Results 1  10
of
12
ConjugateGradient Preconditioning Methods for ShiftVariant PET Image Reconstruction
 IEEE Tr. Im. Proc
, 2002
"... Gradientbased iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian mat ..."
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Cited by 51 (21 self)
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Gradientbased iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shiftinvariant, i.e. for those with approximately blockToeplitz or blockcirculant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantumlimited optical imaging, the Hessian of the weighted leastsquares objective function is quite shiftvariant, and circulant preconditioners perform poorly. Additional shiftvariance is caused by edgepreserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shiftvariant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugategradient (CG) iteration. We also propose a new efficient method for the linesearch step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.
Iterative tomographic image reconstruction using Fourierbased forward and back projectors
 IEEE Trans. Med. Imag
, 2004
"... Fourierbased reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourierbased reprojection methods. We apply a minmax interpolation method for the nonuniform fast Fourier transform (NUFFT) t ..."
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Cited by 23 (4 self)
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Fourierbased reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourierbased reprojection methods. We apply a minmax interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the minmax NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.
Fast Bayesian Matching Pursuit
"... Abstract—A lowcomplexity recursive procedure is presented for minimum mean squared error (MMSE) estimation in linear regression models. A Gaussian mixture is chosen as the prior on the unknown parameter vector. The algorithm returns both an approximate MMSE estimate of the parameter vector and a se ..."
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Cited by 20 (2 self)
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Abstract—A lowcomplexity recursive procedure is presented for minimum mean squared error (MMSE) estimation in linear regression models. A Gaussian mixture is chosen as the prior on the unknown parameter vector. The algorithm returns both an approximate MMSE estimate of the parameter vector and a set of high posterior probability mixing parameters. Emphasis is given to the case of a sparse parameter vector. Numerical simulations demonstrate estimation performance and illustrate the distinctions between MMSE estimation and MAP model selection. The set of high probability mixing parameters not only provides MAP basis selection, but also yields relative probabilities that reveal potential ambiguity in the sparse model. 1 I.
Grouped Coordinate Descent Algorithms for Robust EdgePreserving Image Restoration
 in Proc. SPIE 3071, Im. Recon. and Restor. II
, 1997
"... We present a new class of algorithms for edgepreserving restoration of piecewisesmooth images measured in nonGaussian noise under shiftvariant blur. The algorithms are based on minimizing a regularized objective function, and are guaranteed to monotonically decrease the objective function. The al ..."
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Cited by 17 (11 self)
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We present a new class of algorithms for edgepreserving restoration of piecewisesmooth images measured in nonGaussian noise under shiftvariant blur. The algorithms are based on minimizing a regularized objective function, and are guaranteed to monotonically decrease the objective function. The algorithms are derived by using a combination of two previously unconnected concepts: A. De Pierro's convexity technique for optimization transfer, and P. Huber's iteration for Mestimation. Convergence to the unique global minimum is guaranteed for strictly convex objective functions. The convergence rate is very fast relative to conventional gradientbased iterations. The proposed algorithms are flexibly parallelizable, and easily accommodate nonnegativity constraints and arbitrary neighborhood structures. Implementation in Matlab is remarkably simple, requiring no cumbersome line searches or tolerance parameters. Keywords: Image restoration, nonGaussian noise, deconvolution, Bayesian meth...
Fast Bayesian Matching Pursuit: Model Uncertainty and Parameter Estimation for Sparse Linear Models
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2009
"... A lowcomplexity recursive procedure is presented for model selection and minimum mean squared error (MMSE) estimation in linear regression. Emphasis is given to the case of a sparse parameter vector and fewer observations than unknown parameters. A Gaussian mixture is chosen as the prior on the un ..."
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Cited by 17 (2 self)
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A lowcomplexity recursive procedure is presented for model selection and minimum mean squared error (MMSE) estimation in linear regression. Emphasis is given to the case of a sparse parameter vector and fewer observations than unknown parameters. A Gaussian mixture is chosen as the prior on the unknown parameter vector. The algorithm returns both a set of high posterior probability mixing parameters and an approximate MMSE estimate of the parameter vector. Exact ratios of posterior probabilities serve to reveal potential ambiguity among multiple candidate solutions that are ambiguous due to observation noise or correlation among columns in the regressor matrix. Algorithm complexity is linear in the number of unknown coefficients, the number of observations and the number of nonzero coefficients. If hyperparameters are unknown, a maximum likelihood estimate is found by a generalized expectation maximization algorithm. Numerical simulations demonstrate estimation performance and illustrate the distinctions between MMSE estimation and maximum a posteriori probability model selection.
