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158
Image Quality Assessment: From Error Visibility to Structural Similarity
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2004
"... Objective methods for assessing perceptual image quality have traditionally attempted to quantify the visibility of errors between a distorted image and a reference image using a variety of known properties of the human visual system. Under the assumption that human visual perception is highly adapt ..."
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Cited by 577 (40 self)
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Objective methods for assessing perceptual image quality have traditionally attempted to quantify the visibility of errors between a distorted image and a reference image using a variety of known properties of the human visual system. Under the assumption that human visual perception is highly adapted for extracting structural information from a scene, we introduce an alternative framework for quality assessment based on the degradation of structural information. As a specific example of this concept, we develop a Structural Similarity Index and demonstrate its promise through a set of intuitive examples, as well as comparison to both subjective ratings and stateoftheart objective methods on a database of images compressed with JPEG and JPEG2000.
A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics
 in Proc. 8th Int’l Conf. Computer Vision
, 2001
"... This paper presents a database containing ‘ground truth ’ segmentations produced by humans for images of a wide variety of natural scenes. We define an error measure which quantifies the consistency between segmentations of differing granularities and find that different human segmentations of the s ..."
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Cited by 569 (16 self)
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This paper presents a database containing ‘ground truth ’ segmentations produced by humans for images of a wide variety of natural scenes. We define an error measure which quantifies the consistency between segmentations of differing granularities and find that different human segmentations of the same image are highly consistent. Use of this dataset is demonstrated in two applications: (1) evaluating the performance of segmentation algorithms and (2) measuring probability distributions associated with Gestalt grouping factors as well as statistics of image region properties. 1.
Image denoising using a scale mixture of Gaussians in the wavelet domain
 IEEE Trans Image Processing
, 2003
"... Abstract—We describe a method for removing noise from digital images, based on a statistical model of the coefficients of an overcomplete multiscale oriented basis. Neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables: a Gaussi ..."
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Cited by 350 (18 self)
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Abstract—We describe a method for removing noise from digital images, based on a statistical model of the coefficients of an overcomplete multiscale oriented basis. Neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables: a Gaussian vector and a hidden positive scalar multiplier. The latter modulates the local variance of the coefficients in the neighborhood, and is thus able to account for the empirically observed correlation between the coefficient amplitudes. Under this model, the Bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable. We demonstrate through simulations with images contaminated by additive white Gaussian noise that the performance of this method substantially surpasses that of previously published methods, both visually and in terms of mean squared error.
Learning Overcomplete Representations
, 2000
"... In an overcomplete basis, the number of basis vectors is greater than the dimensionality of the input, and the representation of an input is not a unique combination of basis vectors. Overcomplete representations have been advocated because they have greater robustness in the presence of noise, can ..."
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Cited by 257 (11 self)
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In an overcomplete basis, the number of basis vectors is greater than the dimensionality of the input, and the representation of an input is not a unique combination of basis vectors. Overcomplete representations have been advocated because they have greater robustness in the presence of noise, can be sparser, and can have greater flexibility in matching structure in the data. Overcomplete codes have also been proposed as a model of some of the response properties of neurons in primary visual cortex. Previous work has focused on finding the best representation of a signal using a fixed overcomplete basis (or dictionary). We present an algorithm for learning an overcomplete basis by viewing it as probabilistic model of the observed data. We show that overcomplete bases can yield a better approximation of the underlying statistical distribution of the data and can thus lead to greater coding efficiency. This can be viewed as a generalization of the technique of independent component analysis and provides a method for Bayesian reconstruction of signals in the presence of noise and for blind source separation when there are more sources than mixtures.
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
Statistics of Natural Images and Models
"... Large calibrated datasets of `random' natural images have recently become available. These make possible precise and intensive statistical studies of the local nature of images. We report results ranging from the simplest single pixel intensity to joint distribution of 3 Haar wavelet responses. Some ..."
