This dissertation is motivated by the increasing applications of high-dimensional large-scale data sets in various fields and lack of theoretical understanding of the existing algorithms as well as lack of efficient algorithms in many cases. Hence, identifying the geometrical properties of data sets is essential for many data processing tasks, such as data retrieval and denoising. In Part I, we derive the fundamental limits on the sampling rate required to study three important problems (i) low-rank data completion, (ii) rank estimation, and (iii) data clustering. In Chapter 2 we characterize the geometrical conditions on the sampling pattern, i.e., locations of the sampled entries, for finite and unique completability of a low-rank tensor, assuming that its rank vector is given or estimated. To this end, we propose a manifold analysis and study the independence of a set of polynomials defined based on the sampling pattern. Then, using the polynomial analysis, we derive a lower bound on the sampling rate such that it guarantees that the proposed conditions on the sampling patterns for finite and unique completability hold true with high probability. Then, in Chapter 3, we study the problem of rank estimation, where a data structure is partially sampled and we propose a geometrical analysis on the sampling pattern to estimate the true value of rank for various data structures by providing extremely tight lower and upper bounds on the rank value. And in Chapters 4 and 5, we make use of the developed tools to obtain a lower bound on the sampling rate to be able to correctly cluster a union of sampled matrices or tensors by identifying their corresponding unknown subspaces. In Part II, first in Chapter 6, motivated by the algebraic tools developed in Part I, we develop a data completion algorithm based on solving a set of polynomial equations using Newton's method, that is effective especially when the sampling rate is low. Then, in Chapter 7, we consider a data structure consisting of a union of nested low-rank matrix or tensor subspaces, and develop a structured alternating minimization-based approach for completing such data, that is capable of taking advantage of multiple rank constraints simultaneously to achieve faster convergence and higher recovery accuracy.

x, 81 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We give an exposition of Deligne's tensor category Rep(St) where t is not necessarily an integer. Thereafter, we give a complete description of the blocks in Rep(St) for arbitrary t. Finally, we use our result on blocks to decompose tensor products and classify tensor ideals in Rep(St). / Committee in charge: Victor Ostrik, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Jonathan Brundan, Member, Mathematics; Alexander Kleshchev, Member, Mathematics; Michael Kellman, Outside Member, Chemistry

How to handle large multi-dimensional datasets such as hyperspectral images and video information both efficiently and effectively plays an important role in big-data processing. The characteristics of tensor low-rank decomposition in recent years demonstrate the importance of capturing the tensor structure adequately which usually yields efficacious approaches. In this dissertation, we first aim to explore the tensor singular value decomposition (t-SVD) with the nonconvex regularization on the multi-view subspace clustering (MSC) problem, then develop two new tensor decomposition models with the Bayesian inference framework on the tensor completion and tensor robust principal component analysis (TRPCA) and tensor completion (TC) problems. Specifically, the following developments for multi-dimensional datasets under the mathematical tensor framework will be addressed. (1) By utilizing the t-SVD proposed by Kilmer et al. \cite{kilmer2013third}, we unify the Hyper-Laplacian (HL) and exclusive $\ell_{2,1}$ (L21) regularization with Tensor Log-Determinant Rank Minimization (TLD) to identify data clusters from the multiple views' inherent information. Whereby the HL regularization maintains the local geometrical structure that makes the estimation prune to nonlinearities, and the mixed $\ell_{2,1}$ and $\ell_{1,2}$ regularization provides the joint sparsity within-cluster as well as the exclusive sparsity between-cluster. Furthermore, a log-determinant function is used as a tighter tensor rank approximation to discriminate the dimension of features. (2) By considering a tube as an atom of a third-order tensor and constructing a data-driven learning dictionary from the observed noisy data along the tubes of a tensor, we develop a Bayesian dictionary learning model with tensor tubal transformed factorization to identify the underlying low-tubal-rank structure of the tensor substantially with the data-adaptive dictionary for the TRPCA problem. With the defined page-wise operators, an efficient variational Bayesian dictionary learning algorithm is established for TPRCA that enables to update of the posterior distributions along the third dimension simultaneously. (3) With the defined matrix outer product into the tensor decomposition process, we present a new decomposition model for a third-order tensor. The fundamental idea is to decompose tensors mathematically in a compact manner as much as possible. By incorporating the framework of Bayesian probabilistic inference, the new tensor decomposition model on the subtle matrix outer product (BPMOP) is developed for the TC and TRPCA problems. Extensive experiments on synthetic data and real-world datasets are conducted for the multi-view clustering, TC, and TRPCA problems to demonstrate the desirable effectiveness of the proposed approaches, by detailed comparison with currently available results in the literature.

While diffusion MRI promises an insight into white matter microstructure in vivo, the axonal pathways that connect different brain regions together can only partially be segmented using current methods. Here we present a novel method for estimating the tissue composition of each voxel in the brain from diffusion MRI data, thereby providing a foundation for computing the volume of different pathways in both health and disease. With the tissue dependent diffusion model described in this thesis, white matter is segmented by removing the ambiguity caused by the isotropic partial volumes: both grey matter and cerebrospinal fluid. Apart from the volume fractions of all three tissue types, we also obtain estimates of fibre orientations for tractography as well as diffusivity and anisotropy parameters which serve as proxy indices of pathway coherence. We assume Gaussian diffusion of water molecules for each tissue type. The resulting three-tensor model comprises one anisotropic (white matter) compartment modelled by a cylindrical tensor and two isotropic compartments (grey matter and cerebrospinal fluid). We model the measurement noise using a Rice distribution. Markov chain Monte Carlo sampling techniques are used to estimate posterior distributions over the model’s parameters. In particular, we employ a Metropolis Hastings sampler with a custom burn-in and proposal adaptation to ensure good mixing and efficient exploration of the high-probability region. This way we obtain not only point estimates of quantities of interest, but also a measure of their uncertainty (posterior variance). The model is evaluated on synthetic data and brain images: we observe that the volume maps produced with our method show plausible and well delineated structures for all three tissue types. Estimated white matter fibre orientations also agree with known anatomy and align well with those obtained using current methods. Importantly, we are able to disambiguate the volume and anisotropy information thus alleviating partial volume effects and providing measures superior to the currently ubiquitous fractional anisotropy. These improved measures are then applied to study brain differences in a cohort of healthy volunteers aged 25-65 years. Lastly, we explore the possibility of using prior knowledge of the spatial variability of our parameters in the brain to further improve the estimation by pooling information among neighbouring voxels.

published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

No description available.

This dissertation addresses the problem of adaptive image filtering. Although the topic has a long history in the image processing community, researchers continuously present novel methods to obtain ever better image restoration results. With an expanding market for individuals who wish to share their everyday life on social media, imaging techniques such as compact cameras and smart phones are important factors. Naturally, every producer of imaging equipment desires to exploit cheap camera components while supplying high quality images. One step in this pipeline is to use sophisticated imaging software including, e.g., noise reduction to reduce manufacturing costs, while maintaining image quality. This thesis is based on traditional formulations such as isotropic and tensor-based anisotropic diffusion for image denoising. The difference from main-stream denoising methods is that this thesis explores the effects of introducing contextual information as prior knowledge for image denoising into the filtering schemes. To achieve this, the adaptive filtering theory is formulated from an energy minimization standpoint. The core contributions of this work is the introduction of a novel tensor-based functional which unifies and generalises standard diffusion methods. Additionally, the explicit Euler-Lagrange equation is derived which, if solved, yield the stationary point for the minimization problem. Several aspects of the functional are presented in detail which include, but are not limited to, tensor symmetry constraints and convexity. Also, the classical problem of finding a variational formulation to a given tensor-based partial differential equation is studied. The presented framework is applied in problem formulation that includes non-linear domain transformation, e.g., visualization of medical images. Additionally, the framework is also used to exploit locally estimated probability density functions or the channel representation to drive the filtering process. Furthermore, one of the first truly tensor-based formulations of total variation is presented. The key to the formulation is the gradient energy tensor, which does not require spatial regularization of its tensor components. It is shown empirically in several computer vision applications, such as corner detection and optical flow, that the gradient energy tensor is a viable replacement for the commonly used structure tensor. Moreover, the gradient energy tensor is used in the traditional tensor-based anisotropic diffusion scheme. This approach results in significant improvements in computational speed when the scheme is implemented on a graphical processing unit compared to using the commonly used structure tensor. / VIDI / NACIP / GARNICS / EMC^2

