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Accelerated model-based T1, T2*and proton density mapping using a Bayesian approach with automatic hyperparameter estimation

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MAGNETIC RESONANCE IN MEDICINE 2025年第2期93卷 563-583页

作者： Huang, Shuai Lah, James J. Allen, Jason W. Qiu, Deqiang Emory Univ Dept Radiol & Imaging Sci Atlanta GA 30322 USA Emory Univ Dept Neurol Atlanta GA 30322 USA Indiana Univ Dept Radiol & Imaging Sci Indianapolis IN 46204 USA

Purpose: To achieve automatic hyperparameter estimation for the model-based recovery of quantitative MR maps from undersampled data, we propose a Bayesian formulation that incorporates the signal model and sparse priors among multiple image contrasts. Theory: We introduce a novel approximate message passing framework "AMP-PE" that enables the automatic and simultaneous recovery of hyperparameters and quantitative maps. Methods: We employed the variable-flip-angle method to acquire multi-echo measurements using gradient echo sequence. We explored undersampling schemes to incorporate complementary sampling patterns across different flip angles and echo times. We further compared AMP-PE with conventional compressed sensing approaches such as the l(1)-norm minimization, PICS and other model-based approaches such as GraSP, MOBA. Results: Compared to conventional compressed sensing approaches such as the l(1)-norm minimization and PICS, AMP-PE achieved superior reconstruction performance with lower errors in T*(2) mapping and comparable performance in T-1 and proton density mappings. When compared to other model-based approaches including GraSP and MOBA, AMP-PE exhibited greater robustness and outperformed GraSP in reconstruction error. AMP-PE offers faster speed than MOBA. AMP-PE performed better than MOBA at higher sampling rates and worse than MOBA at a lower sampling rate. Notably, AMP-PE eliminates the need for hyperparameter tuning, which is a requisite for all the other approaches. Conclusion: AMP-PE offers the benefits of model-based recovery with the additional key advantage of automatic hyperparameter estimation. It works adeptly in situations where ground-truth is difficult to obtain and in clinical environments where it is desirable to automatically adapt hyperparameters to individual protocol, scanner and patient.

关键词： approximate message passing compressed sensing complementary undersampling pattern hyperparameter estimation multi-echo gradient echo sequence quantitative MRI Poisson disc variable density variable flip angle

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hyperparameter estimation in image restoration problems with partially known blurs

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OPTICAL ENGINEERING 2002年第8期41卷 1845-1854页

作者： Galatsanos, NP Mesarovic, VZ Molina, R Katsaggelos, AK Mateos, J IIT Dept Elect & Comp Engn Chicago IL 60616 USA Crystal Semicond Corp Austin TX 78744 USA Univ Granada Dept Ciencias Computac & IA E-18071 Granada Spain Northwestern Univ Dept Elect & Comp Engn Evanston IL 60208 USA

This work is motivated by the observation that it is not possible to reliably estimate simultaneously all the necessary hyperparameters in an image restoration problem when the point-spread function is assumed to be the sum of a known deterministic and an unknown random component. To solve this problem we propose to use gamma hyperpriors for the unknown hyperparameters. Two iterative algorithms that simultaneously restore the image and estimate the hyperparameters are derived, based on the application of evidence analysis within the hierarchical Bayesian framework. Numerical experiments are presented that show the benefits of introducing hyperpriors for this problem. (C) 2002 Society of Photo-Optical Instrumentation Engineers.

关键词： image restoration partially known blur hyperparameter estimation gamma hyperpriors Bayesian estimation

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hyperparameter estimation for uncertainty quantification in mesoscale carbon dioxide inversions

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TELLUS SERIES B-CHEMICAL AND PHYSICAL METEOROLOGY 2013年第1期65卷 Article: 20894页

作者： Wu, Lin Bocquet, Marc Chevallier, Frederic Lauvaux, Thomas Davis, Kenneth IPSL CEA CNRS UVSQ Lab Sci Climat & Environm Gif Sur Yvette France Univ Paris Est CEREA Joint Lab Ecole Ponts ParisTech EDF R&D Marne La Vallee France Paris Rocquencourt Res Ctr INRIA Paris France Penn State Univ Dept Meteorol State Coll PA USA

Uncertainty quantification is critical in the inversion of CO2 surface fluxes from atmospheric concentration measurements. Here, we estimate the main hyperparameters of the error covariance matrices for a priori fluxes and CO2 concentrations, that is, the variances and the correlation lengths, using real, continuous hourly CO2 concentration data in the context of the Ring 2 experiment of the North American Carbon Program Mid Continent Intensive. Several criteria, namely maximum likelihood (ML), general cross-validation (GCV) and chi(2) test are compared for the first time under a realistic setting in a mesoscale CO2 inversion. It is shown that the optimal hyperparameters under the ML criterion assure perfect chi(2) consistency of the inverted fluxes. Inversions using the ML error variances estimates rather than the prescribed default values are less weighted by the observations, because the default values underestimate the model-data mismatch error, which is assumed to be dominated by the atmospheric transport error. As for the spatial correlation length in prior flux errors, the Ring 2 network is sparse for GCV, and this method fails to reach an optimum. In contrast, the ML estimate (e. g. an optimum of 20 km for the first week of June 2007) does not support long spatial correlations that are usually assumed in the default values.

