منا خاكزاد قره تپه

Regularisation is a fundamental area within mathematics, aimed at providing efficient an‎d effective methods to approximate real answers accurately. Iterative methods are considered as a regularisation method, where the iteration vector serves as a regularised answer, an‎d the iteration index acts as a parameter for regularisation. In this thesis, we introduce iterative methods employed in addressing the linear convex feasibility problem arising from image reconstruction. We categorise these methods into three main groups an‎d assess their relative advantages an‎d drawbacks. Specifically, we delve into the Iterative Ran‎domised Kaczmarz method. While this method typically reduces error when applied to solving ill-posed linear equations, it may exhibit an increase in error after several iterations, contingent upon the level of noise in the data an‎d the degree of ill-posedness. This phenomenon is known as semi-convergence. In this study, we scrutinise the Ran‎domised Kaczmarz method utilising a constant relaxation parameter an‎d derive an upper bound for the expected noise-error component. Leveraging this upper bound, we propose a relaxation parameter dependent on k to mitigate the issue of semi-convergence. Additionally, we demonstrate the efficacy of the suggested relaxation parameter using samples obtained from tomographic imaging. Subsequently, we investigate the block ran‎domised Kaczmarz method. Here, we explore a variant of the method employing a constant relaxation parameter an‎d employing two types of probability to solve linear systems of equations. We establish upper bounds for the expected noise-error component an‎d propose a relaxation parameter dependent on k to address semi-convergence. Furthermore, we validate the performance of the proposed relaxation parameter using samples acquired from tomographic imaging.

روش كاكزمارز تصادفي , خطاي اختلال , تصويربرداري توموگرافي

Randomized Kaczmarz method , Noise-error , tomographic imaging

Mona khakzad

Dr Touraj Mohammadi