锥束CT迭代算法中投影排序与子集划分的研究
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摘要
在CT图像重建中,与常用的解析算法相比,迭代算法具有较强的抗噪能力,但计算量大、重建速度慢。锥束迭代算法中的投影排序和子集的选择,对锥束CT图像重建速度和重建结果有很大影响。本文以投影排序和子集划分作为研究对象,展开分析讨论。文章研究内容主要分为以下几个方面：
(1)投影使用顺序对代数重建技术(ART)收敛性有影响。分析了锥束圆轨迹扫描模式下射线间的相关性,给出了投影视角排序方案和投影视角下的射线排序方案。实验结果表明：投影视角排序和投影视角下的射线排序都可以提高算法的收敛速度和重建图像质量,但投影视角的排序提高算法收敛速度和重建图像质量的效果显著。将投影视角排序方案推广应用到螺旋轨迹和变旋转轴圆弧轨迹上,取得了较好的重建结果。
(2)射线投影存在相关性导致子集内包含统计信息量有所减少。为了使子集包含统计信息量达到最大,本文先用加权距离投影排序方案(WDS)和按子集水平间隔投影排序方案(SIS)对投影进行排序,使得相邻投影间的相关性尽可能小,然后将投影平均分配在子集中。仿真实验和实际实验结果表明：先投影排序后划分子集的方法可以提高有序子集算法收敛速度和重建图像质量。
(3)投影数据的噪声水平较大时,重建图像低频平滑区域的灰度变化频率变高,灰度变化幅度变大。针对该问题提出分域子集算法,该算法选择受噪声影响比较大的LAPLACE边缘检测算子对重建后的图像进行区域划分。高频区域选择小的子集水平,低频区域选择大的子集水平,进行图像重建。仿真实验和实际实验结果表明：在噪声水平比较大的情况下,该算法既具有小的子集水平的算法的抑制噪声扩散能力,又可以保持大的子集水平算法的收敛速度,并且能够重建出较好的实际图像。
In the CT image reconstruction, compared with the commonly used analytic method, iterative algorithm has strong anti-noise ability, but its calculation is complex, and the speed of reconstruction is slow. The order of applied projections and the selection of subset in the cone beam iterative algorithm have a great effect on speed of convergence and accuracy in reconstructed images. Taking applied projections sequence and order subset selection as research object, this paper proceeds to study. The mainly studying content of the paper follows aspects:
(1) Projection applying sequence of algebraic reconstruction technique (ART) has effect on the convergence. In the mode of cone-beam scan and circular track, the correlation between ray-projections is analyzed. Projection angle sort scheme and projection applying sort scheme on a projection angle are given. The results show that:Projection angle sort scheme and projection applying sort scheme on a projection angle can all improve the convergence speed and image quality, but projection angle sort scheme the results of improving the convergence speed and image quality are obvious. Projection angle sort scheme is used to the helical trajectory and circular arc changing axis of rotation trajectory and good results also are reconstructed.
(2) The correlation between ray-projections leads to the reduction of amount of the statistics information that the subset contains. In order to make amount of statistics information the subset contains maximum, this paper, firstly, sorts projections by the Weight Distance Scheme (WDS) and the Subset-level Interval-projection Scheme (SIS), to make the correlation between ray-projections as small as possible, and then projections are evenly distributed in the subsets. Simulation and actual experimental results show that:the method that projections are distributed in subsets after projections sort can improve convergence speed of the order subset algorithm and quality of images.
(3) When Projection data has high noise level, grayscale in the smooth low-frequency region of the reconstructed images will become higher frequency, and the grayscale fluctuation becomes larger. The sub-region subset algorithm is proposed. This algorithm selects LAPLACE edge detection operator seriously affected by noise to separate reconstruction images. The small level subset is selected in high-frequency region and the large level subset is selected in low-frequency region, and then images are being reconstructed. Simulation and actual experimental results show that:in the case of large noise level, this algorithm has the same ability of noise suppression with the small level subset, also maintains the convergence rate of the large level subset, and the great quality actual images are reconstructs.
