摘要
锥束CT在数据采集的速度、空间分辨率以及射线的利用率等方面，明显优于二维平行束和扇束CT，这对医学影像诊断和工业无损检测有着重要的意义。大面积非晶硅平板探测器在工业透射射线成像领域得到越来越多的应用，本文针对平板探测器下采集到的锥束投影数据的特点，系统地研究了三维锥束CT图像重建方法，主要研究内容包括：
1.根据笛卡尔坐标与极坐标的转换关系，利用正、余弦函数性质，提出了扇束等距CT极坐标反投影重建快速算法。实验表明该算法的重建速度为传统滤波反投影重建算法的4倍以上，而且不会带来新的重建误差，已成功地将此方法延伸应用到三维锥束重建。
2.研究了三维锥束FDK重建算法,分析了其重建特点，并推导了计算机实现的数值化计算方法。根据该算法各重建步骤的特点，提出了FDK优化算法，实验结果表明本文提出的FDK优化算法不但有效地缩短了重建时间，而且其重建质量优越于标准FDK重建算法。
3.研究了几种常用的滤波函数，分析了各滤波函数采样序列的特点，在此基础上提出了R-L和Cosine新型混合滤波器，该滤波器结合了R-L和Cosine两种滤波器的优点，在能够保证图像空间分辨率的同时，相对地提高了密度分辨率，对高低频信息作了很好的折衷。实验结果表明该滤波器重建图像误差相对较小，同时具有平滑作用。
4.深入研究了一种基于平板探测器的数据重排重建算法——F-FDK算法。该算法借鉴P-FDK算法的思路，并吸收了T-FDK算法的优点，将锥束射线重排为垂直面内为扇形的平行束射线，然后利用平行束重建原理进行重建。该算法对m、n两个方向的投影数据都进行了转换，实现了真正的数据重排。实验结果表明该算法减少了由于锥角增大而导致的伪影，重建图像的效果相对较好。
Cone-beam CT outperform 2D parallel-beam and fan-beam CT in the aspects ofacquisition data speed, spatial resolution and ray utilization rate, which are very significant formedical diagnosis and industrial non-destructive testing. Large-areas amorphous siliconflat-panel detector becomes more and more popular in the field of industry ray imaging. Inthis paper, 3D cone-beam CT image reconstruction method is researched systematically,according to the characteristics of the cone-beam projection data based on the flat-paneldetector. The main contents of the paper include the following:
According to the transformation relationship between Cartesian coordinates and polarcoordinates, this paper proposes a fast reconstruction algorithm for the back-projectionalgorithm of the polar coordinates for fan-beam collinear equispaced CT based on the featuresof sine and cosine function. The experimental results show that, compared with the traditionalfiltered back-projection algorithm, this method can improve the reconstruction speed by morethan 4 times, and will not bring new reconstruction error. This method has been successfullyapplied to the reconstruction of 3D cone-beam.
The 3D cone-beam FDK reconstruction algorithm is discussed and the characteristics ofthis algorithm are analyzed. The numerical calculation method of the computer realization isalso derived in this paper. Moreover, the FDK optimization algorithm is proposed accordingto the characteristics of the reconstruction steps. The experimental results show that theproposed optimization algorithm can effectively shorten the reconstruction time and it issuperior to the standard FDK reconstruction algorithm in the reconstruction image quality.
On the basis of discussing several common filter functions and analyzing thecharacteristics of the sample sequence for each filter function, a mixed filter functioncombining with the R-L and Cosine filer function is proposed in this paper. And the proposed filter integrates with the advantages of R-L and Cosine filter function. While this filter canguarantee the image spatial resolution, it relatively increases the density. This filter makes agood compromise between the high and low frequency information. The experimental resultsshow that the error of the reconstructed image by the proposed filter is relatively small. At thesame time, it smoothed the image.
