近年來利用統計重建法重建正子斷層掃描影像，受到相當的重視。因為統計重建法可利用一精確的模型，來描述射源分布與投影資料之間的對應關係，並且可將影響重建影像品質的因素，像是系統幾何、衰減、偵檢器偵測效率等，加入系統偵檢模型中，使得重建影像有較佳的品質。重建過程中需要利用機率矩陣來計算期望值，由於它的矩陣很大，增加儲存及運算上的困難與負擔。為了解決這個問題，我們提出一新的機率矩陣，利用機率矩陣是稀疏與對稱的特性，將其縮小到完整大小的0.006%，改善原本機率矩陣龐大、冗餘的缺點。OSEM重建法是將投影資料區分成數個序列子集之後，依次作疊代運算，所以本文也會針對四種序列子集，比較其重建影像的結果，找出較合適的序列子集。OSEM重建法有效地解決MLEM重建法收斂速度緩慢，需要較長運算時間的缺點。當疊代次數增加到某一程度時，重建影像的雜訊也會逐漸增加，並且收斂的速度越快，雜訊增加的速度也越快。為了解決這個問題，本文提出modified OSEM重建法，將MREM重建法中多解析度的概念，運用在OSEM重建法的序列子集中，使其具備OSEM重建法的加速優點，以及改善OSEM重建法在影像通過誤差最低點之後，影像品質急速下降的缺點。

In the recent years, statistical PET image reconstruction is gaining attention. The statistical image reconstruction can describe the relationship between the source and the projection data by an accurate probability model which includes the parameters such as geometric, attenuation factor and detector efficiency. The size of the matrix is very huge, and it increases the burden for storing and computing. In order to overcome this problem, we presented a new probability matrix by exploiting its sparseness and the high degree of symmetry. We reduced the size to 0.006% of the full size for an image with size of 128 128. OSEM method groups projection data into an ordered sequence of subsets. We effectively improve the slow convergence and long computation time of MLEM methods. One problem in employing the method is the fact that, after a certain stage in the iterative process, the noise properties of the reconstructed images start to deteriorate. The noise in the reconstructed image increased with the convergent rate. To overcome this problem, we presented a modified OSEM method which exploits the multi-resolution approach of MREM method to the ordered subsets of OSEM methods. The modified OSEM method not only has the advantage of fast convergence, but also slows down the deterioration of the reconstructed images after certain stage.

[1] S. Vandenberghe, Y. D’Asseler, R. Van de Walle, T. Kauppinen, M. Koole, L. Bouwens, K. Van Laere, I. Lemahieu, and R.A. Dierckx, “Iterative Reconstruction Algorithms in Nuclear Medicine,” Computerized Medical Imaging and Graphics, vol. 25 pp. 105-111, 2001.
[2] G.L. Zeng, “Image Reconstruction --- A Tutorial,” Computerized Medical Imaging and Graphics, vol. 25, pp. 97-103, 2001.
[3] L. A. Shepp and Y. Verdi, “Maximum Likelihood Reconstruction for Emission Tomography,” IEEE Trans. Med. Imaging, vol. MI-1, no. 2, pp. 113-122, 1982.
[4] Y. Verdi, L. A. Shepp, and L. Kaufman, “A Statistical Model for Positron Emission Tomography,” J. Amer. Statist. Soc., vol. 80, no. 389, pp. 8-20, 1985.
[5] H. Malcolm Hudson and Richard S. Larkin, “Accelerated Image Reconstruction Using Ordered Subsets of Projection Data,” IEEE Trans. Med. Imag., vol. 13, no. 4, pp. 601-609, 1994.
[6] John M. Ollinger and Jeffrey A. Fessler, “Positron-Emission Tomography,” IEEE Sig. Proc. Mag., vol. 14, pp. 43-55, 1997.
[7] Kenneth Lange and Richard Carson, “EM Reconstruction Algorithms for Emission and Transmission Tomography,” J Comput. Assist. Tomogr., vol. 8, no. 2, pp. 306-316, 1984.
[8] Avinash C. Kak, and Malcolm Slaney, “Principles of Computerized Tomographic Imaging,” chapter 3, pp. 49-112, 1988.
[9] Kathleen M. Brennan, William W. Moses and Stephen E. Derenzo, “Advances in Positron Tomography for Oncology,” Nulc. Med. Biol., vol. 23, pp. 659-667, 1996.
[10] Brain F. Hutton, H. Malcolm Hudson, and Freek J. Beekman, “A Clinical Perspective of Accelerated Statistical Reconstruction,” Eur. J. Nucl. Med., vol. 24, pp. 797-808, 1997.
[11] C. Comtat, P.E. Kinahan, M. Defrise, C. Michel, and D.W. Townsend, “Fast Reconstruction of 3D PET Data with Accurate Statistical Modeling,” IEEE Trans. Nucl. Sci., vol. 45, no. 3, pp. 1083-1089, 1998.
