Thesis supervisor: Vincent A. Magnotta. Includes bibliographic references (p. 93-101).

Heidelberg, Universiẗat, Diss., 2004.

SILVA, A. P. Tensor techniques in signal processing: algorithms for the canonical polyadic decomposition (PARAFAC). 2016. 124 f. Tese (Doutorado em Engenharia de Teleinformática) - Centro de Tecnologia, Universidade Federal do Ceará, Fortaleza, 2016. / Submitted by Marlene Sousa (mmarlene@ufc.br) on 2016-09-01T18:41:38Z No. of bitstreams: 1 2016_tese_apsilva.pdf: 1648271 bytes, checksum: be3747d533837939c3a410d2f017ddfa (MD5) / Approved for entry into archive by Marlene Sousa (mmarlene@ufc.br) on 2016-09-01T18:42:06Z (GMT) No. of bitstreams: 1 2016_tese_apsilva.pdf: 1648271 bytes, checksum: be3747d533837939c3a410d2f017ddfa (MD5) / Made available in DSpace on 2016-09-01T18:42:06Z (GMT). No. of bitstreams: 1 2016_tese_apsilva.pdf: 1648271 bytes, checksum: be3747d533837939c3a410d2f017ddfa (MD5) Previous issue date: 2016-06-29 / Low rank tensor decomposition has been playing for the last years an important role in many applications such as blind source separation, telecommunications, sensor array processing, neuroscience, chemometrics, and data mining. The Canonical Polyadic tensor decomposition is very attractive when compared to standard matrix-based tools, manly on system identification. In this thesis, we propose: (i) several algorithms to compute specific low rank-approximations: finite/iterative rank-1 approximations, iterative deflation approximations, and orthogonal tensor decompositions. (ii) A new strategy to solve multivariate quadratic systems, where this problem is reduced to a best rank-1 tensor approximation problem. (iii) Theoretical results to study and proof the performance or the convergence of some algorithms. All performances are supported by numerical experiments

Tensors can be viewed as multilinear arrays or generalizations of the notion of matrices. Tensor decompositions have applications in various fields such as psychometrics, signal processing, numerical linear algebra and data mining. When the data are nonnegative, the nonnegative tensor factorization (NTF) better reflects the underlying structure. With NTF it is possible to extract information from a given dataset and construct lower-dimensional bases that capture the main features of the set and concisely describe the original data. Nonnegative tensor factorizations are commonly computed as the solution of a nonlinear bound-constrained optimization problem. Some inherent difficulties must be taken into consideration in order to achieve good solutions. Many existing methods for computing NTF optimize over each of the factors separately; the resulting algorithms are often slow to converge and difficult to control. We propose an all-at-once approach to computing NTF. Our method optimizes over all factors simultaneously and combines two regularization strategies to ensure that the factors in the decomposition remain bounded and equilibrated in norm. We present numerical experiments that illustrate the effectiveness of our approach. We also give an example of digital-inpainting, where our algorithm is employed to construct a basis that can be used to restore digital images. / Science, Faculty of / Computer Science, Department of / Graduate

Tensors are of great interest to many applications in engineering and in medical imaging, but a proper analysis and visualization remains challenging. It already has been shown that, by employing the metaphor of a fabric structure, tensor data can be visualized precisely on surfaces where the two eigendirections in the plane are illustrated as thread-like structures. This leads to a continuous visualization of most salient features of the tensor data set. We introduce a novel approach to compute such a visualization from tensor field data that is motivated by image-space line integral convolution (LIC). Although our approach can be applied to arbitrary, non-selfintersecting surfaces, the main focus lies on special surfaces following important features, such as surfaces aligned to the neural pathways in the human brain. By adding a postprocessing step, we are able to enhance the visual quality of the of the results, which improves perception of the major patterns.

