Research On Acceleration Algorithm Of Split Feasibility Problem For CT Image Reconstruction

There is a problem in many fields such as physics,mathematics and engineering technology(signal processing and image reconstruction)etc.,that is to find a point contained in a convex set and the image of this point contained in another convex set.This problem is called Split Feasible Problem(SFP).In the CT image reconstruction of medical imaging,the goal of image reconstruction is to find the vector of each image,which is the original CT image to be reconstructed.Therefore,CT image reconstruction is an application of SFP in mathematics.During the whole scanning process of CT imaging,X-ray will cause radiation and certain physical injury to the patients undergoing scanning.Therefore,it is necessary to construct a fast and effective algorithm for SFP,so as to apply it to CT image reconstruction and minimize the radiation received by patients during CT scanning.This dissertation focuses on designing efficient and feasible acceleration algorithms for different problems.Inertial Halpern interative algorithm for SFP with maximal monotone operator and fixed point,inertial adaptive CT image reconstruction acceleration algorithm and random sets block adaptive extrapolation projection CT image reconstuction acceleration algorithm for Multiple-output-sets SFP(MFSP)are proposed respectively.The convergence(or strong convergence)of these algorithms is proved under appropriate conditions.Experiments verify the feasibility and effectiveness of the above three algorithms.And try to apply it to CT image reconstruction.In this dissertation,my research work includes the following three parts:Aiming at the deficiency that the general SFP needs to calculate the spectral radius(maximum eigenvalue)of matrix A~TA,a class of SFP with maximal monotone operator and fixed point in Banach space is proposed,and a new iterative algorithm is given.The algorithm adopts inertial technology to realize Halpern iterative algorithm(inertial Halpern iterative algorithm for short),which avoids the complexity of calculation.Under the assumption of monotonicity of mapping,the strong convergence of the algorithm is proved.Experiments also verify the feasibility and effectiveness of the algorithm.In the second part,in order to construct an algorithm suitable for CT image reconstruction,an inertial adaptive acceleration algorithm is proposed to solve MSFP.On the one hand,the algorithm does not directly calculate the projection of the closed convex set,but the projection of the relaxed set(semi-space)of the closed convex set instead,so the feasibility of the algorithm is improved.On the other hand,the algorithm frees computer from calculating the matrix inverse,thereby reduces occupying its memory capacity.The algorithm also improves its convergence by combining the inertial technology and adaptive rules,while its strong convergence is proved under certain conditions.Furthermore,in the second part,the algorithm has been employed to SFP and MSFP which broadens its application circumstances.Numerical experiments show that the convergence of the proposed algorithm exceeds the existing algorithms.The experimental results of CT image reconstruction also show that the computational efficiency is significantly improved under the influence of inertial operator and adaptive step-size.In the third part,a random sets block CT image reconstruction algorithm with adaptive extrapolation step-size is proposed for MFSP based on the problem of further reducing the amount of computer memory and CPU calculation time required to calculate the adaptive step-size by the algorithm suggested in the second section.The algorithm includes random selection rules of set blocks,random conditions of feasibility problems and adaptive extrapolation step-size selection strategies.It is proved that the extrapolation projection algorithm of random sets blocks converges linearly under expected value when the sets satisfy linear regularity,and the convergence rate depends upon the number of conditions of the feasible problem and the size of blocks.It is also proved that the algorithm is sub-linear convergence when the condition of linear regularity is not satisfied.Finally,results of CT image reconstruction show that the proposed algorithm is effective and superior to other algorithms in reconstruction speed and quality.To sum up,this dissertation put forward different acceleration algorithm,inertial technology,adaptive technology and random sampling technology respectively for SFP containing maximal monotone operators and fixed points and MFSP,.The strong convergence and convergence speed of algorithms are proved under appropriate conditions.Experiments show that the new algorithms we proposed can effectively improve CT image reconstruction speed and quality.