关键词： hyperparameter estimation uncertainty quantification mesoscale carbon dioxide inversions

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hyperparameter estimation Based on Gaussian Process and its Application in Injection Molding

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International Conference on Mechatronics and Materials Processing (ICMMP 2011)

作者： Ma, Junyan Liao, Xiaoping Xia, Wei Yan, Xuelian Guangxi Univ Sch Mech & Engn Nanning 530004 Guangxi Provinc Peoples R China

ISBN: (纸本)9783037852385

As a powerful modeling tool, Gaussian process (GP) employs a Bayesian statistics approach and adopts a highly nonlinear regression technique for general scientific and engineering tasks. In the first step of constructing Gaussian process model is to estimate the best value of the hyperparameter which turned to be used in the second step where a satisfactory nonlinear model was fitted. In this paper, a modified Wolfe line search approach for hyperparameters estimation by maximizing the marginal likelihood based on conjugate gradient method is proposed. And then we analyze parameter correlation according to the value of hyperparameters to control the warpage which is a main defect for a thin shell structure part in injection molding.

关键词： Gaussian Process hyperparameter estimation Modified Wolfe Line Search Parameter Correlation Injection Molding

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hyperparameter estimation of a variational model using a stochastic gradient method

Hyperparameter estimation of a variational model using a sto...

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Conference on Bayesian Inference for Inverse Problems

作者： Zerubia, J Blanc-Féraud, L UNSA INRIA CNRS Ariana Project F-06902 Sophia Antipolis France

ISBN: (纸本)0819429147

This paper addresses the problem of hyperparameter estimation for image deconvolution. The inversion is obtained by minimizing a criterion derived from a variational approach. The regularization term is based on the well-known phi-function model. This model induces linear and nonlinear parameters which have to be estimated. The estimation requires some knowledge about the model behaviour. Herein, we present qualitative studies showing the presence of phase transitions. For incomplete data and in the case of Gibbs distributions, Maximum Likelihood is too hard to compute due to the intractability of the partition function. More precisely, to get the gradient of the functional, we need to sample at the same time both from the prior distribution and the posterior distribution. In the case of deblurring, sampling from posterior distribution is impossible due to the support of the Point Spread Function (PSF). To avoid this difficulty, an adaptive method is presented. We use Laksmanan & Derin method based on Generalized Maximum Likelihood estimation (GML). The maximization with respect to (w.r.t.) the unknown image and the hyperparameters is conducted alternately Hence at each iteration, we use a stochastic gradient algorithm to compute the Maximum Likelihood w.r.t. the hyperparameters. In particular, we have tackled the problem of a nonlinear parameter estimation, this parameter being related to the scaling factor in the phi-function model. Finally the image is estimated using a fast deterministic algorithm (ARTUR) already studied in a previous paper. Unfortunately, we cannot ensure the global convergence of the algorithm in this case, as it optimizes alternately with respect to both arguments of the functional. However we obtain experimental convergence on both synthetic and real data (SPOT image).

关键词： hyperparameter estimation nonlinear estimation stochastic gradient variational method

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hyperparameter estimation using hyperpriors for hierarchical Bayesian image restoration from partially-known blurs

Hyperparameter estimation using hyperpriors for hierarchical...

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Conference on Bayesian Inference for Inverse Problems

作者： Galatsanos, NP Mesarovic, VZ Molina, R Katsaggelos, AK IIT Dept Elect & Comp Engn Chicago IL 60616 USA

ISBN: (纸本)0819429147

In this paper we examine the problem of estimating the hyperparameters in image restoration when the point-spread function (PSF) of the degradation system is partially known. For this problem the PSF is assumed to be the sum of a known deterministic and an unknown random component. In this paper two iterative algorithms are proposed that simultaneously restore the image and estimate the hyperparameters of the restoration filter using hyperpriors. These algorithms are based on evidence analysis (EA) within the hierarchical Bayesian framework. This work was motivated by the observation that it is not possible to simultaneously estimate all the necessary hyperparameters for this problem without any prior knowledge about them. More specificaly, we observed in our previous work that we cannot estimate accurately at the same time the hyperparameters that account for the PSF and the additive in the data noise. Hyperpriors introduce prior knowledge about the hyperparameters and thus facilitate this estimation problem. The proposed iterative algorithms can be derived in the discrete Fourier transform (DFT) domain, therefore, they are computationally efficient even for large images. Numerical experiments are presented where the benefits of introducing hyperpriors are demonstrated.

关键词： image restoration partially-known blur hyperparameter estimation gamma hyperpriors Bayesian estimation

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A cumulant expansion technique for simultaneous Markov random Field image restoration and hyperparameter estimation

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INTERNATIONAL JOURNAL OF COMPUTER VISION 2000年第3期37卷 275-293页

作者： Sigelle, M Ecole Natl Super Telecommun Dept Traitement Signal & Images 46 Rue Barrault F-75634 Paris 13 France.

