attributes to its capability of tuning the interpolation coefficients to match an arbitrarily oriented step edge. A hybrid approach of switching between bilinear interpolation and covariancebased adaptive interpolation is proposed to reduce the overall computational complexity. Two important applications of the new interpolation algorithm are studied: resolution enhancement of grayscale images and reconstruction of color images from CCD samples. Simulation results demonstrate that our new interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional linear interpolation.