(1)投影使用顺序对代数重建技术(ART)收敛性有影响。分析了锥束圆轨迹扫描模式下射线间的相关性,给出了投影视角排序方案和投影视角下的射线排序方案。实验结果表明：投影视角排序和投影视角下的射线排序都可以提高算法的收敛速度和重建图像质量,但投影视角的排序提高算法收敛速度和重建图像质量的效果显著。将投影视角排序方案推广应用到螺旋轨迹和变旋转轴圆弧轨迹上,取得了较好的重建结果。
(2)射线投影存在相关性导致子集内包含统计信息量有所减少。为了使子集包含统计信息量达到最大,本文先用加权距离投影排序方案(WDS)和按子集水平间隔投影排序方案(SIS)对投影进行排序,使得相邻投影间的相关性尽可能小,然后将投影平均分配在子集中。仿真实验和实际实验结果表明：先投影排序后划分子集的方法可以提高有序子集算法收敛速度和重建图像质量。
(3)投影数据的噪声水平较大时,重建图像低频平滑区域的灰度变化频率变高,灰度变化幅度变大。针对该问题提出分域子集算法,该算法选择受噪声影响比较大的LAPLACE边缘检测算子对重建后的图像进行区域划分。高频区域选择小的子集水平,低频区域选择大的子集水平,进行图像重建。仿真实验和实际实验结果表明：在噪声水平比较大的情况下,该算法既具有小的子集水平的算法的抑制噪声扩散能力,又可以保持大的子集水平算法的收敛速度,并且能够重建出较好的实际图像。
In the CT image reconstruction, compared with the commonly used analytic method, iterative algorithm has strong anti-noise ability, but its calculation is complex, and the speed of reconstruction is slow. The order of applied projections and the selection of subset in the cone beam iterative algorithm have a great effect on speed of convergence and accuracy in reconstructed images. Taking applied projections sequence and order subset selection as research object, this paper proceeds to study. The mainly studying content of the paper follows aspects:
(1) Projection applying sequence of algebraic reconstruction technique (ART) has effect on the convergence. In the mode of cone-beam scan and circular track, the correlation between ray-projections is analyzed. Projection angle sort scheme and projection applying sort scheme on a projection angle are given. The results show that:Projection angle sort scheme and projection applying sort scheme on a projection angle can all improve the convergence speed and image quality, but projection angle sort scheme the results of improving the convergence speed and image quality are obvious. Projection angle sort scheme is used to the helical trajectory and circular arc changing axis of rotation trajectory and good results also are reconstructed.
(2) The correlation between ray-projections leads to the reduction of amount of the statistics information that the subset contains. In order to make amount of statistics information the subset contains maximum, this paper, firstly, sorts projections by the Weight Distance Scheme (WDS) and the Subset-level Interval-projection Scheme (SIS), to make the correlation between ray-projections as small as possible, and then projections are evenly distributed in the subsets. Simulation and actual experimental results show that:the method that projections are distributed in subsets after projections sort can improve convergence speed of the order subset algorithm and quality of images.
(3) When Projection data has high noise level, grayscale in the smooth low-frequency region of the reconstructed images will become higher frequency, and the grayscale fluctuation becomes larger. The sub-region subset algorithm is proposed. This algorithm selects LAPLACE edge detection operator seriously affected by noise to separate reconstruction images. The small level subset is selected in high-frequency region and the large level subset is selected in low-frequency region, and then images are being reconstructed. Simulation and actual experimental results show that:in the case of large noise level, this algorithm has the same ability of noise suppression with the small level subset, also maintains the convergence rate of the large level subset, and the great quality actual images are reconstructs.
引文
[1]Herman G T. Image reconstruction from projections:The fundamentals computerized tomogr-aphy[M]. New York:Academic Press,1980.
[2]谢强.计算机断层成像技术：原理、设计、伪像和进展[M].张朝宗,郭志平,王贤刚等译.北京：科学出版社,2006.
[3]庄天戈.生物医学放射成像与IT的互动发展[J].计算机应用与软件,2006,23(10)：7-10.
[4]Kalender W A. X-ray computed tomography[J]. Physics in Medicine and Biology,2006, 51(13):29-423.
[5]霍修坤.锥束CT直接三维成像算法研究[D].合肥：安徽大学,2005.
[6]Taguehi, Aradate. Algorithm for image reconstruction in multi-slice CT[J]. Medical Physics, 1998,25(4):550-561.
[7]Mccollough C H, Zink F E. Performance evaluation of a multi-slice CT system[J]. Medical Physics,1999,26(11):2223-2230.
