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Abstract

Time that an imaging device needs to produce results is one of the most crucial factors in medical imaging. Shorter scanning duration causes fewer artifacts such as those created by the patient motion. In addition, it increases patient comfort and in the case of some imaging modalities also decreases exposure to radiation. There are some possibilities, hardware-based or software-based, to improve the imaging speed. One way is to speed up the scanning process by acquiring fewer measurements. A recently developed mathematical framework called compressed sensing shows that it is possible to accurately recover undersampled images provided a suitable measurement matrix is used and the image itself has a sparse representation. Nevertheless, not only measurements are important but also good reconstruction models are required. Such models are usually expressed as optimization problems. In this thesis, we concentrated on the reconstruction of the undersampled Magnetic Resonance (MR) images. For this purpose a complex-valued reconstruction model was provided. Since the reconstruction should be as quick as possible, fast methods to find the solution for the reconstruction problem are required. To meet this objective, three popular algorithms FISTA, Augmented Lagrangian and Non-linear Conjugate Gradient were adopted to work with our model. By changing the complex-valued reconstruction model slightly and dualizing the problem, we obtained an instance of the quadratically constrained quadratic program where both the objective function and the constraints are twice differentiable. Hence new model opened doors to two other methods, the first order method which resembles FISTA and is called in this thesis Normed Constrained Quadratic FGP, and the second order method called Truncated Newton Primal Dual Interior Point. Next, in order to compare performance of the methods, we set up the experiments and evaluated all presented methods against the problem of reconstructing undersampled MR images. In the experiments we used a number of invocations of the Fourier transform to measure the performance of all algorithms. As a result of the experiments we found that in the context of the original model the performance of Augmented Lagrangian is better than the other two methods. Performance of Non-linear Conjugate Gradient and FISTA are about the same. In the context of the extended model Normed Constrained Quadratic FGP beats the Truncated Newton Primal Dual Interior Point method.