Solving Least Squares Problems

Lawson, Charles L.; Hanson, Richard J.

doi:10.1137/1.9781611971217

Cited by 4,435 publications

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“…T 2 distributions were created using 120 T 2 times logarithmically spaced between 0.5 × first echo time and 2 × last echo time, resulting in a range of 5-2240 ms. NNLS was used to fit a basis of T 2 times to the decays [26,5] and regularization was performed using a generalized cross-validation approach [27,28].…”

Section: Resultsmentioning

confidence: 99%

Temporal phase correction of multiple echo T2 magnetic resonance images

Bjarnason

Laule

Bluman

et al. 2013

Journal of Magnetic Resonance

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Typically, magnetic resonance imaging (MRI) analysis is performed on magnitude data, and multiple echo T 2 data consist of numerous images of the same slice taken with different echo spacing, giving voxel-wise temporal sampling of the noise as the signals decay according to T 2 relaxation. Magnitude T 2 decay data has Rician distributed noise which is characterized by a change in the noise distribution from Gaussian, through a transitional region, to Rayleigh as the signal to noise ratio decreases with increasing echo time. Non-Gaussian noise distributions may produce errors in the commonly applied non-negative least squares (NNLS) algorithm that is used to assess multiple echo decays for compartmentalized water environments through the creation of T 2 distributions. Typically, Gaussian noise is sought by performing spatial-based phase correction on the MRI data; however, these methods cannot capitalize on the temporal information available from multiple echo T 2 acquisitions. Here we describe a temporal phase correction (TPC) algorithm that utilizes the temporal noise information available in multiple echo T 2 acquisitions to put the relevant decay information in the Real portion of the decay data and leave only noise in the Imaginary portion. We apply this TPC algorithm to create real-valued multiple echo T 2 data from human subjects measured at 1.5 T. We show that applying TPC causes changes in the T 2 distribution estimates; notably the possible resolution of separate extracellular and intracellular water environments, and the disappearance of the commonly labeled cerebrospinal fluid peak, which might be an artefact observed in many previously published multiple echo T 2 analyses.

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Section: Resultsmentioning

confidence: 99%

Temporal phase correction of multiple echo T2 magnetic resonance images

Bjarnason

Laule

Bluman

et al. 2013

Journal of Magnetic Resonance

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“…One problem with the standard LS method of branch length estimation (e.g., Rzhetsky and Nei 1993) is that some branch length estimates may become negative. We therefore suggest that the ordinary LS method with the constraint of nonnegative branches be used (e.g., Lawson and Hanson 1974;Felsenstein 1995).…”

Section: Distance Methodsmentioning

confidence: 99%

Accuracies of ancestral amino acid sequences inferred by the parsimony, likelihood, and distance methods

1997

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Abstract. Information about protein sequences of ancestral organisms is important for identifying critical amino acid substitutions that have caused the functional change of proteins in evolution. Using computer simulation, we studied the accuracy of ancestral amino acids inferred by two currently available methods (maximumparsimony [MP] and maximum-likelihood [ML] methods) in addition to a distance method, which was newly developed in this paper. All three methods give reliable inference when the divergence of amino acid sequences is low. When the extent of sequence divergence is high, however, the ML and distance methods give more accurate results than the MP method, particularly when the phylogenetic tree includes long branches. The accuracy of inferred ancestral amino acids does not change very much when a few present-day sequences are added or eliminated. When an incorrect model of amino acid substitution is used for the ML and distance methods, the accuracy decreases, but it is still higher than that for the MP method. When the tree topology used is partially incorrect, the accuracy in the correct part of the tree is virtually unaffected. The posterior probability of inferred ancestral amino acids computed by the ML and distance methods is an unbiased estimate of the true probability when a correct substitution model is used but may become an overestimate when a simpler model is used.

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“…Through computer simulation, the performance of the proposed method, abbreviated as Twomey, was examined both qualitatively and quantitatively in comparison with lead field normalized minimum norm approach (also known as "weighted minimum norm"; Lawson and Hanson, 1974) and the 90% fMRI-constrained Wiener estimation, denoted as WMN and Wiener, respectively. The simulation was conducted on a realistic-geometry BEM-model reconstructed from the T1-weighted MRI images of a human subject (128 slices, 256×256 pixels; 1.6 mm thickness, 1.17×1.17 mm in-plane resolution).…”

Section: Computer Simulationmentioning

confidence: 99%

Effects of fMRI–EEG mismatches in cortical current density estimation integrating fMRI and EEG: A simulation study

Li¹,

Kecman²,

He³

2006

Clinical Neurophysiology

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Objective-Multimodal functional neuroimaging by combining functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) has been studied to achieve high-resolution reconstruction of the spatiotemporal cortical current density (CCD) distribution. However, mismatches between these two imaging modalities may occur due to their different underlying mechanisms. The aim of the present study is to investigate the effects of different types of fMRI-EEG mismatches, including fMRI invisible sources, fMRI extra regions and fMRI displacement, on the fMRI-constrained cortical imaging in a computer simulation based on realistic-geometry boundary-element-method (BEM) model. Methods-Two methods have been adopted to integrate the synthetic fMRI and EEG data for CCD imaging. In addition to the well-known 90% fMRI-constrained Wiener filter approach (Liu AK, Belliveau JW and Dale AM, PNAS, 95: 8945-8950, 1998), we propose a novel two-step algorithm (referred to as "Twomey algorithm") for fMRI-EEG integration. In the first step, a "hard" spatial prior derived from fMRI is imposed to solve the EEG inverse problem with a reduced source space; in the second step, the fMRI constraint is removed and the source estimate from the first step is reentered as the initial guess of the desired solution into an EEG least squares fitting procedure with Twomey regularization. Twomey regularization is a modified Tikhonov technique that attempts to simultaneously minimize the distance between the desired solution and the initial estimate, and the residual errors of fitness to EEG data. The performance of the proposed Twomey algorithm has been evaluated both qualitatively and quantitatively along with the lead-field normalized minimum norm (WMN) and the 90% fMRI-weighted Wiener filter approach, under repeated and randomized source configurations. Point spread function (PSF) and localization error (LE) are used to measure the performance of different imaging approaches with or without a variety of fMRI-EEG mismatches.Results-The results of the simulation show that the Twomey algorithm can successfully reduce the PSF of fMRI invisible sources compared to the Wiener estimation, without losing the merit of having much lower PSF of fMRI visible sources relative to the WMN solution. In addition, the existence of fMRI extra sources does not significantly affect the accuracy of the fMRI-EEG integrated CCD estimation for both the Wiener filter method and the proposed Twomey algorithm, while the Twomey algorithm may further reduce the chance of occurring spurious sources in the extra fMRI regions. The fMRI displacement away from the electrical source causes enlarged localization error in the imaging results of both the Twomey and Wiener approaches, while Twomey gives smaller LE than Wiener with the fMRI displacement ranging from 1-cm to 2-cm. With less than 2-cm fMRI displacement, the LEs for the Twomey and Wiener approaches are still smaller than in the WMN solution. Conclusions-The present study suggests that the presence of fMRI invi...

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Solving Least Squares Problems

Cited by 4,435 publications

References 0 publications

Temporal phase correction of multiple echo T2 magnetic resonance images

Temporal phase correction of multiple echo T2 magnetic resonance images

Accuracies of ancestral amino acid sequences inferred by the parsimony, likelihood, and distance methods

Effects of fMRI–EEG mismatches in cortical current density estimation integrating fMRI and EEG: A simulation study

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