[8]Radon J. On the determination of function from their integrals along certain manifolds[J]. Mathematisch-Physicsche Klasse,1917,69:262-277.
[9]Zou Y, Pan X C. Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT[J]. Phys. Med. Biol.,2004,49:941-959.
[10]Smith B. Image reconstruction from cone-beam projections:necessary and sufficient conditions and reconstruction methods[J]. IEEE Trans. Med. Imaging,1985,4:14-25.
[11]Feldkamp L A, Davis L C, Kress J W. Practical cone-beam algorithm[J]. Journal of the Optical Society of America,1984,1(6):612-619.
[12]Katsevich A. Theoretically exact filtered back-projection-type inversion algorithm for Spiral CT[J]. SIAM Journal on Applied Mathematics,2002,62:2012-2026.
[13]Gordon R, Bender R, Herman G T. Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography[J]. J.Theo.Biol,1970,29: 471-481.
[14]Gilbert P. Iterative methods for the three-dimensional reconstruction of an object from projection[J]. J.Theo.Biol,1972,36:105-107.
[15]Dempster A P, Laird N M, Rubin D B. Maximum likelihood from incomplete data via the EM algorithm[J]. J.Royal.Stat.Soc.Series.B,1977,39:1-38.
[16]M. D. Altschuler, G. T. Herman, A. Lent. Fully three-dimensional image reconstruction from cone-beam sources[J]. in Proc. Conf. Pattern Recognition Image Processing,1978:194-199.
[17]M. D. Altschuler, G. T. Herman. Fully three-dimensional image reconstruction using series expansion methods in A Review of Information Processing in Medical Imaging[J]. A. B. Brill et al., Eds. Oak Ridge National Laboratory,1977:125-142.
[18]M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman. Y. H. Kuo, R. M. Lewitt. M. McKay, H. K. Tuy, J. K. Udupa, M. M. Yau. Demonstration of a software package for the reconstruction of a dynamically changing structure of the human heart from cone-beam X-ray projections[J]. J. Med. Syst.,1980,4(2):289-304.
[19]Ying Liu. Half-scan Cone-beam CT fluoroscopy with multiple X-ray sourse[J]. Med Phys, 2001,28:1466.
[20]Grass M.3-D Cone-beam CT reconstruction for circular trajectories [J]. Med Phys Biol, 2000,45:329.
[21]Wang G. Half-scan cone-beam x-ray microtomography formula[J]. Scanning, J.,1994,16(4): 216-220.
[22]J. G. Colsher. Iterative three-dimensional image reconstruction form tomography projections [J]. Comput. Grap. Image Processing,1977,6:513-537.
[23]M. Schlindwein. Iterative three-dimensional reconstruction form twin-cone beam project-tions[J]. IEEE trans. Nucl.Sci.,1978,(35)5:1135-1143.
[24]S. Matej, G. T. Herman, T. K. Narayan, S. S. Furuie, R. M. Lewitt, P. E. Kinahan. Evaluation of task-oriented performance of several fully 3-D PET reconstruction algorithms[J]. Phys, Med. Biol,1994,39:355-367.
[25]G. T. Herman, A relaxation method for reconstruction objects form noisy X-ray[J]. Math. Programming,1975,8:1-19.
[26]G.T. Herman, L.B. Meyer. Algebraic reconstruction technique can be made computationally efficient[J]. IEEE Trans. Med. Imag.,1993,12(3):600-609.
[27]Guan H, Gordon R. A projection access order for speedy convergence of ART:A multilevel scheme for computed tomography[J]. Phys Med Biol,1994,39:1005-1022.
[28]Mueller K. The weighted distance scheme:A globally optimizing projuction ordering method for ART[J]. Appeared in Transactions on Medical Imaging,1997,16(2):1-14.
[29]T. Kohler. A Projection Access Scheme for Iterative Reconstruction Based on the Golden Section[J] in Proc. of the 2004 IEEE Nuclear Science and Medical Imaging Conference, CDROM M10-200, Rome, Italy,2004,12(16-22):3961-3965.
[30]王宏钧,路宏年，傅健.代数重建技术中投影序列选择次序的研究[J].光学技术，2006,32(3)：389-392.
[31]H. M. Hudson, R. S. Larkin. Accelerated image reconstruction technique using ordered subsets ofprojection data[J]. IEEE Tram. Medical Imaging,1994,13(4):601-609.
