锥形束三维XCT重建算法研究
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摘要
X射线CT (Computed Tomography计算机断层摄影术)自1972年问世以来得到了越来越广泛的应用。CT的广泛应用又反过来推动了对它的研究,使它得到进一步的发展,在30年的发展过程中,它基本上经历了几次大的变革。CT的这些发展变化主要体现在两个方面,一是提高扫描速度,二是改善图像的质量。
X射线CT图像是由计算机对物体的投影数据进行计算而获得的,因此CT的成像过程可以分为两大步,一是用X光系统进行投影数据的采集,获得充分的投影数据;二是图像重建,用计算机对投影数据进行计算来获得断层图像。扫描模式可以分为三种,平行束、扇形束和锥形束。平行束投影在工程上是不能直接实现的。和扇形束投影相比,锥形束投影一次性获得的信息量更大,因此锥形束投影结构有利于提高扫描速度和图像质量。但是锥形束投影数据和断层图像之间的关系很复杂,这会使计算的复杂度大大增加。近十年来,出现了很多针对锥形束CT的研究,但基本上都不是很成熟,到目前为止还没有出现真正意义上的商业化锥形束CT。针对CT目前存在的问题,本文主要研究的内容有以下几点:
1.三维重建算法的可靠性和准确性的研究过程中,模型设计以及投影数据模拟是必不可少的一部分。本文以三维Shepp-Logan头部模型为例,详细介绍了其模型设计思路和仿真投影数据的计算方法,并利用经典FDK重建算法的重建结果验证了模型设计思路的准确性和投影数据计算的可行性。
2.针对当前螺旋锥束CT存在算法实现复杂,重建相当耗时的问题,本论文提出了一种螺旋锥束近似重建算法,并在提高重建速度方面进行了研究。主要从算法结构、实现技巧及代码优化等方面研究了几种快速重建算法,计算机模拟实验证明了这些方法可有效提高图像重建的速度。
3.在无约束最优化为基础的图像重建问题中,为了提高迭代效率及重建图像质量,首次提出将变度量法应用到图像重建中,并且介绍了最速下降法和共轭梯度法等算法。通过模拟实验,在迭代的次数和重建图像的质量方面比较了几种无约束最优化重建算法,得出变度量法在迭代效率和重建图像质量方面优于其它算法。
本论文还对目前的一些常见的锥形束扫描重建算法进行了分析和比较,按照轨道类型、是否精确等进行了分类,并且讨论了它们各自的特点。介绍的锥束扫描重建算法有FDK、P-FDK、T-FDK、HT-FDK和S-FDK等。
X-ray CT has been widely used in the world, since the birth of the first CT (Computed Tomography) in 1972。The broad using of CT also generates great impetus for CT research. Several big changes have been taken place during the past 30 years. These changes mainly involve in two aspects--speed and image quality.
X-ray CT image is calculated from projection data by computer. So there are two steps to get image. Firstly, use X-ray to collect projection data. Secondly, calculate projection data to get image. There are three kinds of scan mode: parallel-beam, fan-beam and cone-beam. Parallel-beam projection data can’t be realized in the project directly. Compared with the fan-beam projection data, the cone-beam projection data get more information once. So it is in favor of improving scan speed and image quality. But the relationship between cone-beam projection data and the cross section image is very complex, and it increases the complexity of reconstruction algorithm and scan frame. In recent years, many research in the cone-beam scan technology appeared .But many of the algorithms are not very perfect, so the commercial cone-beam CT is not appeared at present. Three main questions have been researched in this thesis in order to solve the problem existed in CT.
1. In the process of research in the accuracy of model computation and the feasibility of three-dimension reconstruction algorithm, the design of model and computer simulation of projection data is a necessary part. 3D Shepp-Logan head phantom is used as an example. The thesis introduces its design and the projection simulation method in detail. The projection data simulation is performed based on 3D Shepp-Logan head phantom. The results of image reconstruction using FDK algorithm indicate the accuracy of model design and the feasibility of computing projection data.