This paper also focuses on the study of the projection data rebinning algorithm namedF-FDK based on flat panel detector. This algorithm which refers to the idea of P-PDKalgorithm and absorbs the advantages of T-FDK algorithm, rearranges the cone-beam datainto fan-beam data in the vertical plane, then reconstructs image by the principle of theparallel-beam. This algorithm truly achieves the data rebinning by converting projection datain both directions of m and n. The experimental results show that F-FDK algorithm reducedthe artifact which caused by the cone angle increasing, and the reconstruction image qualitycan be obviously improved.
引文
[1]庄天戈.CT原理和算法[M].上海：上海交通大学出版社，1992：1-99.
[2]Jiang Hsieh. Computed Tomography: Principle, Design, Artifacts and RecentAdvances[M].北京：科学出版社，2006：1-71.
[3]A.C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging[M]. Piscataway,NJ: IEEE Service Center, 1988.
[4]E1441_00. Standard guide for computed tomography[S]. West Conshohocken: ASTM,2000.
[5]刘德镇.现代射线检测技术[M].北京：中国标准出版社，1999：250-252.
[6]J.Radon. Berichte S?chsische Akademie der Wissenschaft. Math.Phys, 1917, 30:262-277.
[7]Cormark A.M. Representation of a function by its line integrates with some radiologicalapplication[J]. Journal of Applied Physics,1963, 34(9):2722-2727.
[8]G.N. Hounsfield. Computerized transverse axial scanning(tomography):Part 1. Descriptionof system[J]. British Journal of Applied of Radiology, 1973, 46:1016-1022.
[9]G.T.Herman. Image reconstruction from projections, the fundamentals of computerizedtomography[M]. New York: Computer Science and Applied Math, 1980.
[10]Tuy H.K. An inversion formula for cone-beam reconstruction[J]. SIAM Journal ofApplied Mathematics, 1983, 43:546-552.
[11]Bruce D. Smith, JiXing Chen. Improvement of a Novel Cone-Beam ReconstructionMethod[J]. IEEE Transactions on Medical Imaging, June 1992, 11(2):260-266.
[12]Pierre Grangeat. Mathematical framework of cone beam 3D reconstruction Via firstderivative of the radon transform. In Mathematical Methods in Tomography (Eds. HermanCT.Louis AK.Natterer F). Leature Notes in Mathematics. Springer-Verlag, 1990:66-96.
[13]Rolf Clark, Michel Defrise. Overview of reconstruction algorithms for exact cone-beamtomography[J]. Spie , 1994, 2299:230-241.
[14]Rolf Clark, Michel Defrise. A filtered-backprojection cone-beam algorithm for generalorbits with implementation details for the two-orthogonal-circles orbit[J]. IEEE, 1993,3:1590-1594.
[15]M. Defrise, R. Clack. A Cone-Beam Reconstruction Algorithm Using Shift-VariantFiltering and cone-beam Backprojection[J]. IEEE Transactions On Medical Imaging, March1994, 13(1):186-194.
[16]Fredecie Noo, Rolf Clark. A Cone-Beam Reconstruction from General Discrete VertexSets using Radon Rebinning Algorithms[J]. IEEE Transactions On Nuclear Science, June1997, 44(3):1309-1316.
[17]F. Noo, M. Defrise, R.Clack et al. Stable and efficient shift-variant algorithm forcircle-plus-linesorbits in cone-beam CT[J]. IEEE, 1996, 3:539-542.
[18]Fredecie Noo, Michel Defrise, Rolf Clark. Direct Reconstruction of Cone-Beam DataAcquired with a Vertex Path Containing a Circle[J]. IEEE Transactions on Image Processing,June 1998, 7(6):854-867.
[19]Katsevich. Ana1ysis of an exact inversion algorithm for spiral cone-beam CT [J]. Phys.Med.Bio. ,2002, 47:2583-2597.
[20]Katsevich. Theoretically exact filetered back-projection-type inversion algorithm forspiral CT[J]. SIAM J. Appl, Math. ,2002, 62(7):2012-2026.