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Diffusion tensor imaging provides the ability to study white matter connectivity and integrity noninvasively. The information contained in the diffusion tensors is very complex. Therefore a simple way of dealing with tensors is to compute rotationally invariant scalar quantities. These scalar indices have been used to perform population studies between controls and patients with neurological and psychiatric disorders. Implementing the scalar values may reduce the information contained in the whole tensor. A group analysis using the full tensors may give better estimate of white matter changes that occur in the diseased subjects. For spatial normalization of diffusion tensors, it is necessary to interpolate the tensor representation as well as rotate the diffusion tensors after transformation to keep the tensors consistent with the tissue reorientation. Existing reorientation methods cannot be directly used for higher order diffusion models (e.g. q-ball imaging). A novel technique called gradient rotation is introduced where the rotation is directly applied to the diffusion sensitizing gradients providing a voxel by voxel estimate of the diffusion gradients instead of a volume of by volume estimate. The technique is validated by comparing it with an existing method where the transformation is applied to the resulting diffusion tensors. For better matching of diffusion tensors a novel multichannel registration method is proposed based on a non-parametric diffeomorphic demons algorithm. The channels used for the registration include T1-weighted volume and tensor components. A fractional anisotropy (FA) channel is used for defining the contribution of each channel. Including the anatomical data together with the tensors, allows the registration to accurately match the global brain shape and the underlying white matter architecture simultaneously. Using this multichannel registration framework, 10 healthy controls and 9 patients of schizophrenia were spatially normalized. For the group analysis, the tensors were transformed to log-euclidean space. Linear regression analysis was performed on the transformed tensors. Results show that there is a significant difference in the anisotropy between patients and controls especially in the anterior regions that include genu of the corpus callosum and anterior and superior corona radiata, forceps minor and anterior limb on the internal capsule.

[EN] At present, different methods for magnetostatic force densities in a continuous medium are used. Some of them are based on models of interaction of matter with an external field and others are derived from Maxwell's tensors in matter. All of these densities are different. Current models of magnetomechanical interactions are conditioned by the choice of one of these densities. In this dissertation a new model of magnetomechanical interaction that is independent on force density adopted is proposed. The Cauchy stress tensor obtained is constituted by the addition of three components: elastic, magnetic and magnetoelastic tensors. The magnetic component is the only one dependent on force density adopted. It is found that the magnetoelastic Cauchy stress tensor is not in general symmetric and an extended magnetostatic force density that takes into account the effects of elastic interaction is defined. / [ES] En la actualidad, se emplean diferentes métodos para obtener las densidades de fuerza magnetostáticas en un medio continuo. Algunos de ellos se fundamentan en modelos de interacción de la materia con un campo externo y otros se derivan de tensores de Maxwell en la materia. Todas estas densidades son diferentes. Los modelos actuales de interacción magnetomecánica están condicionados por la elección de una de estas densidades. En esta Tesis se propone un nuevo modelo de interacción independiente de la densidad de fuerza adoptada. El tensor de Cauchy obtenido está constituido por un tensor elástico, otro magnético y otro magnetoelástico. La componente magnética es la única que depende del tensor de Maxwell o densidad de fuerza adoptada. Se constata que el tensor de Cauchy no es en general simétrico y se define una densidad de fuerza magnetostática extendida que tiene en cuenta los efectos de interacción elástica. / [CA] En l'actualitat, s'utilitzen diferents mètods per a obtenir les densitats de forces magnetostàtiques en un medi continu. Alguns d'ells tenen els seus fonaments en models d'interacció de la matèria amb un camp extern y d'altres es deriven de tensors de Màxwell en la matèria. Totes les densitats son diferents. Els models actuals d'interacció magnetomecànica estan condicionats per l'elecció d'una d'aquestes densitats. En la present Tesi es propasa un nou model d'interacció independent de la densitat de força adoptada. El tensor de Cauchy obtingut està constituït per un tensor elàstic, un magnètic i un magnetoelàstic. La component magnètica és l'única que depend del tensor de Màxwell o densitat de força adoptada. Es constata que el tensor de Cauchy no és en general simètric i es defineix una densitat de força magnetostàtica estesa que té en compte els efectes d'interacció elàstica. / Usieto Galve, A. (2016). Modelización de la interacción magnetomecánica bajo un tensor de Cauchy de tensiones magnéticas obtenido por procedimientos energéticos [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/63240

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