2. Cone-beam helical CT is complex and time-consuming in reconstruction algorithm for the moment. The helical Cone-Beam approximate reconstruction algorithm is introduced in the thesis. And the thesis researches on reconstruction speed from the structure of algorithm、the technique of implementation and the optimization of code. Computer simulation experiment express that the methods can efficiently improve the speed of image reconstruction.
3. In the image reconstruction basing on optimization without constraint,in order to improve iterative efficiency and reconstructed quality, the thesis firstly applies the variable metric method to reconstruct image, and describes variable metric method、steepest descent Method、conjugate gradient method respectively. The thesis compared three algorithms above at iterative efficiency and reconstruction quality respectively by simulating experiment. The result shows that variable metric method is better than other algorithms in the aspects of iterative efficiency and reconstruction quality.
In this thesis, some familiar cone-beam scan reconstruction algorithms are also analyzed. These algorithms can be sorted by scan mode, reconstruction precision. The cone-beam scan reconstruction algorithms are introduced in this paper include FDK、P-FDK、T-FDK、HT-FDK and S-FDK, etc.
X射线CT图像是由计算机对物体的投影数据进行计算而获得的,因此CT的成像过程可以分为两大步,一是用X光系统进行投影数据的采集,获得充分的投影数据;二是图像重建,用计算机对投影数据进行计算来获得断层图像。扫描模式可以分为三种,平行束、扇形束和锥形束。平行束投影在工程上是不能直接实现的。和扇形束投影相比,锥形束投影一次性获得的信息量更大,因此锥形束投影结构有利于提高扫描速度和图像质量。但是锥形束投影数据和断层图像之间的关系很复杂,这会使计算的复杂度大大增加。近十年来,出现了很多针对锥形束CT的研究,但基本上都不是很成熟,到目前为止还没有出现真正意义上的商业化锥形束CT。针对CT目前存在的问题,本文主要研究的内容有以下几点:
1.三维重建算法的可靠性和准确性的研究过程中,模型设计以及投影数据模拟是必不可少的一部分。本文以三维Shepp-Logan头部模型为例,详细介绍了其模型设计思路和仿真投影数据的计算方法,并利用经典FDK重建算法的重建结果验证了模型设计思路的准确性和投影数据计算的可行性。
2.针对当前螺旋锥束CT存在算法实现复杂,重建相当耗时的问题,本论文提出了一种螺旋锥束近似重建算法,并在提高重建速度方面进行了研究。主要从算法结构、实现技巧及代码优化等方面研究了几种快速重建算法,计算机模拟实验证明了这些方法可有效提高图像重建的速度。
3.在无约束最优化为基础的图像重建问题中,为了提高迭代效率及重建图像质量,首次提出将变度量法应用到图像重建中,并且介绍了最速下降法和共轭梯度法等算法。通过模拟实验,在迭代的次数和重建图像的质量方面比较了几种无约束最优化重建算法,得出变度量法在迭代效率和重建图像质量方面优于其它算法。
本论文还对目前的一些常见的锥形束扫描重建算法进行了分析和比较,按照轨道类型、是否精确等进行了分类,并且讨论了它们各自的特点。介绍的锥束扫描重建算法有FDK、P-FDK、T-FDK、HT-FDK和S-FDK等。
X-ray CT has been widely used in the world, since the birth of the first CT (Computed Tomography) in 1972。The broad using of CT also generates great impetus for CT research. Several big changes have been taken place during the past 30 years. These changes mainly involve in two aspects--speed and image quality.
X-ray CT image is calculated from projection data by computer. So there are two steps to get image. Firstly, use X-ray to collect projection data. Secondly, calculate projection data to get image. There are three kinds of scan mode: parallel-beam, fan-beam and cone-beam. Parallel-beam projection data can’t be realized in the project directly. Compared with the fan-beam projection data, the cone-beam projection data get more information once. So it is in favor of improving scan speed and image quality. But the relationship between cone-beam projection data and the cross section image is very complex, and it increases the complexity of reconstruction algorithm and scan frame. In recent years, many research in the cone-beam scan technology appeared .But many of the algorithms are not very perfect, so the commercial cone-beam CT is not appeared at present. Three main questions have been researched in this thesis in order to solve the problem existed in CT.