[21]陈志强.锥束CT精确重建算法研究最新进展[J].CT理论与应用研究，2005.8,14(3)：65-70.
[22]L.A.Feldkamp, L.C.Davis, and J.W.Kress. Practical cone-beam algorithm[J]. Opt. Soc.Am. A. 1984, 1(6):612-619.
[23]肖永顺.大型工业CT的快速迭代重建算法及可视化技术研究[D].北京:清华大学,2003.
[24]Ge Wang, Tein-Hsiang Lin, Ping-chin Cheng, Douglas M. Shinozaki. A generalCone-beam Reconstruction Algorithm[J]. IEEE Trans. on Medical Imaging, 1993,12(3):486-495.
[25]Ben Wang, Hong Liu, Ge Wang. Generalized Feldkamp Image Reconstruction fromEquiangular Cone-Beam Projection Data[C]. Computer-Based Medical Systems, IEEESymposium on. U.S.:2000:123-128.
[26]Henrik Turbell. Cone-beam reconstruction using filtered backprojection[D].Sweden:Departmet of Electrical Engineering Linkoping University, 2001.
[27]Grass, M., Kohler, Th., Proksa, R.. Weighted hybrid cone beam reconstruction for circulartrajectories[C]. IEEE Medical Imaging. Lyon, France. U.S. IEEE, 2000, 2:15-1, 15-2.
[28]张剑，陈志强.三维锥形束CT成像FDK重建算法发展综述[J].中国体视学与图像分析，2005,10(2):116-120.
[29]曾凯,陈志强,张丽,赵自然.基于同心圆轨道的锥形束CT重建算法[J].清华大学学报(自然科学版),2004,44(6):725-727,731.
[30]Wang G., Liu Y., Lin T.H., Cheng P C.Half-scan cone-beam x-ray microtomographyformula[J]. Scanning Microscopy, 1994, 16(4):216-220.
[31]Liu Y.,Liu H., Wang G. Half-scan cone-beam CT fluoroscopy with multiple x-raysources[J]. Med.Phys., 2001, 28(7):1466-1471.
[32]傅健,路宏年,张全红.扇束工业CT重建算法速度优化[J].CT理论与应用研究,2002,11(3):16-19.
[33]傅健,路宏年.扇束工业CT滤波反投影重构算法的快速实现[J].计算机应用研究,2003,(3):51-53.
[34]R.Gordon, R.Bender, G.T.herman. Algebraic Reconstruction Techniques(ART) for threeDimensional Electron Micoscopy and X-ray Photography[J]. J.Theor.Biol., 1970, 29:471-481.
[35]P.Gilbert. Iterative methods for the three-dimensional reconstruction of an object fromprojection[J]. J.theor. Biol., 1972, 36:105-107.
[36]P.P.B.Eggermont, G.T.Herman, A.Lent. Iterative Algorithms for Large Partitioned LinearSystems with Applications to Image Reconstruction[J]. Linear Algebra Appl, 1981, 40:37-67.
[37]Y.Censor. Parallel Application of Block-Iterative Methods in Medical Imaging andRadiation Therapy[J]. Mathematical Programming, 1988, 42:307-325.
[38]G.T.herman, H.Levkowitz, H.K.Tuy, S.Mcromick. Multilevel Image Reconstruction inMulti-resolution Image Processing and Analysis[J]. Springer-Verlag, Berlin, 1984, 121-135.
[39]R.L.Kashyap, M.C.Mittal. Picture reconstruction from projections[J]. IEEE. Trans.Computer, 1975, 24: 915-923.
[40]王小璞.一种块迭代的快速代数重建算法[J].CT理论与应用研究,2002.9,9:10-12.
[41]张朋.几种CT图像重建算法的研究和比较[J].CT理论与应用研究,2001,10(4):4-9.
[42]张剑等.三维锥形束CT成像FDK重建算法发展综述[J].中国体视学与图像分析，2005,10(2):116-121.
