疊代影像重建法在正子斷層掃描之應用不斷的蓬勃發展，列運算影像重建法即是其中一種有效率的演算法。在列運算影像重建法的應用上，最重要的一項控制因素是決定適當的鬆弛係數，以發揮列運算的效能與穩定性，但實際上鬆弛係數受到許多因素的影響，以致於非常不容易決定出適當值。所以本研究提出應用前置調整矩陣計算鬆弛係數初始值，並搭配有效率的運算方式決定遞減序列，以幫助列運算疊代法在鬆弛係數的選擇更有效率，並增加列運算疊代法的應用性。簡化鬆弛係數初始值的選擇，可以幫助我們進一步分析鬆弛序列的影響，鬆弛序列所決定的遞減程度是幫助列運算穩定收斂的關鍵，但並沒有一定的選擇標準。經由測試分析可以瞭解，當係數隨著每一次疊代的遞減程度大時，可以產生更穩定的收斂效果，但卻犧牲了列運算的加速性。藉由前置調整矩陣所決定的鬆弛係數初始值，因為其過渡鬆弛的加速性，依然可以幫助整體的運算效能維持在高效能的水準，也就是同時保持列運算演算法的高效率與穩定性，幫助列運算演算法能有更好的應用性。

Row-action algorithms were developed as faster alternative to the conventional expectation-maximization (EM) algorithm for maximizing the Poisson likelihood function of statistical image reconstruction in positron emission tomography. The major advantage of row-action algorithms is the use of a relaxation parameter that controls the amount of image updates during reconstruction process. This unique characteristic makes row-action algorithm sharing the similar convergence rate as ordered subset expectation-maximization (OSEM) algorithm, while maintaining better stability. However, the selection of appropriate relaxation parameter depends on many physical factors. In this research, we propose a novel flexible number generation scheme for relaxation parameter. The idea is to incorporate a pre-conditioned matrix, approximating the Hessian matrix of the objective function, into the original reconstruction algorithm so that the original likelihood function will become more well-condition. This step can also alleviate the difficulty in selecting initial value of the relaxation parameter. We then adopt the relaxation sequence suggested by Tanaka and Kudo for stable convergence. Experimental results indicate that the proposed relaxation scheme for row-action algorithms can achieve stable convergence for larger subsets, while the corresponding OSEM algorithms fail to converge.

1. H. Malcolm Hudson and Richard S. Larkin, “ Accelerated image reconstruction using ordered subsets of projection data, ” IEEE Transactions on Medical Imaging, Vol. 13, No. 4, pp.601-609, Dec. 1994
2. Y. Censor, “Row-action methods for huge and sparse systems and their applications,” SIAM Review, Vol. 23, Issue 4, pp. 444-466, October 1981.
3. Jolyon Browne, and Alvaro R. De Pierro, "A Row-Action Alternative to the EM Algorithm for Maximizing Likelihoods in Emission Tomography," IEEE Trans. Med. Imag. Vol. 15, October 1996
4. J. Qi, R. Leahy, C. H. Hsu, T. Farquhar and S. Cherry, "Fully 3D Bayesian Image Reconstruction for ECAT EXACT HR+," IEEE Transaction on Nuclear Science, Vol. 45, No. 3, pp. 1096-1103, 1998.
5. E. Levitan and G. T. Herman, “A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Transactions on Medical Imaging, Vol. MI-6, No. 3, pp. 185-192, September 1987.
6. Alvaro R. De Pierro, and Michel Eduardo Beleza Yamagishi, "Fast EM-Like Methods for Maximum "A Posteriori" Estimates in Emission Tomography," IEEE transaction on medical imaging, vol. 20, no. 4, April 2001
7. M. E. Daube-Witherspoon, S. Matej, J. S. Karp, and R. M. Lewitt, “Application of the row action maximum likelihood algorithm with spherical basis functions to clinical PET imaging,” IEEE Transactions on Nuclear Science, Vol. 48, No. 1, pp. 24-30, February 2001.
8. Erkan Mumcuoglu, Richard Leahy, and Simon R. Cherry, "Bayesian reconstruction of PET images: methodology and performance analysis," Phys. Med. Biol. Vol.41, pp.1777–1807, 1996.
9. Erkan Mumcuoglu, Richard Leahy, Simon R. Cherry, and Zhenyu Zhou, "Fast Gradient-Based Methods for Bayesian Reconstruction of Transmission and Emission PET Image," IEEE Transactions on Medical Imaging, Vol.13, No. 4, pp.687-701, Dec. 1994.
10. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Transactions on Medical Imaging, Vol. MI-1, No. 2, pp. 113-122, Oct. 1982.
11. G. T. Herman and L. B. Meyer, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Transactions on Medical Imaging, Vol. 12, No. 3, pp. 600-609, September 1993.
12. Curtis F. Gerald and Patrick O. Wheatley, Applied numerical analysis, 6th, Addison Wesley, 1999.
13. P. J. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm,” IEEE Transactions on Medical Imaging, Vol. 9, No. 1, pp. 84-93, March 1990.
14. David G. Luenberger, Linear and nonlinear programming, 2nd, Addison Wesley, 1984.
15. G. Chinn and S. C. Huang, “A general class of preconditioners for statistical iterative reconstruction of emission computed tomography,” IEEE Transactions on Medical Imaging, Vol. 16, No. 1, pp. 1-10, February 1997.
16. G. Chinn and S. C. Huang, “A fast preconditioned conjugate gradient algorithm for regularized WLS reconstruction for PET,” Nuclear Science Symposium and Medical Imaging Conference Record, 1995 IEEE , Vol. 2, pp. 1297-1301, October 1995.
17. N. H. Clinthorne, T. S. Pan, P. C. Choao, W. L. Rogers, and J. A. Stamos, “Preconditioning methods for improved convergence rates in iterative reconstructions,” IEEE Transactions on Medical Imaging, Vol. 12, No. 1, pp. 78-83, March 1993.
18. Sangtae Ahn and Jeffrey A. Fessler, "Globally convergent ordered subsets algorithms: application to tomography," 2001 IEEE Nucl. Sci. Symp and Med. Imag. Conf. Rec.
19. J. A. Fessler and W. L. Rogers, “Spatial resolution properties of penalized likelihood image reconstruction: space-invariant tomographs,” IEEE Transactions on Image Processing, Vol. 5, No. 9, pp. 1346-58, September 1996.