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507
Compressive sensing
 IEEE Signal Processing Mag
, 2007
"... The Shannon/Nyquist sampling theorem tells us that in order to not lose information when uniformly sampling a signal we must sample at least two times faster than its bandwidth. In many applications, including digital image and video cameras, the Nyquist rate can be so high that we end up with too m ..."
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Cited by 700 (62 self)
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The Shannon/Nyquist sampling theorem tells us that in order to not lose information when uniformly sampling a signal we must sample at least two times faster than its bandwidth. In many applications, including digital image and video cameras, the Nyquist rate can be so high that we end up with too many samples and must compress in order to store or transmit them. In other applications, including imaging systems (medical scanners, radars) and highspeed analogtodigital converters, increasing the sampling rate or density beyond the current stateoftheart is very expensive. In this lecture, we will learn about a new technique that tackles these issues using compressive sensing [1, 2]. We will replace the conventional sampling and reconstruction operations with a more general linear measurement scheme coupled with an optimization in order to acquire certain kinds of signals at a rate significantly below Nyquist. 2
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 427 (35 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
Image denoising by sparse 3D transformdomain collaborative filtering
 IEEE TRANS. IMAGE PROCESS
, 2007
"... We propose a novel image denoising strategy based on an enhanced sparse representation in transform domain. The enhancement of the sparsity is achieved by grouping similar 2D image fragments (e.g., blocks) into 3D data arrays which we call “groups.” Collaborative filtering is a special procedure d ..."
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Cited by 424 (32 self)
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We propose a novel image denoising strategy based on an enhanced sparse representation in transform domain. The enhancement of the sparsity is achieved by grouping similar 2D image fragments (e.g., blocks) into 3D data arrays which we call “groups.” Collaborative filtering is a special procedure developed to deal with these 3D groups. We realize it using the three successive steps: 3D transformation of a group, shrinkage of the transform spectrum, and inverse 3D transformation. The result is a 3D estimate that consists of the jointly filtered grouped image blocks. By attenuating the noise, the collaborative filtering reveals even the finest details shared by grouped blocks and, at the same time, it preserves the essential unique features of each individual block. The filtered blocks are then returned to their original positions. Because these blocks are overlapping, for each pixel, we obtain many different estimates which need to be combined. Aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy. A significant improvement is obtained by a specially developed collaborative Wiener filtering. An algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail; an extension to colorimage denoising is also developed. The experimental results demonstrate that this computationally scalable algorithm achieves stateoftheart denoising performance in terms of both peak signaltonoise ratio and subjective visual quality.
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 352 (22 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.
Fields of experts: A framework for learning image priors
 In CVPR
, 2005
"... We develop a framework for learning generic, expressive image priors that capture the statistics of natural scenes and can be used for a variety of machine vision tasks. The approach extends traditional Markov Random Field (MRF) models by learning potential functions over extended pixel neighborhood ..."
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Cited by 292 (4 self)
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We develop a framework for learning generic, expressive image priors that capture the statistics of natural scenes and can be used for a variety of machine vision tasks. The approach extends traditional Markov Random Field (MRF) models by learning potential functions over extended pixel neighborhoods. Field potentials are modeled using a ProductsofExperts framework that exploits nonlinear functions of many linear filter responses. In contrast to previous MRF approaches all parameters, including the linear filters themselves, are learned from training data. We demonstrate the capabilities of this Field of Experts model with two example applications, image denoising and image inpainting, which are implemented using a simple, approximate inference scheme. While the model is trained on a generic image database and is not tuned toward a specific application, we obtain results that compete with and even outperform specialized techniques. 1.
Image information and visual quality
 IEEE Trans. IP
, 2006
"... Abstract—Measurement of visual quality is of fundamental importance to numerous image and video processing applications. The goal of quality assessment (QA) research is to design algorithms that can automatically assess the quality of images or videos in a perceptually consistent manner. Image QA a ..."
