Research On Regularization-based MRI Reconstruction Models And Fast Algorithms

Magnetic resonance(MR)imaging technology is an image reconstruction tech-nique that utilizes the hydrogen nucleus under the influence of an external magnetic field to get a magnetic resonance signal on K space,and then obtains the internal structure of an object by using mathematical methods.This technique belongs to bio-magnetic spin imaging technology,and is often used for detection in the fields of physics,chemistry,biology,medicine,etc.In the process of MR imaging,due to the large amount of sam-pled data and the complexity of algorithms during data inversion process,the imaging speed is relatively slow.However,the efficient and fast imaging technologies are needed in practical application.Therefore,how to improve the imaging speed is an important research topic.Among the methods of improving the imaging speed,the improvement of the physical performance for the magnetic resonance equipments has reached a bottleneck.Therefore,the reconstruction methods that use the prior information of the image struc-ture to establish the appropriate energy functional models and then propose the efficient and stable numerical methods to solve the models have been widely concerned in recent years.These reconstruction methods have two main research directions:One is to seek suitable sparse bases(such as wavelet transform,shear wave transform,PBDW trans-form,curve wave,etc.)to sample sparsely on the K spatial data and then to establish a model to depict the structural features of images;Second is to propose an effective nu-merical method to solve the proposed energy functional model.For example,utilizing splitting type algorithms,augmented Lagrangian methods,proximity methods,primal-dual methods,fixed-point method,etc.,and seeking some acceleration methods to im-prove the imaging speed.In order to effectively solve numerical computing problems with regularization terms and multi-channel large data processing and to promote the rapid development of fast calculation methods,many MR image reconstruction methods based on partial differential equations(PDEs)have been proposed,including augment-ed Lagrangian methods,primal-dual methods,variable separation technologies,alterna-tive minimization methods,etc.In this dissertation,we study regularization-based MR imaging reconstruction models and fast algorithms,the main research work and results are as follows:1.Based on the advantages which the Lysake-Osher-Tai(LOT)model and its de-formations can effectively preserve image details and edge structure features,in this dissertation,we propose a coupling model for MR image reconstruction based on total variation(TV),and by using operator splitting techniques,we develop an alternate di-rection multiplier method(ADMM method)to solve the proposed coupling model.In the process of solving the sub-problems of algebraic equations,since the corresponding coefficient matrix has a cyclic structure,the fast Fourier transform can be used to ob-tain the solutions,and a non-monotonic Barzilai-Barwein step size selection scheme is proposed to improve the efficiency of the algorithm.By selecting the different sampling rates and comparing with the classical numerical algorithms,we can verify the feasi-bility and validity of the proposed model and algorithm,as well as the stability of the algorithm.At the same time,we extend the proposed model and algorithm to the case of the anisotropic TV.2.In processing the nonsmooth regions of MR images,the models based on the traditional TV will give rise to the stair casing artifacts.By utilizing the property which the high order total variational model can effectively preserve the smooth regions of the images,in this dissertation,we propose a coupling model for MR image reconstruction based on the second-order total variation(TV~2),and by using the variable separation techniques twice,we can transform the solution of the subproblems based on the fourth order PDEs into the solution of the subproblems based on the second order PDEs,thus reducing the complexity of solving the original model.On this basis,by combining the split Bregman iterative scheme,alternate minimization method with fast Fourier trans-form,an ADMM method for solving the coupling model is proposed.The experimental results show that the proposed model and algorithm can effectively suppress the stair casing artifacts,thus obtaining high quality reconstructed images.At the same time,we also extend the proposed model and algorithm to the case of the anisotropic TV~2.3.Because the shearlet has better structures in sparse signal processing on K space,in this dissertation,based on the shearlet regularization,we propose a MR image recon-struction model coupling the generalized total variation and the shearlet regularization.This model combines the advantages of shearlet in sparse signal processing on K space,the reconstructed images have better image detail structures.By using the variable sep-aration techniques to avoid solving the sub-problems based on the high-order PDEs,and by combining the split Bregman iterative scheme with the alternative minimization method,we propose an ADMM method for solving the coupling model.At the same time,the Fourier transform is applied to simplify the calculation by utlizing the special structure of the relating operators.By testing at different sampling rates and comparing with the other two algorithms,the results show that we can stably obtain better quality MRI reconstructed images based on the proposed model and algorithm.This dissertation is typeset by software L~AT_EX2_?.