Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

Diffusion Tensor Imaging in Alzheimer's disease

Attentional control and Information processing speed are central concepts in cognitive psychology and neuropsychology. Functional neuroimaging and neuropsychological assessment have depicted theoretical models considering attention as a complex and non-unitary process. One of its component processes, Attentional set-shifting ability, is commonly assessed using the Trail Making Test (TMT). Performance in the TMT decreases with increasing age in adults, Mild Cognitive Impairment (MCI) and Alzheimer’s Disease (AD). Besides, speed of information processing (SIP) seems to modulate attentional performance. While neural correlates of attentional control have been widely studied, there are few evidences about the neural substrates of SIP in these groups of patients. Different authors have suggested that it could be a property of cerebral white matter, thus, deterioration of the white matter tracts that connect brain regions related to set-shifting may underlie the age-related, MCI and AD decrease in performance. The aim of this study was to study the anatomical dissociation of attentional and speed mechanisms. Diffusion tensor imaging (DTI) provides a unique insight into the cellular integrity of the brain, offering an in vivo view into the microarchitecture of cerebral white matter. At the same time, the study of ageing, characterized by white matter decline, provides the opportunity to study the anatomical substrates speeded or slowed information processing. We hypothesized that FA values would be inversely correlated with time to completion on Parts A and B of the TMT, but not the derived scores B/A and B-A.

Aspects of a noncommutative scalar/tensor duality

We study the noncommutative massless Kalb-Ramond gauge field coupled to a dynamical U(1) gauge field in the adjoint representation together with a compensating vector field. We derive the Seiberg-Witten map and obtain the corresponding mapped action to first order in theta. The (emergent) gravity structure found in other situations is not present here. The off-shell dual scalar theory is derived and it does not coincide with the Seiberg-Witten mapped scalar theory. Dispersion relations are also discussed. The p-form generalization of the Seiberg-Witten map to order theta is also derived.

One-loop energy-momentum tensor in QED with electriclike background

We have obtained nonperturbative one-loop expressions for the mean-energy-momentum tensor and current density of Dirac's field on a constant electriclike back-round. One of the goals of this calculation is to give a consistent description of backreaction in such a theory. Two cases of initial states are considered: the vacuum state and the thermal equilibrium state. First, we perform calculations for the vacuum initial state. In the obtained expressions, we separate the contributions due to particle creation and vacuum polarization. The latter contribution,, are related to the Heisenberg-Euler Lagrangian. Then, we Study the case of the thermal initial state. Here, we separate the contributions due to particle creation, vacuum polarization, and the contributions due to the work of the external field on the particles at the initial state. All these contributions are studied in detail, in different regimes of weak and strong fields and low and high temperatures. The obtained results allow us to establish restrictions on the electric field and its duration under which QED with a strong constant electric field is consistent. Under such restrictions, one can neglect the backreaction of particles created by the electric field. Some of the obtained results generalize the calculations of Heisenberg-Euler for energy density to the case of arbitrary strong electric fields.

An evaluation of the time dependence of the anisotropy of the water diffusion tensor in acute human ischemia

We have performed MRI examinations to determine the water diffusion tensor in the brain of six patients who were admitted to the hospital within 12 h after the onset of cerebral ischemic symptoms. The examinations have been carried out immediately after admission, and thereafter at varying intervals up to 90 days post admission. Maps of the trace of the diffusion tensor, the fractional anisotropy and the lattice index, as well as maps of cerebral blood perfusion parameters, were generated to quantitatively assess the character of the water diffusion tensor in the infarcted area. In patients with significant perfusion deficits and substantial lesion volume changes, four of six cases, our measurements show a monotonic and significant decrease in the diffusion anisotropy within the ischemic lesion as a function of time. We propose that retrospective analysis of this quantity, in combination with brain tissue segmentation and cerebral perfusion maps, may be used in future studies to assess the severity of the ischemic event. (C) 1999 Elsevier Science Inc.

