A family of alternating minimization algorithms for finding maximum-likelihood estimates of attenuation functions in transmission X-ray tomography is described. The model from which the algorithms are derived includes...
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A family of alternating minimization algorithms for finding maximum-likelihood estimates of attenuation functions in transmission X-ray tomography is described. The model from which the algorithms are derived includes polyenergetic photon spectra, background events, and nonideal point spread functions. The maximum-likelihood image reconstruction problem is reformulated as a double minimization of the I-divergence. A novel application of the convex decomposition lemma results in an alternatingminimization algorithm that monotonically decreases the objective function. Each step of the minimization is in closed form. The family of algorithms includes variations that use ordered subset techniques for increasing the speed of convergence. Simulations demonstrate the ability to correct the cupping artifact due to beam hardening and the ability to reduce streaking artifacts that arise from beam hardening and background events.
In this paper, we propose algorithm to restore blurred and noisy images based on the discretized total variation minimization technique. The proposed method is based on an alternating technique for image deblurring an...
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ISBN:
(纸本)9781538695760
In this paper, we propose algorithm to restore blurred and noisy images based on the discretized total variation minimization technique. The proposed method is based on an alternating technique for image deblurring and denoising. Start by finding an approximate image using a Tikhonov regularization method. This corresponds to a deblurring process with possible artifacts and noise remaining. In the denoising step, we use fast iterative shrinkage-thresholding algorithm (SFISTA) or fast gradient-based algorithm (FGP). Besides, we prove the convergence of the proposed algorithm. Numerical results demonstrate the efficiency and viability of the proposed algorithm to restore the degraded images.
The problem of image formation for X-ray transmission tomography is formulated as a statistical inverse problem. The maximum likelihood estimate of the attenuation function is sought. Using convex optimization methods...
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ISBN:
(纸本)0819452025
The problem of image formation for X-ray transmission tomography is formulated as a statistical inverse problem. The maximum likelihood estimate of the attenuation function is sought. Using convex optimization methods, maximizing the log-likelihood functional is equivalent to a double minimization of I-divergence, one of the minimizations being over the attenuation function. Restricting the minimization over the attenuation function to a coarse grid component forms the basis for a multigrid algorithm that is guaranteed to monotonically decrease the I-divergence at every iteration on every scale.
We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x, y) = f(x) + Q(x, y) + g(y), where f and g are proper lower semicontinuous func...
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We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x, y) = f(x) + Q(x, y) + g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Lojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to "metrically regular" problems. Our main result can be stated as follows: If L has the Kurdyka-Lojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to Q(x, y) = parallel to x - y parallel to(2) and to f, g indicator functions, the algorithm is an alternating projection mehod ( a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with "regular" intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l(1) algorithm for compressive sensing and an application to rank reduction problems.
alternating minimization algorithms are developed to solve two variational models, for image colorization based on chromaticity and brightness color system. Image colorization is a task of inpainting color from a smal...
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alternating minimization algorithms are developed to solve two variational models, for image colorization based on chromaticity and brightness color system. Image colorization is a task of inpainting color from a small region of given color information. While the brightness is defined on the entire image domain, the chromaticity components are only given on a small subset of image domain. The first model is the edge-weighted total variation (TV) and the second one is the edge-weighted harmonic model that proposed by Kang and March (IEEE Trans. Image Proc. 16(9):2251-2261, 2007). Both models minimize a functional with the unit sphere constraints. The proposed methods are based on operator splitting, augmented Lagrangian, and alternating direction method of multipliers, where the computations can take advantage of multi-dimensional shrinkage and fast Fourier transform under periodic boundary conditions. Convergence analysis of the sequence generated by the proposed methods to a Karush-Kahn-Tucker point and a minimizer of the edge-weighted TV model are established. In several examples, we show the effectiveness of the new methods to colorize gray-level images, where only small patches of colors are given. Moreover, numerical comparisons with quadratic penalty method, augmented Lagrangian method, time marching, and/or accelerated time marching algorithms demonstrate the efficiency of the proposed methods.
We propose an alternatingminimization (AM) algorithm for estimating attenuation functions in x-ray transmission tomography using priors that promote sparsity in the pixel/voxel differences domain. As opposed to stand...
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ISBN:
(纸本)9781628415032
We propose an alternatingminimization (AM) algorithm for estimating attenuation functions in x-ray transmission tomography using priors that promote sparsity in the pixel/voxel differences domain. As opposed to standard maximum-a-posteriori (MAP) estimation, we use the automatic relevance determination (ARD) framework. In the ARD approach, sparsity (or compressibility) is promoted by introducing latent variables which serve as the weights of quadratic penalties, with one weight for each pixel/voxel;these weights are then automatically learned from the data. This leads to an algorithm where the quadratic penalty is reweighted in order to effectively promote sparsity. In addition to the usual object estimate, ARD also provides measures of uncertainty (posterior variances) which are used at each iteration to automatically determine the trade-off between data fidelity and the prior, thus potentially circumventing the need for any tuning parameters. We apply the convex decomposition lemma in a novel way and derive a separable surrogate function that leads to a parallel algorithm. We propose an extension of branchless distance-driven forward/back-projections which allows us to considerably speed up the computations associated with the posterior variances. We also study the acceleration of the algorithm using ordered subsets.
Three-dimensional image reconstruction for scanning baggage in security applications is becoming increasingly important. Compared to medical x-ray imaging, security imaging systems must be designed for a greater varie...
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ISBN:
(纸本)9781628414912
Three-dimensional image reconstruction for scanning baggage in security applications is becoming increasingly important. Compared to medical x-ray imaging, security imaging systems must be designed for a greater variety of objects. There is a lot of variation in attenuation and nearly every bag scanned has metal present, potentially yielding significant artifacts. Statistical iterative reconstruction algorithms are known to reduce metal artifacts and yield quantitatively more accurate estimates of attenuation than linear methods. For iterative image reconstruction algorithms to be deployed at security checkpoints, the images must be quantitatively accurate and the convergence speed must be increased dramatically. There are many approaches for increasing convergence;two approaches are described in detail in this paper. The first approach includes a scheduled change in the number of ordered subsets over iterations and a reformulation of convergent ordered subsets that was originally proposed by Ahn, Fessler et. al.(1) The second approach is based on varying the multiplication factor in front of the additive step in the alternatingminimization (AM) algorithm, resulting in more aggressive updates in iterations. Each approach is implemented on real data from a SureScan (TM) x1000 Explosive Detection System* and compared to straightforward implementations of the alternatingminimization algorithm of O'Sullivan and Benac(2) with a Huber-typeedge-preserving penalty, originally proposed by Lange.(3)
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