چكيده به لاتين
A very common problem in mathematics is convex feasibilty problem which is used in different areas of sciences such as: medical imaging and radiation therapy treatment planning (computerized tomography), Electron microscopy, Signal processing, Minimization of non smooth and convex functions. A common solution approach
to such problems is using iterative algorithmic operators. These iterative methods have different algorithmic structures as full or block sequential (simultaneous) methods and a more general construction which is based on string averaging. Our main
algorithmic scheme is based on the string-averaging which is particularly suitable for parallel computing and therefore has the ability to handle huge-size problems.
This thesis focuses on the use of nonlinear algorithmic operators in convex feasibility problems and on their applications to medical imaging. We describe various extensions of the notion of operators in a Hilbert space. These include projections, subgradient projections, relaxec cutter operators and strictly quasi-nonexpansive operators. First of all, for a subfamily of strictly quasi-nonexpansive operators, we replace the paracontraction property with a weaker property called GPP which does not require continuity of the operators. We show that the sequence generated by a fixed-point iteration method based on a finite pool of operators in this subfamily, converges to a feasible point. Furthermore, we discuss about influence of using optimal weights instead of optimal relaxation parameters which were not considered before. This contribution unifies several existing results on the convergence of iterative methods in Hilbert spaces.
In the following, we introduce a step size function to accelerate a fixed point process which is constructed based on string averaging of strictly relaxed cutter operators and show it converges to the solution. We reproduce mentioned results for class of strictly quasi-Nonexpansive operators. We study Projected non-stationary Simultaneous Iterative Reconstruction Techniques which are common techniques in the field of medical image reconstruction and give their convergence result.
At the end, we model CT-scanners process as a consistent linear system of equations. We use clipped statistical weights to reduce Lipschitz constant and compare different block iterative algorithms for solving this linear system.