[12] Chris Kamphuis, Freek J. Beekman and Max A. Viergever, “Evaluation of OS-EM vs. ML-EM for 1D, 2D and Fully 3D SPECT Reconstruction,” IEEE Trans. Nucl. Sci., vol. 43, no. 3, pp. 2018-2023, 1996.
[13] Jinyi Qi, Richard M. Leahy, Chinghan Hsu, Thomas H. Farquhar, and Simon R. Cherry, “Fully 3D Bayesian Image Reconstruction for the ECAT EXACT HR+,” IEEE Trans. Nucl. Sci., vol. 45, no. 3, pp. 1096-1103, 1998.
[14] C. M. Chen, S.-Y. Lee, and Z. H. Cho, “Parallelization of the EM Algorithm for 3-D PET Image Reconstruction,” IEEE Trans. Med. Imag., vol. 10, no. 4, pp. 513-522, 1991.
[15] Todd K. Moon, “The Expectation Maximization Algorithm,” IEEE Signal Processing, pp. 47-60, 1996.
[16] D. Visvikis, C. Cheze-LeRest, D.C. Costa, J. Bomanji, S. Gacinovic, and P.J. Ell, “Influence of OSEM and Segmented Attenuation Correction in the Calculation of Standardised Uptake Values for [18F]FDG PET,” Eur. J. Nucl. Med., vol. 28, pp. 1326-1335, 2001.
[17] Chin-Tu Chen, Valen E. Johnson, Wing H. Wong, Xiaoping Hu, and Charles E. Metz, “Bayesian Image reconstruction in Positron Emission Tomography,” IEEE Trans. Nucl. Sci., vol. 37, no. 2, pp. 636-641, 1990.
[18] Chris Kamphuis, Freek J. Beekman, Peter P. van Rijk, Max A. Viergever, “Dual Martrix Ordered Subsets Reconstruction for Accelerated 3D Scatter Compensation in Single- Photon Emission Tomography,” Eur. J. Nucl. Med., vol. 25, pp. 8-18, 1998.
[19] L. A. Shepp, Y. Vardi, J. B. Ra, S. K. Hilal, and Z. H. Cho, “Maximum Likelihood PET with Real Data,” IEEE Trans. Nucl. Sci., vol. NS-31, no. 2, pp. 910-913, 1984.
[20] A.J. Rockmore and A. Macovski, “A Maximum Likelihood Approach to Emission Image Reconstruction from Projections,” IEEE Trans. Nucl. Sci., vol. NS-23, pp. 1428-1432, 1976.
[21] Jinyi Qi, Richard M Leahy, Simon R Cherry, Arion Chatziioannou and Thomas H Farquhar, “ High-resolution 3D Bayesian Image Reconstruction Using the microPET Small-animal Scanner,” Phys. Med. Biol., vol. 43, pp. 1001-1013, 1998.
[22] Peter Schmidlin, Matthias E Bellemann and Gunnar Brix, “Subsets and Overrelaxation in Iterative Image Reconstruction,” Phys. Med. Biol., vol. 44, pp. 1385-1396, 1999.
[23] Linda Kaufman, “Implementing and Accelerating the EM Algorithm for Positron Emission Tomography,” IEEE Trans. Med. Imag., vol. MI-6, no. 1, pp. 37-51, 1987.
[24] Frederik A.A. de Jonge, and Koos (J.) A.K. Blokland, “Statistical Tomographic Reconstruction: How Many More Iterations to Go?” Eur. J. Nucl. Med., vol. 26, pp. 1247-1250, 1999.
[25] M. V. Ranganath, Atam P. Dhawan, and N. Mullani, “A Multigrid Expectation Maximization Reconstruction Algorithm for Positron Emission Tomography,” IEEE Trans. Med. Imag., vol. 7, no. 4, pp. 273-278, 1988.
[26] Amar Raheja, Timothy F. Doniere and Atam P. Dhawan, “Multiresolution Expectation Maximization Reconstruction Algorithm for Positron Emission Tomography using Wavelet Processing,” IEEE Trans. Nucl. Sci., vol. 46, no. 3, pp. 594-602, 1999.
[27] L.A. Shepp and B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci., vol. NS-21, pp. 21-43, 1974.
[28] A. P. Dempster, N. M. Larid, and D. B. Rubin, “Maximum Likelihood from Incomplete Data via the EM Algorithm,” J. Royal Statiscal Soc., Ser. B, vol. 39, no. 1, pp. 1-38, 1977.
[29] Jianying Li, Ronald J Jaszczak, Kim L Greer and R Edward Coleman, “Implementation of an Accelerated Iterative Algorithm for Cone-beam SPECT,” Phys. Med. Biol. vol. 39, no. 3, pp. 643-653, 1994.
[30] K. J. Kearfott, “Practical Considerations,” Comment in J. Amer. Statist. Ass., vol. 80, no. 389, pp. 26-28, 1985.