1. In the process of research in the accuracy of model computation and the feasibility of three-dimension reconstruction algorithm, the design of model and computer simulation of projection data is a necessary part. 3D Shepp-Logan head phantom is used as an example. The thesis introduces its design and the projection simulation method in detail. The projection data simulation is performed based on 3D Shepp-Logan head phantom. The results of image reconstruction using FDK algorithm indicate the accuracy of model design and the feasibility of computing projection data.
2. Cone-beam helical CT is complex and time-consuming in reconstruction algorithm for the moment. The helical Cone-Beam approximate reconstruction algorithm is introduced in the thesis. And the thesis researches on reconstruction speed from the structure of algorithm、the technique of implementation and the optimization of code. Computer simulation experiment express that the methods can efficiently improve the speed of image reconstruction.
3. In the image reconstruction basing on optimization without constraint,in order to improve iterative efficiency and reconstructed quality, the thesis firstly applies the variable metric method to reconstruct image, and describes variable metric method、steepest descent Method、conjugate gradient method respectively. The thesis compared three algorithms above at iterative efficiency and reconstruction quality respectively by simulating experiment. The result shows that variable metric method is better than other algorithms in the aspects of iterative efficiency and reconstruction quality.
In this thesis, some familiar cone-beam scan reconstruction algorithms are also analyzed. These algorithms can be sorted by scan mode, reconstruction precision. The cone-beam scan reconstruction algorithms are introduced in this paper include FDK、P-FDK、T-FDK、HT-FDK and S-FDK, etc.
引文
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[2] Smith B, Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods, IEEE Trans. Med. Imaging, 4:14-25, 1985.
[3] Grangeat G, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, Mathematical Methods in Tomography. Lecture Notes in Math, Vol. 1497 (G. T. Herman, A. K. Louis, and F. Natterer, eds.), 1991.
[4] Defrise M, Noo F and Kudo H, A solution to the long-object problem in helical cone-beam tomography, Phys. Med. Biol. 45:623-643, 2000.
[5] Kudo H, Noo F and Defrise M, Cone-beam filtered-backprojection algorithm for truncated helical data, Phys. Med. Biol., 43:2885-2909, 1998.
[6] Kudo H, Noo F and Defrise M, Quasi-exact filtered backprojection algorithm for long object problem in helical cone-beam tomography, IEEE Trans. Med. Imaging, 19:902-921, 2000.
[7] Tam K C, Smarasekera S and Sauer F, Exact cone-beam CT with a spiral scan. Phys. Med. Biol., 43:1015-1024, 1998.
[8] Tam K C, Exact local regions-of-interest reconstruction in spiral cone-beam filtered backprojection CT, Proc. SPIE, 3979: 506-519, 2000.
[9] Danielsson P E, Edholm P and Seger M, Towards exact 3D-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement and a new completeness condition, Proc. Int. Meet. On Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Pittsburgh) ed D W Townsend and P E Kinahan pp. 141-144, 1997.
[10] Schaller S, Noo F, Sauer F, Tam K C, lauritschi G and Flohr T, Exact radon rebinning algorithm using local regions-of-interest for helical cone-beam CT, IEEE Trans. Med.Imaging, 19:361-375, 2000.
[11] Katsevich A, Theoretically exact filtered backprojection-type inversion algorithm for Spiral CT, SIAM Journal on Applied Mathematics, 62:2012-2026, 2002.
[12] Katsevich A, Improved Exact FBP algorithm for Spiral CT, Adv. Appl. Math. 32(2004), 681-697.
[13] Katsevich A, A General Scheme for Constructing Inversion Algorithm for cone beam CT. Int. J. Math. Sci. 21, 1305–1321, 2003.