We investigate hyperparameter estimation for incomplete data in Markov random Field image restoration. Assuming linear dependence of energies with respect to hyperparameters framework, we use a cumulant expansion technique widely known in Statistical Physics and Signal Processing. New insight is given on Maximum Likelihood estimation of hyperparameters of the prior, regularization and contour probability distribution functions (pdfs) for an explicit joint boundary-pixel process aimed to preserve discontinuities. In particular the case where the prior regularization potential is an homogeneous function of pixels is fully analyzed. A Generalized Stochastic Gradient (GSG) algorithm with a fast sampling technique is devised aiming to achieve simultaneous hyperparameter estimation and pixel restoration. Image restoration performances of Posterior Mean performed during GSG convergence and of Simulated Annealing performed after GSG convergence are compared experimentally. Results and perspectives are given.

关键词： Markov random fields image restoration boundary process hyperparameter estimation stochastic gradient cumulants

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Efficient iterative methods for hyperparameter estimation in large-scale linear inverse problems

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2024年第6期50卷 1-33页

作者： Hall-Hooper, Khalil A. Saibaba, Arvind K. Chung, Julianne Miller, Scot M. North Carolina State Univ Dept Math 2311 Sullivan Dr Raleigh NC 27695 USA Emory Univ Dept Math 400 Dowman Dr Atlanta GA 30322 USA Johns Hopkins Univ Dept Environm Hlth & Engn 3400 N Charles St Baltimore MD 21218 USA

We study Bayesian methods for large-scale linear inverse problems, focusing on the challenging task of hyperparameter estimation. Typical hierarchical Bayesian formulations that follow a Markov Chain Monte Carlo approach are possible for small problems but are not computationally feasible for problems with a very large number of unknown inverse parameters. In this work, we describe an empirical Bayes (EB) method to estimate hyperparameters that maximize the marginal posterior, i.e., the probability density of the hyperparameters conditioned on the data, and then we use the estimated hyperparameters to compute the posterior of the unknown inverse parameters. For problems where the computation of the square root and inverse of prior covariance matrices are not feasible, we describe an approach based on the generalized Golub-Kahan bidiagonalization to approximate the marginal posterior and seek hyperparameters that minimize the approximate marginal posterior. Numerical results from seismic and atmospheric tomography demonstrate the accuracy, robustness, and potential benefits of the proposed approach.

关键词： Bayesian inverse problems hyperparameter estimation Golub-Kahan bidiagonalization Low-rank approximation

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hyperparameter estimation for Sparse Bayesian Learning Models\ast

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SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION 2024年第3期12卷 759-787页

作者： Yu, Feng Shen, Lixin Song, Guohui Univ Minnesota Dept Math Minneapolis MN 55455 USA Syracuse Univ Dept Math Syracuse NY 13244 USA Old Dominion Univ Dept Math & Stat Norfolk VA 23529 USA

Sparse Bayesian learning (SBL) models are extensively used in signal processing and machine learning for promoting sparsity through hierarchical priors. The hyperparameters in SBL models are crucial for the model's performance, but they are often difficult to estimate due to the nonconvexity and the high-dimensionality of the associated objective function. This paper presents a comprehensive framework for hyperparameter estimation in SBL models, encompassing well-known algorithms such as the expectation-maximization, MacKay, and convex bounding algorithms. These algorithms are cohesively interpreted within an alternating minimization and linearization (AML) paradigm, distinguished by their unique linearized surrogate functions. Additionally, a novel algorithm within the AML framework is introduced, showing enhanced efficiency, especially under low signal noise ratios. This is further improved by a new alternating minimization and quadratic approximation paradigm, which includes a proximal regularization term. The paper substantiates these advancements with thorough convergence analysis and numerical experiments, demonstrating the algorithm's effectiveness in various noise conditions and signal-to-noise ratios.

关键词： sparse Bayesian learning hyperparameter estimation alternating minimization

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Simultaneous image restoration and hyperparameter estimation by a cumulant analysis

Simultaneous image restoration and hyperparameter estimation...

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Conference on Bayesian Inference for Inverse Problems

作者： Sigelle, M Ecole Natl Super Telecommun F-75634 Paris 13 France

ISBN: (纸本)0819429147

hyperparameter estimation for incomplete data in Markov random Field image restoration is investigated. Assuming linear dependence of energies w.r.t. hyperparameters framework, we use a classical cumulant expansion technique for Maximum Likelihood estimation of hyperparameters of the prior, pixel regularization probability density function (pdf). The particular case where the prior potential is an homogeneous function of pixels is fully analyzed. This approach is then extended to an explicit joint boundary-pixel process aimed to preserve discontinuities. A Generalized Stochastic Gradient (GSG) algorithm with a fast sampling technique is devised aiming to achieve simultaneous hyperparameter estimation, pixel and boundary restoration. Image restoration performances of Posterior Mean performed during GSG convergence and of Simulated Annealing performed after GSG convergence are compared experimentally. Results and perspectives are given.

关键词： Markov random fields image restoration hyperparameter estimation stochastic gradient cumulants

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