[32]C.Kamphuis, F.J. Beekman. Accelerated iterative transmission CT reconstruction using an ordered subsets convex algorithm[J]. IEEE Ttans. Medical Imaging,1998,11(17):1101-1105.
[33]C.Byrne. Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods[J]. IEEE Trans. Imag.,1998,7:100-109.
[34]C. L. Byrne. Block-iterative methods for image reconstruction from projections [J]. IEEE Trans. Image Processing,1996,3(5):792-794.
[35]R. M. Lewitt, G. Muehllehner. Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation[J]. IEEE Trans. Med. Imaging,1986,3(5):16-22.
[36]Dan J.Kadrmas. Statistically regulated and adaptive EM reconstruction for emission computed tomography[J]. IEEE Trans. Nuclear Science,2001,48:790-797.
[37]孔慧华.加速图像重建的迭代算法研究[D].太原：中北大学,2006.
[38]冀东江.三维锥束CT迭代重建算法的研究[D].重庆：重庆大学,2008.
[39]薛震.锥束螺旋CT的迭代算法研究[D].太原：中北大学,2009.
[40]吕林根,许子道.解析几何[M].北京：高等教育出版社,1982.
[41]李春葆,苏光奎.数据结构与算法教程[M].北京：清华大学出版社,2005.
[42]Kuhn H W, Tucker A W. Nonlinear Programming [D]. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkely, California,1951.
[43]吴琨,潘晋孝.三维锥束代数重建算法的投影排序方案[J].CT理论与应用研究,2010,19(4)：61-70.
[44]张德丰.详解MATLAB数字图像处理[M].北京：电子工业出版社,2010.
[2]谢强.计算机断层成像技术：原理、设计、伪像和进展[M].张朝宗,郭志平,王贤刚等译.北京：科学出版社,2006.
[3]庄天戈.生物医学放射成像与IT的互动发展[J].计算机应用与软件,2006,23(10)：7-10.
[4]Kalender W A. X-ray computed tomography[J]. Physics in Medicine and Biology,2006, 51(13):29-423.
[5]霍修坤.锥束CT直接三维成像算法研究[D].合肥：安徽大学,2005.
[6]Taguehi, Aradate. Algorithm for image reconstruction in multi-slice CT[J]. Medical Physics, 1998,25(4):550-561.
[7]Mccollough C H, Zink F E. Performance evaluation of a multi-slice CT system[J]. Medical Physics,1999,26(11):2223-2230.
[8]Radon J. On the determination of function from their integrals along certain manifolds[J]. Mathematisch-Physicsche Klasse,1917,69:262-277.
[9]Zou Y, Pan X C. Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT[J]. Phys. Med. Biol.,2004,49:941-959.
[10]Smith B. Image reconstruction from cone-beam projections:necessary and sufficient conditions and reconstruction methods[J]. IEEE Trans. Med. Imaging,1985,4:14-25.
[11]Feldkamp L A, Davis L C, Kress J W. Practical cone-beam algorithm[J]. Journal of the Optical Society of America,1984,1(6):612-619.
[12]Katsevich A. Theoretically exact filtered back-projection-type inversion algorithm for Spiral CT[J]. SIAM Journal on Applied Mathematics,2002,62:2012-2026.
[13]Gordon R, Bender R, Herman G T. Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography[J]. J.Theo.Biol,1970,29: 471-481.
[14]Gilbert P. Iterative methods for the three-dimensional reconstruction of an object from projection[J]. J.Theo.Biol,1972,36:105-107.
[15]Dempster A P, Laird N M, Rubin D B. Maximum likelihood from incomplete data via the EM algorithm[J]. J.Royal.Stat.Soc.Series.B,1977,39:1-38.
[16]M. D. Altschuler, G. T. Herman, A. Lent. Fully three-dimensional image reconstruction from cone-beam sources[J]. in Proc. Conf. Pattern Recognition Image Processing,1978:194-199.
[17]M. D. Altschuler, G. T. Herman. Fully three-dimensional image reconstruction using series expansion methods in A Review of Information Processing in Medical Imaging[J]. A. B. Brill et al., Eds. Oak Ridge National Laboratory,1977:125-142.