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Cited by 284 (41 self)
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Abstract—Measurement of visual quality is of fundamental importance to numerous image and video processing applications. The goal of quality assessment (QA) research is to design algorithms that can automatically assess the quality of images or videos in a perceptually consistent manner. Image QA algorithms generally interpret image quality as fidelity or similarity with a “reference ” or “perfect ” image in some perceptual space. Such “fullreference ” QA methods attempt to achieve consistency in quality prediction by modeling salient physiological and psychovisual features of the human visual system (HVS), or by signal fidelity measures. In this paper, we approach the image QA problem as an information fidelity problem. Specifically, we propose to quantify the loss of image information to the distortion process and explore the relationship between image information and visual quality. QA systems are invariably involved with judging the visual quality of “natural ” images and videos that are meant for “human consumption. ” Researchers have developed sophisticated models to capture the statistics of such natural signals. Using these models, we previously presented an information fidelity criterion for image QA that related image quality with the amount of information shared between a reference and a distorted image. In this paper, we propose an image information measure that quantifies the information that is present in the reference image and how much of this reference information can be extracted from the distorted image. Combining these two quantities, we propose a visual information fidelity measure for image QA. We validate the performance of our algorithm with an extensive subjective study involving 779 images and show that our method outperforms recent stateoftheart image QA algorithms by a sizeable margin in our simulations. The code and the data from the subjective study are available at the LIVE website. Index Terms—Image information, image quality assessment (QA), information fidelity, natural scene statistics (NSS). I.
The DualTree Complex Wavelet Transform  A coherent framework for multiscale signal and image processing
, 2005
"... The dualtree complex wavelet transform (CWT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties: It is nearly shift invariant and directionally selective in two and higher dimensions. It achieves this with a redundancy factor of only 2 ..."
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Cited by 270 (29 self)
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The dualtree complex wavelet transform (CWT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties: It is nearly shift invariant and directionally selective in two and higher dimensions. It achieves this with a redundancy factor of only 2 d for ddimensional signals, which is substantially lower than the undecimated DWT. The multidimensional (MD) dualtree CWT is nonseparable but is based on a computationally efficient, separable filter bank (FB). This tutorial discusses the theory behind the dualtree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing. We use the complex number symbol C in CWT to
Sparse representation for color image restoration
 the IEEE Trans. on Image Processing
, 2007
"... Sparse representations of signals have drawn considerable interest in recent years. The assumption that natural signals, such as images, admit a sparse decomposition over a redundant dictionary leads to efficient algorithms for handling such sources of data. In particular, the design of well adapted ..."
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Cited by 219 (30 self)
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Sparse representations of signals have drawn considerable interest in recent years. The assumption that natural signals, such as images, admit a sparse decomposition over a redundant dictionary leads to efficient algorithms for handling such sources of data. In particular, the design of well adapted dictionaries for images has been a major challenge. The KSVD has been recently proposed for this task [1], and shown to perform very well for various grayscale image processing tasks. In this paper we address the problem of learning dictionaries for color images and extend the KSVDbased grayscale image denoising algorithm that appears in [2]. This work puts forward ways for handling nonhomogeneous noise and missing information, paving the way to stateoftheart results in applications such as color image denoising, demosaicing, and inpainting, as demonstrated in this paper. EDICS Category: COLCOLR (Color processing) I.
A New TwIST: TwoStep Iterative Shrinkage/Thresholding Algorithms for Image Restoration
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2007
"... Iterative shrinkage/thresholding (IST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic ..."
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Cited by 184 (26 self)
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Iterative shrinkage/thresholding (IST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic regularizer (e.g., total variation or waveletbased regularization). It happens that the convergence rate of these IST algorithms depends heavily on the linear observation operator, becoming very slow when this operator is illconditioned or illposed. In this paper, we introduce twostep IST (TwIST) algorithms, exhibiting much faster convergence rate than IST for illconditioned problems. For a vast class of nonquadratic convex regularizers ( norms, some Besov norms, and total variation), we show that TwIST converges to a minimizer of the objective function, for a given range of values of its parameters. For noninvertible observation operators, we introduce a monotonic version of TwIST (MTwIST); although the convergence proof does not apply to this scenario, we give experimental evidence that MTwIST exhibits similar speed gains over IST. The effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples.