Thesis (Ph. D.)--University of Alberta, 2010. / Title from pdf file main screen (viewed on June 24, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta. Includes bibliographical references.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2005. / Includes bibliographical references (leaves 91-100). Also available in electronic version.

Techn. University, Diss., 2003--Darmstadt.

Systems of large sparse linear equations frequently arise in engineering and science. Therefore, there is a great need for methods that can solve these systems. In this thesis we will present three of the earliest and simplest iterative methods and also look at two more sophisticated methods. We will study their rate of convergence and illustrate them with examples.

A Tomografia por Emissão de Pósitrons (PET) está se definindo como um dos métodos preferidos para diagnóstico e seguimento de inúmeras doenças em Oncologia, Neurologia e Cardiologia. Esta modalidade é realizada com sistemas dedicados e sistemas baseados em câmaras de cintilação, que podem ser também usados em tomografia por emissão de fótons únicos (SPECT). Neste trabalho, efetuamos uma avaliação dos fatores que favorecem a quantificação em imagens PET com câmaras de cintilação em coincidência, caracterizadas por urna menor sensibilidade em relação a sistemas dedicados. Avaliamos as condições de quantificação de imagens sob os modos 2D e 3D de aquisição, obtidas por métodos de reconstrução 2D e 3D diversos e correções associadas. Dados de aquisição foram simulados por método de Monte Carlo empregando parâmetros realistas. Objetos de interesse diversos foram modelados. Imagens foram reconstruídas pelos métodos FBP, ART, MLEM e OSEM e consideramos correções de sensibilidade, normalização de detector, espalhamento e atenuação de radiação. Estabelecemos uma metodologia de avaliação de detectabilidade e recuperação de contrastes em imagens que contemplam, a partir de dois parâmetros mensuráveis, os aspectos mais relevantes em quantificação. Análises visuais também foram consideradas. Verificamos que o modo 3D é mais adequado que 2D na recuperação de baixos contrastes no objeto com a aplicação das correções selecionadas. A detectabilidade de pequenas estruturas está limitada pelos efeitos de volume parcial e pela resolução espacial finita dos sistemas de detecção. Os métodos ART, MLEM e, em particular, OSEM com 8 subconjuntos, apresentam-se adequados para estudos quantitativos no modo 3D. Os parâmetros definidos para avaliação podem ser empregados como indicadores de condições propícias a estudos quantitativos. / Volumetric reconstruction in gamma camera based PET imaging Positron Emission Tomography (PET) is considered as a very useful tool for diagnosing and following several diseases in Oncology, Neurology and Cardiology. Two types of systems are available for this imaging modality: the dedicated systems and those based on gamma camera technology. In this work, we assessed a number of factors affecting the quantitation of gamma camera based PET imaging, characterized by a lower sensitivity compared to those of dedicated systems. We also evaluated image quantitation conditions under 2D and 3D acquisition/reconstruction modes, for different reconstruction methods and associated corrections. Acquisition data were simulated by Monte Carla method, using realistic parameters. Several objects of interest were modelled. We reconstructed slices and volumes using FBP, ART, MLEM and OSEM and also included four corrections: detector sensitivity, detector normalization, scatter and attenuation of annihilation photons. We proposed a method to assess detectability and object contrast recovery by using two measurable parameters. Visual analysis was also considered. We found that 3D mode is more effective than 2D for low contrast recovery when the selected (J corrections are applied. Detectability of small structures is limited by partial volume effects and device finite spatial resolution. ART, MLEM and specially 8-subsets OSEM are the most adequate methods for quantitative studies in 3D mode. The parameter that we have defined may also be used as indicators of suitable conditions for quantitation in images.

