One of the most important challenges in dynamic magnetic resonance imaging (MRI) is to achieve high spatial and temporal resolution when it is limited by system performance. It is desirable to acquire data fast enough to capture the dynamics in the image time series without losing high spatial resolution and signal to noise ratio. Many techniques have been introduced in the recent decades to achieve this goal. Newly developed algorithms like Highly Constrained Backprojection (HYPR) and Compressed Sensing (CS) reconstruct images from highly undersampled data using constraints. Using these algorithms, it is possible to achieve high temporal resolution in the dynamic image time series with high spatial resolution and signal to noise ratio (SNR). In this thesis we have analyzed the performance of HYPR to CS algorithm. In assessing the reconstructed image quality, we considered computation time, spatial resolution, noise amplification factors, and artifact power (AP) using the same number of views in both algorithms, and that number is below the Nyquist requirement. In the simulations performed, CS always provides higher spatial resolution than HYPR, but it is limited by computation time in image reconstruction and SNR when compared to HYPR. HYPR performs better than CS in terms of SNR and computation time when the images are sparse enough. However, HYPR suffers from streaking artifacts when it comes to less sparse image data.

Ultracold neutron (UCN) projection imaging is demonstrated using a boron-coated back-illuminated CCD camera and the Los Alamos UCN source. Each neutron is recorded through the capture reactions with10B. By direct detection at least one of the byproducts α, 7Li and γ (electron recoils) derived from the neutron capture and reduction of thermal noise of the scientific CCD camera, a signal-to-noise improvement on the order of 104 over the indirect detection has been achieved. Sub-pixel position resolution of a few microns is confirmed for individual UCN events. Projection imaging of test objects shows a spatial resolution less than 100μm by an integrated UCN flux one the order of 106 cm−2. The bCCD can be used to build UCN detectors with an area on the order of 1 m2. The combination of micrometer scale spatial resolution, low readout noise of a few electrons, and large area makes bCCD suitable for quantum science of UCN.

The work deals with physical principles of X-ray generation and development of image during projection and projection reconstruction. A proposal of user’s application in a Matlab – Guide is given, which can be used as a laboratory exercise of the simulation of the projection- and projection image reconstruction. The computer program involves an evaluation of a X-ray quality of CT RTG ZS – quantitative assessment of spatial resolution and as well as the acquisition contrast as a function on an object size. The main aim of the work was the comparison of the acquisition contrast at various acquisition projection and projection-reconstruction parameters. Also, the work is illustrated by some results achieved.

This thesis is concerned with the development of novel numerical methods for solving nondifferentiable convex optimization problems in real Hilbert spaces and with the investigation of their asymptotic behavior. To this end, we are also making use of monotone operator theory as some of the provided algorithms are originally designed to solve monotone inclusion problems. After introducing basic notations and preliminary results in convex analysis, we derive two numerical methods based on different smoothing strategies for solving nondifferentiable convex optimization problems. The first approach, known as the double smoothing technique, solves the optimization problem with some given a priori accuracy by applying two regularizations to its conjugate dual problem. A special fast gradient method then solves the regularized dual problem such that an approximate primal solution can be reconstructed from it. The second approach affects the primal optimization problem directly by applying a single regularization to it and is capable of using variable smoothing parameters which lead to a more accurate approximation of the original problem as the iteration counter increases. We then derive and investigate different primal-dual methods in real Hilbert spaces. In general, one considerable advantage of primal-dual algorithms is that they are providing a complete splitting philosophy in that the resolvents, which arise in the iterative process, are only taken separately from each maximally monotone operator occurring in the problem description. We firstly analyze the forward-backward-forward algorithm of Combettes and Pesquet in terms of its convergence rate for the objective of a nondifferentiable convex optimization problem. Additionally, we propose accelerations of this method under the additional assumption that certain monotone operators occurring in the problem formulation are strongly monotone. Subsequently, we derive two Douglas–Rachford type primal-dual methods for solving monotone inclusion problems involving finite sums of linearly composed parallel sum type monotone operators. To prove their asymptotic convergence, we use a common product Hilbert space strategy by reformulating the corresponding inclusion problem reasonably such that the Douglas–Rachford algorithm can be applied to it. Finally, we propose two primal-dual algorithms relying on forward-backward and forward-backward-forward approaches for solving monotone inclusion problems involving parallel sums of linearly composed monotone operators. The last part of this thesis deals with different numerical experiments where we intend to compare our methods against algorithms from the literature. The problems which arise in this part are manifold and they reflect the importance of this field of research as convex optimization problems appear in lots of applications of interest.