Determination of chemical shielding tensor of an indole carbon and application of tryptophan orientation of a membrane peptide

Using tryptophan C-13-enriched at the C-4 (C epsilon(3)) of the indole, the orientation of the C epsilon(3) chemical shift tensor relative to the C epsilon(3)-H dipolar axis was determined from the C-13 chemical shift/C-13-H-1 dipolar 2D NMR powder pattern. The principal values obtained were 208, 137 and 15 ppm with sigma(33) perpendicular to the indole plane, and sigma(11) (least shielded direction) 5 degrees off the C epsilon(3)-H bond toward C xi(3). The side off the C epsilon(3)-H bond was determined by comparing the reduced chemical shift anisotropies obtained by solid-state NMR and from molecular dynamics calculations of [4-C-13] tryptophans in gramicidin A aligned in phospholipid membranes. (C) 1999 Elsevier Science B.V. All rights reserved.

Loss of connectivity in Alzheimer's disease: an evaluation of white matter tract integrity with colour coded MR diffusion tensor imaging

A novel MRI method-diffusion tensor imaging-was used to compare the integrity of several white matter fibre tracts in patients with probable Alzheimer's disease. Relative to normal controls, patients with probable Alzheimer's disease showed a highly significant reduction in the integrity of the association white matter fibre tracts, such as the splenium of the corpus callosum, superior longitudinal fasciculus, and cingulum. By contrast, pyramidal tract integrity seemed unchanged. This novel finding is consistent with the clinical presentation of probable Alzheimer's disease, in which global cognitive decline is a more prominent feature than motor disturbance.

Completion of the isomorphous Ln(trensal) series

The CeIII, PrIII, NdIII, GdIII and YbIII complexes of the heptadentate ligand 2,2´,2´´-tris(salicylideneimino) triethylamine, H3trensal (in its trianionic form), have been synthesized and characterized structurally by X-ray crystallography. These five [Ln(trensal)] structures complete a rare isomorphous and isostructural series of lanthanoid complexes in the trigonal P–3c1 space group with a ≈ 13.1 and c ≈ 16.5 Å

R-matrices and the tensor product graph method

A systematic method for constructing trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra has been developed by the Brisbane group and its collaborators. This method has been referred to as the Tensor Product Graph Method. Here we describe applications of this method to untwisted and twisted quantum affine superalgebras.

A comparative study of telephone versus onsite completion of the WOMAC 3.0 Osteoarthritis Index

Objective. Outcome assessment in clinical trials using the Western Ontario and McMaster University (WOMAC 3.0) Osteoarthritis Index is traditionally achieved through self-administration of the Index. However, in other areas of clinical measurement, telephone administration has been shown to be a reliable method of acquiring data that are both accurate and complete. To address this issue in knee osteoarthritis (OA), we conducted a comparative study of telephone administration by interviewer of WOMAC LK3.0 versus onsite self-completion at the hospital. Methods. Fifty consenting patients with knee OA were randomized to complete the WOMAC LK3.0 Index by telephone interview one day, followed by onsite completion the following day, or vice versa. Neither patients nor interviewers had access to any prior scores. Results. The mean age of the 50 patients was 66.3 years (range 44-82); 34 (68%) were female and 16 (32%) male. There was excellent agreement between the mean office and telephone scores, with mean differences for the WOMAC LK3.0 pain, stiffness, and function subscale scores and total score of 0.09, 0.12, 0.78, and 0.98, respectively. These differences were well within the respective protocol defined equivalence criteria of +/- 1.7, +/- 0.9, +/- 6.4, and +/- 9.1, and represented differences from office scores of 0.9, 2.6, 2.4, and 2.2%, respectively. Conclusion. The use of telephone interviews for the WOMAC LK3.0 Index is a valid method of obtaining OA outcome measurements. These observations have important implications for designing data acquisition strategies for future OA clinical trials and for longterm observational studies.

Completion problems for real matrices, II

Thirty years ago, G.N. de Oliveira has proposed the following completion problems: Describe the possible characteristic polynomials of [C-ij], i,j is an element of {1, 2}, where C-1,C-1 and C-2,C-2 are square submatrices, when some of the blocks C-ij are fixed and the others vary. Several of these problems remain unsolved. This paper gives the solution, over the field of real numbers, of Oliveira's problem where the blocks C-1,C-1, C-2,C-2 are fixed and the others vary.