For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems and medical imaging problems such as greenhouse gas tracking and computational tomography reconstruction. This dissertation describes advancements in computational tools for solving large-scale inverse problems and for uncertainty quantification. Oftentimes, inverse problems are ill-posed and large-scale. Iterative projection methods have dramatically reduced the computational costs of solving large-scale inverse problems, and regularization methods have been critical in obtaining stable estimations by applying prior information of unknowns via Bayesian inference. However, by combining iterative projection methods and variational regularization methods, hybrid projection approaches, in particular generalized hybrid methods, create a powerful framework that can maximize the benefits of each method. In this dissertation, we describe various advancements and extensions of hybrid projection methods that we developed to address three recent open problems. First, we develop hybrid projection methods that incorporate mixed Gaussian priors, where we seek more sophisticated estimations where the unknowns can be treated as random variables from a mixture of distributions. Second, we describe hybrid projection methods for mean estimation in a hierarchical Bayesian approach. By including more than one prior covariance matrix (e.g., mixed Gaussian priors) or estimating unknowns and hyper-parameters simultaneously (e.g., hierarchical Gaussian priors), we show that better estimations can be obtained. Third, we develop computational tools for a respirometry system that incorporate various regularization methods for both linear and nonlinear respirometry inversions. For the nonlinear systems, blind deconvolution methods are developed and prior knowledge of nonlinear parameters are used to reduce the dimension of the nonlinear systems. Simulated and real-data experiments of the respirometry problems are provided. This dissertation provides advanced tools for computational inversion and uncertainty quantification. / Doctor of Philosophy / For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems such as greenhouse gas tracking where the problem of estimating the amount of added or removed greenhouse gas at the atmosphere gets more difficult. The number of observations has been increased with improvements in measurement technologies (e.g., satellite). Therefore, the inverse problems become large-scale and they are computationally hard to solve. Another example of an inverse problem arises in tomography, where the goal is to examine materials deep underground (e.g., to look for gas or oil) or reconstruct an image of the interior of the human body from exterior measurements (e.g., to look for tumors). For tomography applications, there are typically fewer measurements than unknowns, which results in non-unique solutions. In this dissertation, we treat unknowns as random variables with prior probability distributions in order to compensate for a deficiency in measurements. We consider various additional assumptions on the prior distribution and develop efficient and robust numerical methods for solving inverse problems and for performing uncertainty quantification. We apply our developed methods to many numerical applications such as greenhouse gas tracking, seismic tomography, spherical tomography problems, and the estimation of CO2 of living organisms.

Perfusion magnetic resonance imaging is modern diagnostic method used mainly in oncology. In this method, contrast agent is injected to the subject and then is continuously monitored the progress of its concentration in the affected area in time. Correct determination of the arterial input function (AIF) is very important for perfusion analysis. One possibility is to model AIF by multichannel blind deconvolution but the estimated AIF is necessary to be scaled. This master´s thesis is focused on description of scaling methods and their influence on perfussion parameters in dependence on used model of AIF in different tissues.

Slepá dekonvoluce je metoda, při které je rozptylová funkce a skutečný obraz rekonstruován zároveň. Cílem této práce je představit různé metody slepé dekonvoluce a najít optimální metodu rekonstrukce původního obrazu a rozptylové funkce. Jako nejvhodnější metoda slepé dekonvoluce byl zvolen algoritmus střídavé minimalizace, který byl upraven a testován. Vlastnosti navrženého algoritmu byly testovány na uměle degradovaných datech a na reálných datech pořízených skenovacím transmisním elektronovým mikroskopem. Účinnost algoritmu byla hodnocena hned několika hodnotícími kritérii. Byla zjištěna omezení algoritmu a tím specifikováno jeho využití.

Perfusion magnetic resonance imaging is modern diagnostic method used mainly in oncology. In this method, contrast agent is injected to the subject and then is continuously monitored the progress of its concentration in the affected area in time. Correct determination of the arterial input function (AIF) is very important for perfusion analysis. One possibility is to model AIF by multichannel blind deconvolution but the estimated AIF is necessary to be scaled. This master´s thesis is focused on description of scaling methods and their influence on perfussion parameters in dependence on used model of AIF in different tissues.

This thesis deals with the possibilities of determining perfusion parameters of vascular system, using contrast-enhanced ultrasound imaging, which is non-invasive method. Properties of ultrasonography and use of contrast agents are briefly summarized. The methods selected for perfusions analysis were Bolus-tracking¬¬, Burst-replenishment and both of them combined – Bolus&Burst. Parametric models based on these methods were created for modelling an approximation of set perfusion parameters with the use of blind deconvolution.

This master thesis deals with the contrast agents and their application in the ultrasound perfusion analysis. It is focused on Bolus & Burst method which, as a combination of two approaches that have been used so far, allows an absolute quantification of perfusion parameters in the region of interest. Contrast agent concentration time sequence is modeled as a convolution of the parametrically defined arterial input function and the tissue residual funkction. Thesis discusses different mathematical models of these functions as well as the methods of the parameters estimation. The methods functionality is validated on simulated and also preclinical data.

This Master Thesis deals with image restoration using deconvolution. The terms introducing into deconvolution theory like two-dimensional signal, distortion model, noise and convolution are explained in the first part of thesis. The second part deals with deconvolution methods via utilization of the Bayes approach which is based on the probability principle. The third part is focused on the Alternating Minimization Algorithm for Multichannel Blind Deconvolution. At the end this algorithm is written in Matlab with utilization of the NAG C Library. Then comparison of different optimization methods follows (simplex, steepest descent, quasi-Newton), regularization forms (Tichonov, Total Variation) and other parameters used by this deconvolution algorithm.