Journal Article10.1109/TMI.2018.2870968
A Convergence Proof of MLEM and MLEM-3 With Fixed Background
Koen Salvo,Michel Defrise +1 more
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TL;DR: This paper introduces a “fundamental equality” for the MLEM complete data from which two key properties easily follow that allows it to prove in an elegant and compact way the convergence of MLEm for a forward model with fixed background.
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Abstract: Maximum likelihood expectation-maximization (MLEM) is a popular algorithm to reconstruct the activity image in positron emission tomography. This paper introduces a “fundamental equality” for the MLEM complete data from which two key properties easily follow that allows us to: 1) prove in an elegant and compact way the convergence of MLEM for a forward model with fixed background (i.e., counts such as random and scatter coincidences) and 2) generalize this proof for the MLEM-3 algorithm. Moreover, we give necessary and sufficient conditions for the solution to be unique.
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Citations
Predictive risk estimation for the expectation maximization algorithm with Poisson data
Paolo Massa,Federico Benvenuto +1 more
TL;DR: A Poisson counterpart of the Stein’s Lemma for Gaussian variables is proved, and from this result the proposed estimator is derived, showing its analogies with the well-known Stein's unbiased risk estimator valid for a quadratic loss.
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Cone-beam X-ray luminescence computed tomography based on MLEM with adaptive FISTA initial image
TL;DR: In this article , an adaptive reconstruction algorithm based on the maximum likelihood expectation estimation (MLEM) framework is proposed for CB-XLCT reconstruction from Poisson distributed projections, in which the image reconstructed by fast iterative shrinkage-thresholding algorithm (FISTA) is used as the initial image for MLEM iterations to improve reconstruction accuracy.
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A simple convergence proof of the ML-EM algorithm in the presence of background emission
Mario Bertero,Christine De Mol +1 more
TL;DR: This paper presents an alternative convergence proof, which it is deemed simpler, in a deterministic framework and using only basic tools from convex analysis and optimization theory.
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Predictive risk estimation for the Expectation Maximization algorithm with Poisson data
Paolo Massa,Federico Benvenuto +1 more
TL;DR: In this article, a novel estimator of the predictive risk with Poisson data, when the loss function is the Kullback-Leibler divergence, was introduced to define a regularization parameter's choice rule for the Expectation Maximization (EM) algorithm.
Recent advances in variable metric first-order methods
Silvia Bonettini,Federica Porta,Marco Prato,Simone Rebegoldi,Valeria Ruggiero,Luca Zanni +5 more
- 27 Nov 2019
TL;DR: This work deeply analyze a possible way to include a variable metric in first-order methods for the minimization of a functional which can be expressed as the sum of a differentiable term and a nondifferentiable one.
References
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The EM algorithm and extensions
Geoffrey J. McLachlan,Thriyambakam Krishnan +1 more
- 15 Nov 1996
TL;DR: The EM Algorithm and Extensions describes the formulation of the EM algorithm, details its methodology, discusses its implementation, and illustrates applications in many statistical contexts, opening the door to the tremendous potential of this remarkably versatile statistical tool.
Maximum Likelihood Reconstruction for Emission Tomography
L. A. Shepp,Y. Vardi +1 more
TL;DR: In this paper, the authors proposed a more accurate general mathematical model for ET where an unknown emission density generates, and is to be reconstructed from, the number of counts n*(d) in each of D detector units d. Within the model, they gave an algorithm for determining an estimate? of? which maximizes the probability p(n*|?) of observing the actual detector count data n* over all possible densities?.
4.5K
Bayesian-Based Iterative Method of Image Restoration
TL;DR: An iterative method of restoring degraded images was developed by treating images, point spread functions, and degraded images as probability-frequency functions and by applying Bayes’s theorem.
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