Fast and accurate Polar Fourier transform
 Appl. Comput. Harmon. Anal.
, 2006
"... In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is pr ..."
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Cited by 17 (1 self)
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In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudoPolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudoPolar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudoPolar FFT plays the role of a halfway point—a nearlyPolar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesianbased unequallysampled FFT method to ours, both algorithms using a smallsupport interpolation and no precompensating, and show marked advantage to the use of the pseudoPolar initial grid.
Simultaneously Inpainting in Image and Transformed Domains
"... In this paper, we focus on the restoration of images that have incomplete data in either the image domain or the transformed domain or in both. The transform used can be any orthonormal or tight frame transforms such as orthonormal wavelets, tight framelets, the discrete Fourier transform, the Gabor ..."
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Cited by 9 (4 self)
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In this paper, we focus on the restoration of images that have incomplete data in either the image domain or the transformed domain or in both. The transform used can be any orthonormal or tight frame transforms such as orthonormal wavelets, tight framelets, the discrete Fourier transform, the Gabor transform, the discrete cosine transform, and the discrete local cosine transform. We propose an iterative algorithm that can restore the incomplete data in both domains simultaneously. We prove the convergence of the algorithm and derive the optimal properties of its limit. The algorithm generalizes, unifies, and simplifies the inpainting algorithm in image domains given in [8] and the inpainting algorithms in the transformed domains given in [7,16,19]. Finally, applications of the new algorithm to superresolution image reconstruction with different zooms are presented. 1
Preconditioning Methods for ShiftVariant Image Reconstruction
 in Proc. IEEE Intl. Conf. on Image Processing
, 1997
"... Preconditioning methods can accelerate the convergence of gradientbased iterative methods for tomographic image reconstruction and image restoration. Circulant preconditioners have been used extensively for shiftinvariant problems. Diagonal preconditioners offer some improvement in convergence rate ..."
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Cited by 3 (1 self)
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Preconditioning methods can accelerate the convergence of gradientbased iterative methods for tomographic image reconstruction and image restoration. Circulant preconditioners have been used extensively for shiftinvariant problems. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. For inverse problems that are approximately shiftinvariant (i.e. approximately blockToeplitz or blockcirculant Hessians), circulant or Fourierbased preconditioners can provide remarkable acceleration. However, in applications with nonuniform noise variance (such as arises from Poisson statistics in emission tomography and in quantumlimited optical imaging), the Hessian of the (penalized) weighted leastsquares objective function is quite shiftvariant, and the Fourier preconditioner performs poorly. Additional shiftvariance is caused by edgepreserving regularization methods based on nonquadratic penalty ...
Iterative Methods for Image Reconstruction
, 2008
"... These annotated slides were prepared by Jeff Fessler for attendees of the ISBI tutorial on statistical image reconstruction methods. The purpose of the annotation is to provide supplemental details, and particularly to provide extensive literature references for further study. For a fascinating hist ..."
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Cited by 3 (0 self)
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These annotated slides were prepared by Jeff Fessler for attendees of the ISBI tutorial on statistical image reconstruction methods. The purpose of the annotation is to provide supplemental details, and particularly to provide extensive literature references for further study. For a fascinating history of tomography, see [1]. For broad coverage of image science, see [2]. For further references on image reconstruction, see review papers and chapters, e.g., [3–9].
Fully 3D PET Image Reconstruction Using A Fourier Preconditioned ConjugateGradient Algorithm
 in Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf
, 1996
"... Since the data sizes in fully 3D PET imaging are very large, iterative image reconstruction algorithms must converge in very few iterations to be useful. One can improve the convergence rate of the conjugategradient (CG) algorithm by incorporating preconditioning operators that approximate the inver ..."
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Cited by 2 (1 self)
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Since the data sizes in fully 3D PET imaging are very large, iterative image reconstruction algorithms must converge in very few iterations to be useful. One can improve the convergence rate of the conjugategradient (CG) algorithm by incorporating preconditioning operators that approximate the inverse of the Hessian of the objective function. If the 3D cylindrical PET geometry were not truncated at the ends, then the Hessian of the penalized leastsquares objective function would be approximately shiftinvariant, i.e. G 0 G would be nearly blockcirculant, where G is the system matrix. We propose a Fourier preconditioner based on this shiftinvariant approximation to the Hessian. Results show that this preconditioner significantly accelerates the convergence of the CG algorithm with only a small increase in computation. I. Introduction Statistical methods for tomographic image reconstruction from fully 3D PET scans are particularly promising since the number of unknown parameters (...