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Cited by 198 (5 self)
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Large calibrated datasets of `random' natural images have recently become available. These make possible precise and intensive statistical studies of the local nature of images. We report results ranging from the simplest single pixel intensity to joint distribution of 3 Haar wavelet responses. Some of these statistics shed light on old issues such as the near scaleinvariance of image statistics and some are entirely new. We fit mathematical models to some of the statistics and explain others in terms of local image features. 1
Statistical Models for Images: Compression, Restoration and Synthesis
 In 31st Asilomar Conf on Signals, Systems and Computers
, 1997
"... this paper, we examine the problem of decomposing digitized images, through linear and/or nonlinear transformations, into statistically independent components. The classical approach to such a problem is Principal Components Analysis (PCA), also known as the KarhunenLoeve (KL) or Hotelling transfor ..."
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Cited by 138 (33 self)
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this paper, we examine the problem of decomposing digitized images, through linear and/or nonlinear transformations, into statistically independent components. The classical approach to such a problem is Principal Components Analysis (PCA), also known as the KarhunenLoeve (KL) or Hotelling transform. This is a linear transform that removes secondorder dependencies between input pixels. The most wellknown description of image statistics is that their power spectra take the form of a power law [e.g., 20, 11, 24]. Coupled with a constraint of translationinvariance, this suggests that the Fourier transform is an appropriate PCA representation. Fourier and related representations are widely used in image processing applications.
Multiresolution markov models for signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coheren ..."
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Cited by 122 (18 self)
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This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for selfsimilar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden
Scale Mixtures of Gaussians and the Statistics of Natural Images
 in Adv. Neural Information Processing Systems
, 2000
"... The statistics of photographic images, when represented using multiscale (wavelet) bases, exhibit two striking types of nonGaussian behavior. First, the marginal densities of the coefficients have extended heavy tails. Second, the joint densities exhibit variance dependencies not captured by secon ..."
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Cited by 118 (19 self)
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The statistics of photographic images, when represented using multiscale (wavelet) bases, exhibit two striking types of nonGaussian behavior. First, the marginal densities of the coefficients have extended heavy tails. Second, the joint densities exhibit variance dependencies not captured by secondorder models. We examine properties of the class of Gaussian scale mixtures, and show that these densities can accurately characterize both the marginal and joint distributions of natural image wavelet coefficients. This class of model suggests a Markov structure, in which wavelet coefficients are linked by hidden scaling variables corresponding to local image structure. We derive an estimator for these hidden variables, and show that a nonlinear ``normalization'' procedure can be used to Gaussianize the coefficients.
A probabilistic framework for the adaptation and comparison of image codes
 J. Opt. Soc. Am. A
, 1999
"... We apply a Bayesian method for inferring an optimal basis to the problem of finding efficient image codes for natural scenes. The basis functions learned by the algorithm are oriented and localized in both space and frequency, bearing a resemblance to twodimensional Gabor functions, and increasing ..."
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Cited by 112 (9 self)
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We apply a Bayesian method for inferring an optimal basis to the problem of finding efficient image codes for natural scenes. The basis functions learned by the algorithm are oriented and localized in both space and frequency, bearing a resemblance to twodimensional Gabor functions, and increasing the number of basis functions results in a greater sampling density in position, orientation, and scale. These properties also resemble the spatial receptive fields of neurons in the primary visual cortex of mammals, suggesting that the receptivefield structure of these neurons can be accounted for by a general efficient coding principle. The probabilistic framework provides a method for comparing the coding efficiency of different bases objectively by calculating their probability given the observed data or by measuring the entropy of the basis function coefficients. The learned bases are shown to have better coding efficiency than traditional Fourier and wavelet bases. This framework also provides a Bayesian solution to the problems of image denoising and filling in of missing pixels. We demonstrate that the results obtained by applying the learned bases to these problems are improved over those obtained with traditional techniques. © 1999 Optical Society of America [S07403232(99)031075] OCIS codes: 000.5490, 100.2960, 100.3010.