[43]L.A.Shepp, Y.Vardi. Maximum likelihood reconstruction for mission and tomography[J].IEEE Trans on medical imaging, 1982, MI-1(2):113-122.
[44]H.M.Hudson, R.S.Larkin. Accelerated image reconstruction using ordered subsets ofprojection data[J]. IEEE Trans on medical imaging,1994, 13(4):601-609.
[45]Y.Uchiyama, K.Ogawa. MAP-EM Image Reconstruction using a k-Neighbor Method inEmission CT[C]. Conference Record of IEEE Nuclear Science Symposium and MedicalImaging Conference, 1992, 2:1144-1146.
[46]聂聪.不同平板探测器DR的比较研究[J].医疗设备信息，2007.4，21(4)：87-88.
[47]唐杰,张丽,高文焕.基于平板探测器的锥束CT系统综述[J].中国体视学与图像分析,2004,9(4)：65-70.
[48]陈娟.X射线平板探测器成像条件优化及成像系统建模技术研究[D]，北京：北京航空航天大学，2004.
[49]肖勇顺，陈志强，张丽等.基于数字平板探测器的高能X射线成像实验研究[J].光学技术，2003.11，29(6):660-663.
[50]Varian Medical System. PaxScan2520VIP-9 System & Service Guide. June 2000.
[51]Sylvie Chapuy,Marc Dimcovski. Real-time Flat-Panel Imaging System and Control forX-ray and Neutron Detection[J]. IEEE TRANSACTIONS ON NUCLEAR SCIENCE. 2001,48(6):2357-2364.
[52]Weifu Fang, Kazufumi Ito, David A. Redfern. A homogenization model for laserbeam-induced current imaging and detection of non-uniformities in semiconductor arrays[J].Journal of Computational and Applied Mathematics, October, 2006, 194(2):395-408.
[53]H Gao, L Zhang, Z Chen, Y Xing. Beam Hardening Correction for Middle-EnergyIndustrial Computerized Tomography[J]. IEEE Transactions on Nuclear Science, October,2006, 53(5):796-2807.
[54]J M M Anderson, B A Mair, M Rao, et al. Weighted least-squares reconstruction methodsfor positron emission tomography. IEEE Trans Med Imaging. 1997, 16(2):159-165.
[55]L A Feldkamp, L C Davis, J W Kress. Practical Cone-beam Algorithm[J]. J. Opt. Soc.Am. A, 1984, 1(6):1008-1014.
[56]刘晓平.扇束卷积反投影法的程序优化[J].CT理论与应用研究,1996,5(1):5-37.
[57]张朝宗，郭志平等.用几何参数表方法实现快速重建CT图像[J].清华大学学报，1998，38(7)：47-49.
[58]王贤刚，郭志平等.用几何参数表实现快速三维CT重建图像[J].清华大学学报，2002，42(5):655-658.
[59]S Basu, Y Bresler. O(N3logN) backprojection algorithm for the 3-D Radon transform[J].IEEE Transactions on Medical Imaging, 2002, 21(2):76-88.
[60]G.N. Ramachandran and A.V. Lakshminarayanan. Three-dimensional Reconstructionfrom Radiographs and Electron Micrographs: Application of Convolutions instead of FourierTransforms[J]. Proc. Nat. Acad. Sci. USA, 1971, 68(9):2236-2240.
[61]L.A. Shepp and B.F. Logan. The fourier reconsruction of a head section[J], IEEE Trans.Nucl. Sci., 1974, NS-21:21-43.
[62]刘俊杰.CT投影数据的计算机模拟及预处理研究[D].哈尔滨：哈尔滨工业大学，2004.
[63]翟静.三维锥束CT中滤波反投影算法的研究[D].太原：中北大学，2008.
[64]M. Grass, T. Khler, R. Proksa. 3D cone-beam reconstruction for circular trajectories[J].Phys. Med. Biol., 2000, 45:329-347.