The reading network is only partially understood and even less is known regarding how the network functions when reading is impaired. Dyslexia is characterized by poor phonological processing and affects roughly 5-12% of the population. The Dorsal-Ventral and Cerebellar-Deficit models propose distinct behavioral and structural differences in young adults with dyslexia. Behavioral assessments were used to determine if deficits for young adults with dyslexia were restricted to the literacy domain or dispersed among reading and associated behavioral domains. Diffusion tensor imaging (DTI) was used determine the extent to which white matter pathways and gray matter regions differ structurally in young adults with dyslexia. The present study also investigated whether brain-behavior relationships exist and are consistent with the theoretical models of reading in this population. Findings show that young adults with dyslexia exhibited deficits in both literacy and associated behavioral domains, including verbal working memory and motor function. Structural findings showed increased fractional anisotropy in the left anterior region (the aslant) and decreased fractional anisotropy in left posterior regions (inferior occipital fasciculus and vertical occipital fasciculus) of the reading network for young adults with dyslexia. Brain-behavior associations were found between the right inferior frontal gyrus and decoding for those with dyslexia. These findings provide support for the use of an altered reading network by young adults with dyslexia, as outlined by the Dorsal-Ventral model of reading. Limited structural and behavior findings support of the Cerebellar-Deficit model of reading, findings that warrant additional investigation.

Diffusion tensor imaging (DTI) is a relatively new technology of magnetic resonance imaging, which enables us to observe the insight structure of the human body in vivo and non-invasively. It displays water molecule movement by a 3×3 diffusion tensor at each voxel. Tensor field processing, visualisation and tractography are all based on the diffusion tensors. The accuracy of estimating diffusion tensor is essential in DTI. This research focuses on exploring the potential improvements at the tensor estimation of DTI. We analyse the noise arising in the measurement of diffusion signals. We present robust methods, least median squares (LMS) and least trimmed squares (LTS) regressions, with forward search algorithm that reduce or eliminate outliers to the desired level. An investigation of the criterion to detect outliers is provided in theory and practice. We compare the results with the generalised non-robust models in simulation studies and applicants and also validated various regressions in terms of FA, MD and orientations. We show that the robust methods can handle the data with up to 50% corruption. The robust regressions have better estimations than generalised models in the presence of outliers. We also consider the multiple tensors problems. We review the recent techniques of multiple tensor problems. Then we provide a new model considering neighbours' information, the Bayesian single and double tensor models using neighbouring tensors as priors, which can identify the double tensors effectively. We design a framework to estimate the diffusion tensor field with detecting whether it is a single tensor model or multiple tensor model. An output of this framework is the Bayesian neighbour (BN) algorithm that improves the accuracy at the intersection of multiple fibres. We examine the dependence of the estimators on the FA and MD and angle between two principal diffusion orientations and the goodness of fit. The Bayesian models are applied to the real data with validation. We show that the double tensors model is more accurate on distinct fibre orientations, more anisotropic or similar mean diffusivity tensors. The final contribution of this research is in covariance tensor estimation. We define the median covariance matrix in terms of Euclidean and various non-Euclidean metrics taking its symmetric semi-positive definiteness into account. We compare with estimation methods, Euclidean, power Euclidean, square root Euclidean, log-Euclidean, Riemannian Euclidean and Procrustes median tensors. We provide an analysis of the different metric between different median covariance tensors. We also provide the weighting functions and define the weighted non-Euclidean covariance tensors. We finish with manifold-valued data applications that improve the illustration of DTI images in tensor field processing with defined non-weighted and weighted median tensors. The validation of non-Euclidean methods is studied in the tensor field processing. We show that the root square median estimator is preferable in general, which can effectively exclude outliers and clearly shows the important structures of the brain. The power Euclidean median estimator is recommended when producing FA map.