Abstract Multiuser demodulation algorithms for centralized receivers of asynchronous direct-sequence (DS) spread-spectrum code-division multiple-access (CDMA) systems in frequency-selective fading channels are studied. Both DS-CDMA systems with short (one symbol interval) and long (several symbol intervals) spreading sequences are considered. Linear multiuser receivers process ideally the complete received data block. The approximation of ideal infinite memory-length (IIR) linear multiuser detectors by finite memory-length (FIR) detectors is studied. It is shown that the FIR detectors can be made near-far resistant under a given ratio between maximum and minimum received power of users by selecting an appropriate memory-length. Numerical examples demonstrate the fact that moderate memory-lengths of the FIR detectors are sufficient to achieve the performance of the ideal IIR detectors even under severe near-far conditions. Multiuser demodulation in relatively fast fading channels is analyzed. The optimal maximum likelihood sequence detection receiver and suboptimal receivers are considered. The parallel interference cancellation (PIC) receiver is demonstrated to achieve better performance in known channels than the decorrelating receiver, but it is observed to be more sensitive to channel coefficient estimation errors than the decorrelator. At high channel loads the PIC receiver suffers from bit error rate (BER) saturation, whereas the decorrelating receiver does not. Choice of channel estimation filters is shown to be crucial if low BER is required. Data-aided channel estimation is shown to be more robust than decision-directed channel estimation, which may suffer from BER saturation caused by hang-ups at high signal-to-noise ratios. Multiuser receivers for dynamic CDMA systems are studied. Algorithms for ideal linear detector computation are derived and their complexity is analyzed. The complexity of the linear detector computation is a cubic function of KL, where K and L are the number of users and multipath components, respectively. Iterative steepest descent, conjugate gradient, and preconditioned conjugate gradient algorithms are proposed to reduce the complexity. The computational requirements for one iteration are a quadratic function of KL. The iterative detectors are also shown to be applicable for parallel implementation. Simulation results demonstrate that a moderate number of iterations yields the performance of the corresponding ideal linear detectors. A quantitative analysis shows that the PIC receivers are significantly simpler to implement than the linear receivers and only moderately more complex than the conventional matched filter bank receiver.

Freeform optics provide excellent performance for a wide variety of applications. However, obtaining an accurate freeform surface measurement is highly challenging due to its large aspheric/freeform departure. It has been proven that SCOTS (Software Configurable Optical Test System), an advanced deflectometry system developed at the University of Arizona, can measure the departure of a freeform surface from the desired shape with nanometer accuracy. Here, a new data processing technique was used to measure a freeform surface without any prior knowledge of the shape of the surface. Knowing only the geometry of one point on the test surface, this method can take a blind measurement of a freeform surface and arrive at the true surface through iterative construction.

Phase Retrieval is an inference problem where one seeks to recover an unknown complex-valued 𝓃-dimensional signal vector from the magnitudes of 𝓶 linear measurements. The linear measurements are specified using a 𝓶 × 𝓃 sensing matrix. This problem is a mathematical model for imaging systems arising in X-ray crystallography and other applications where it is infeasible to acquire the phase of the measurements. This dissertation presents some results regarding the analysis of this problem in the high-dimensional asymptotic regime where the number of measurements and the signal dimension diverge proportionally so that their ratio remains fixed. A limitation of existing high-dimensional analyses of this problem is that they model the sensing matrix as a random matrix with independent and identically (i.i.d.) distributed Gaussian entries. In practice, this matrix is highly structured with limited randomness. This work studies a correction to the i.i.d. Gaussian sensing model, known as the sub-sampled Haar sensing model which faithfully captures a crucial orthogonality property of realistic sensing matrices. The first result of this thesis provides a precise asymptotic characterization of the performance of commonly used spectral estimators for phase retrieval in the sub-sampled Haar sensing model. This result can be leveraged to tune certain parameters involved in the spectral estimator optimally. The second part of this dissertation studies the information-theoretic limits for better-than-random (or weak) recovery in the sub-sampled Haar sensing model. The main result in this part shows that appropriately tuned spectral methods achieve weak recovery with the information-theoretically optimal number of measurements. Simulations indicate that the performance curves derived for the sub-sampled Haar sensing model accurately describe the empirical performance curves for realistic sensing matrices such as randomly sub-sampled Fourier sensing matrices and Coded Diffraction Pattern (CDP) sensing matrices. The final part of this dissertation tries to provide a mathematical understanding of this empirical universality phenomenon: For the real-valued version of the phase retrieval problem, the main result of the final part proves that the dynamics of a class of iterative algorithms, called Linearized Approximate Message Passing schemes, are asymptotically identical in the sub-sampled Haar sensing model and a real-valued analog of the sub-sampled Fourier sensing model.

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980. / Includes bibliographical references. / by Seshadri Ramaswamy. / Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980.

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