[18]M. D. Altschuler, Y. Censor, P. P. B. Eggermont, G. T. Herman. Y. H. Kuo, R. M. Lewitt. M. McKay, H. K. Tuy, J. K. Udupa, M. M. Yau. Demonstration of a software package for the reconstruction of a dynamically changing structure of the human heart from cone-beam X-ray projections[J]. J. Med. Syst.,1980,4(2):289-304.
[19]Ying Liu. Half-scan Cone-beam CT fluoroscopy with multiple X-ray sourse[J]. Med Phys, 2001,28:1466.
[20]Grass M.3-D Cone-beam CT reconstruction for circular trajectories [J]. Med Phys Biol, 2000,45:329.
[21]Wang G. Half-scan cone-beam x-ray microtomography formula[J]. Scanning, J.,1994,16(4): 216-220.
[22]J. G. Colsher. Iterative three-dimensional image reconstruction form tomography projections [J]. Comput. Grap. Image Processing,1977,6:513-537.
[23]M. Schlindwein. Iterative three-dimensional reconstruction form twin-cone beam project-tions[J]. IEEE trans. Nucl.Sci.,1978,(35)5:1135-1143.
[24]S. Matej, G. T. Herman, T. K. Narayan, S. S. Furuie, R. M. Lewitt, P. E. Kinahan. Evaluation of task-oriented performance of several fully 3-D PET reconstruction algorithms[J]. Phys, Med. Biol,1994,39:355-367.
[25]G. T. Herman, A relaxation method for reconstruction objects form noisy X-ray[J]. Math. Programming,1975,8:1-19.
[26]G.T. Herman, L.B. Meyer. Algebraic reconstruction technique can be made computationally efficient[J]. IEEE Trans. Med. Imag.,1993,12(3):600-609.
[27]Guan H, Gordon R. A projection access order for speedy convergence of ART:A multilevel scheme for computed tomography[J]. Phys Med Biol,1994,39:1005-1022.
[28]Mueller K. The weighted distance scheme:A globally optimizing projuction ordering method for ART[J]. Appeared in Transactions on Medical Imaging,1997,16(2):1-14.
[29]T. Kohler. A Projection Access Scheme for Iterative Reconstruction Based on the Golden Section[J] in Proc. of the 2004 IEEE Nuclear Science and Medical Imaging Conference, CDROM M10-200, Rome, Italy,2004,12(16-22):3961-3965.
[30]王宏钧,路宏年，傅健.代数重建技术中投影序列选择次序的研究[J].光学技术，2006,32(3)：389-392.
[31]H. M. Hudson, R. S. Larkin. Accelerated image reconstruction technique using ordered subsets ofprojection data[J]. IEEE Tram. Medical Imaging,1994,13(4):601-609.
[32]C.Kamphuis, F.J. Beekman. Accelerated iterative transmission CT reconstruction using an ordered subsets convex algorithm[J]. IEEE Ttans. Medical Imaging,1998,11(17):1101-1105.
[33]C.Byrne. Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods[J]. IEEE Trans. Imag.,1998,7:100-109.
[34]C. L. Byrne. Block-iterative methods for image reconstruction from projections [J]. IEEE Trans. Image Processing,1996,3(5):792-794.
[35]R. M. Lewitt, G. Muehllehner. Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation[J]. IEEE Trans. Med. Imaging,1986,3(5):16-22.
[36]Dan J.Kadrmas. Statistically regulated and adaptive EM reconstruction for emission computed tomography[J]. IEEE Trans. Nuclear Science,2001,48:790-797.
[37]孔慧华.加速图像重建的迭代算法研究[D].太原：中北大学,2006.
[38]冀东江.三维锥束CT迭代重建算法的研究[D].重庆：重庆大学,2008.
[39]薛震.锥束螺旋CT的迭代算法研究[D].太原：中北大学,2009.
[40]吕林根,许子道.解析几何[M].北京：高等教育出版社,1982.
[41]李春葆,苏光奎.数据结构与算法教程[M].北京：清华大学出版社,2005.
[42]Kuhn H W, Tucker A W. Nonlinear Programming [D]. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkely, California,1951.
[43]吴琨,潘晋孝.三维锥束代数重建算法的投影排序方案[J].CT理论与应用研究,2010,19(4)：61-70.
[44]张德丰.详解MATLAB数字图像处理[M].北京：电子工业出版社,2010.