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391
A generalized Gaussian image model for edgepreserving MAP estimation
 IEEE Trans. on Image Processing
, 1993
"... Absfrucf We present a Markov random field model which allows realistic edge modeling while providing stable maximum a posteriori MAP solutions. The proposed model, which we refer to as a generalized Gaussian Markov random field (GGMRF), is named for its similarity to the generalized Gaussian distri ..."
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Cited by 288 (35 self)
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Absfrucf We present a Markov random field model which allows realistic edge modeling while providing stable maximum a posteriori MAP solutions. The proposed model, which we refer to as a generalized Gaussian Markov random field (GGMRF), is named for its similarity to the generalized Gaussian distribution used in robust detection and estimation. The model satisifies several desirable analytical and computational properties for MAP estimation, including continuous dependence of the estimate on the data, invariance of the character of solutions to scaling of data, and a solution which lies at the unique global minimum of the U posteriori loglikeihood function. The GGMRF is demonstrated to be useful for image reconstruction in lowdosage transmission tomography. I.
Restoration of a Single Superresolution Image from Several Blurred, Noisy, and Undersampled Measured Images
, 1997
"... The three main tools in the single image restoration theory are the maximum likelihood (ML) estimator, the maximum a posteriori probability (MAP) estimator, and the set theoretic approach using projection onto convex sets (POCS). This paper utilizes the above known tools to propose a unified methodo ..."
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Cited by 247 (20 self)
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The three main tools in the single image restoration theory are the maximum likelihood (ML) estimator, the maximum a posteriori probability (MAP) estimator, and the set theoretic approach using projection onto convex sets (POCS). This paper utilizes the above known tools to propose a unified methodology toward the more complicated problem of superresolution restoration. In the superresolution restoration problem, an improved resolution image is restored from several geometrically warped, blurred, noisy and downsampled measured images. The superresolution restoration problem is modeled and analyzed from the ML, the MAP, and POCS points of view, yielding a generalization of the known superresolution restoration methods. The proposed restoration approach is general but assumes explicit knowledge of the linear space and timevariant blur, the (additive Gaussian) noise, the different measured resolutions, and the (smooth) motion characteristics. A hybrid method combining the simplicity of the ML and the incorporation of nonellipsoid constraints is presented, giving improved restoration performance, compared with the ML and the POCS approaches. The hybrid method is shown to converge to the unique optimal solution of a new definition of the optimization problem. Superresolution restoration from motionless measurements is also discussed. Simulations demonstrate the power of the proposed methodology.
Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods
, 1994
"... This document is the electronic version of the 2nd edition of the Templates book, which is available for purchase from the Society for Industrial and Applied ..."
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Cited by 233 (5 self)
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This document is the electronic version of the 2nd edition of the Templates book, which is available for purchase from the Society for Industrial and Applied
Preconditioning techniques for large linear systems: A survey
 J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 161 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
A local update strategy for iterative reconstruction from projections
 IEEE Tr. Sig. Proc
, 1993
"... Iterative methods for statisticallybased reconstruction from projections are computationally costly relative to convolution backprojection, but allow useful image reconstruction from sparse and noisy data. We present a method for Bayesian reconstruction which relies on updates of single pixel value ..."
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Cited by 147 (32 self)
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Iterative methods for statisticallybased reconstruction from projections are computationally costly relative to convolution backprojection, but allow useful image reconstruction from sparse and noisy data. We present a method for Bayesian reconstruction which relies on updates of single pixel values, rather than the entire image, at each iteration. The technique is similar to GaussSeidel (GS) iteration for the solution of differential equations on finite grids. The computational cost per iteration of the GS approach is found to be approximately equal to that of gradient methods. For continuously valued images, GS is found to have significantly better convergence at modes representing high spatial frequencies. In addition, GS is well suited to segmentation when the image is constrained to be discretely valued. We demonstrate that Bayesian segmentation using GS iteration produces useful estimates at much lower signaltonoise ratios than required for continuously valued reconstruction. This paper includes analysis of the convergence properties of gradient ascent and GS for reconstruction from integral projections, and simulations of both maximumlikelihood and maximum a posteriori cases.
A unified approach to statistical tomography using coordinate descent optimization
 IEEE Trans. on Image Processing
, 1996
"... Abstract 1 Over the past ten years there has been considerable interest in statistically optimal reconstruction of image crosssections from tomographic data. In particular, a variety of such algorithms have been proposed for maximum a posteriori (MAP) reconstruction from emission tomographic data. ..."
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Cited by 131 (26 self)
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Abstract 1 Over the past ten years there has been considerable interest in statistically optimal reconstruction of image crosssections from tomographic data. In particular, a variety of such algorithms have been proposed for maximum a posteriori (MAP) reconstruction from emission tomographic data. While MAP estimation requires the solution of an optimization problem, most existing reconstruction algorithms take an indirect approach based on the expectation maximization (EM) algorithm. In this paper we propose a new approach to statistically optimal image reconstruction based on direct optimization of the MAP criterion. The key to this direct optimization approach is greedy pixelwise computations known as iterative coordinate decent (ICD). We show that the ICD iterations require approximately the same amount of computation per iteration as EM based approaches, but the new method converges much more rapidly (in our experiments typically 5 iterations). Other advantages of the ICD method are that it is easily applied to MAP estimation of transmission tomograms, and typical convex constraints, such as positivity, are simply incorporated.
Highly accurate optic flow computation with theoretically justified warping
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2006
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Mean and Variance of Implicitly Defined Biased Estimators (such as Penalized Maximum Likelihood): Applications to Tomography
 IEEE Tr. Im. Proc
, 1996
"... Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear leastsquares estimation. For such estimators, exact analyti ..."
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Cited by 99 (30 self)
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Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear leastsquares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore investigators usually resort to numerical simulations to examine properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson sta...
Penalized Weighted LeastSquares Image Reconstruction for Positron Emission Tomography
 IEEE TR. MED. IM
, 1994
"... This paper presents an image reconstruction method for positronemission tomography (PET) based on a penalized, weighted leastsquares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a leastsquares objective function is as approp ..."
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Cited by 97 (38 self)
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This paper presents an image reconstruction method for positronemission tomography (PET) based on a penalized, weighted leastsquares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a leastsquares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. We propose a simple databased method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive overrelaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximumlikelihood expectationmaximization (MLEM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminat...
Efficient numerical methods in nonuniform sampling theory
, 1995
"... We present a new “second generation” reconstruction algorithm for irregular sampling, i.e. for the problem of recovering a bandlimited function from its nonuniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named ..."
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Cited by 90 (10 self)
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We present a new “second generation” reconstruction algorithm for irregular sampling, i.e. for the problem of recovering a bandlimited function from its nonuniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems. The choice of ”adaptive weights” can be seen as a simple but very efficient method of preconditioning. Further substantial acceleration is achieved by utilizing the Toeplitztype structure of the system matrix. This new algorithm can handle problems of much larger dimension and condition number than have been accessible so far. Furthermore, if some gaps between samples are large, then the algorithm can still be used as a very efficient extrapolation method across the gaps.