| The MLEM(Maximum Likelihood Expectation Maximization)algorithm is an algebraic approach estimating variables with maximum likelihood.This approach considers both the physical characteristics of the system and the statistical properties of data,it can achieve superior performance,especially in PET/SPECT.The MLEM algorithm not only has the characteristics of global convergence but also has low computational complexity and strong operability.However,the convergence rate of the iterative scheme of MLEM is quite slow.This paper focuses on studying the accelerated convergence strategy of the MLEM algorithm in image reconstruction.To speed up the convergence rate,we introduce a relaxation parameter in the conventional MLEM iterative formula compared with the gradient descent algorithm.And then we rewrite this formula of the algorithm a new expression that depends on the iterative matrix.Based on the convergent conclusions of the Landweber iteration,in the case where the spectral radius of the updated matrix takes the minimum value,and propose the relaxation strategy that simultaneously relies on the maximum and minimum eigenvalues of the iterative matrix on the condition that the spectral radius of the derived iterative matrix from the iterative scheme with the accelerated parameter reaches a minimum value.Considering the huge dimensional number of the image and the difficulty of finding the minimum positive eigenvalue,we get the maximum eigenvalue according to the nonnegative characteristics of the iteration matrix.On this condition,we also obtain the accelerated strategy that only depends on the maximum eigenvalue.We get the spectral radius of the proposed relaxation strategy is much smaller than that of the original MLEM algorithm.According to the association between the spectral radius of the iterative matrix and the convergence rate,we prove the new method can indeed save computation time in theory.Besides,through further research on the accelerated MLEM algorithm,we prove the accelerated MLEM algorithm also keeps the same properties as the original method,such as monotonicity and convergence.The Shepp-Logan phantom and a crude wood ring image which has a complex texture with complete projection data are used for data simulation.We found that the relaxation strategy which we presented has some good characteristics through experiments,such as simplify iteration steps and the restored image has good quality,the calculation of this new strategy is much faster than the traditional method. |