[14] Katsevich A, Analysis of an exact invertion algorithm for spiral cone-beam CT, Phys. Med. Biol., 47:2583-2597, 2002.
[15] Zou Y and Pan X C, Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT, Phys. Med. Biol., 49:941-959. 2004.
[16] Y. Ye and G. Wang: Exact FBP CT reconstruction along general scanning curves, Med. Phys. 32 (2005), 42-48.
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[21] J.Yang, Q.Kong, T.Zhou and M.Jiang, Cone beam cover method: An approach to performing backprojection in Katsevich’s exact algorithm for spiral cone beam CT, Journal of X-ray Science and Technology 12: 199-214, 2004.
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[2] Smith B, Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods, IEEE Trans. Med. Imaging, 4:14-25, 1985.
[3] Grangeat G, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, Mathematical Methods in Tomography. Lecture Notes in Math, Vol. 1497 (G. T. Herman, A. K. Louis, and F. Natterer, eds.), 1991.
[4] Defrise M, Noo F and Kudo H, A solution to the long-object problem in helical cone-beam tomography, Phys. Med. Biol. 45:623-643, 2000.
[5] Kudo H, Noo F and Defrise M, Cone-beam filtered-backprojection algorithm for truncated helical data, Phys. Med. Biol., 43:2885-2909, 1998.
[6] Kudo H, Noo F and Defrise M, Quasi-exact filtered backprojection algorithm for long object problem in helical cone-beam tomography, IEEE Trans. Med. Imaging, 19:902-921, 2000.
[7] Tam K C, Smarasekera S and Sauer F, Exact cone-beam CT with a spiral scan. Phys. Med. Biol., 43:1015-1024, 1998.
[8] Tam K C, Exact local regions-of-interest reconstruction in spiral cone-beam filtered backprojection CT, Proc. SPIE, 3979: 506-519, 2000.
[9] Danielsson P E, Edholm P and Seger M, Towards exact 3D-reconstruction for helical cone-beam scanning of long objects. A new detector arrangement and a new completeness condition, Proc. Int. Meet. On Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Pittsburgh) ed D W Townsend and P E Kinahan pp. 141-144, 1997.
[10] Schaller S, Noo F, Sauer F, Tam K C, lauritschi G and Flohr T, Exact radon rebinning algorithm using local regions-of-interest for helical cone-beam CT, IEEE Trans. Med.Imaging, 19:361-375, 2000.
[11] Katsevich A, Theoretically exact filtered backprojection-type inversion algorithm for Spiral CT, SIAM Journal on Applied Mathematics, 62:2012-2026, 2002.
[12] Katsevich A, Improved Exact FBP algorithm for Spiral CT, Adv. Appl. Math. 32(2004), 681-697.
[13] Katsevich A, A General Scheme for Constructing Inversion Algorithm for cone beam CT. Int. J. Math. Sci. 21, 1305–1321, 2003.
[14] Katsevich A, Analysis of an exact invertion algorithm for spiral cone-beam CT, Phys. Med. Biol., 47:2583-2597, 2002.
[15] Zou Y and Pan X C, Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT, Phys. Med. Biol., 49:941-959. 2004.
[16] Y. Ye and G. Wang: Exact FBP CT reconstruction along general scanning curves, Med. Phys. 32 (2005), 42-48.
[17] S. Zhao, H. Yu, and G. Wang, A family of analytic algorithms for conebeam CT, Proc. SPIE Vol. 5535, Developments in X-Ray Tomography.
[18] F. Noo, J. Pack and D. Heuscher, Exact helical reconstruction using native cone-beam geometries, Phys. Med. Biol. 48 (2003) , 3787-3818.
[19] H. Yu, G. Wang, Studies on Implementation of the Katsevich Algorithm for Spiral Cone-beam CT, Journal of X-ray Science and Technology 12(2004), 97-116.
[20] H. Yu, G. Wang, Studies on Artifacts of the Katsevich Algorithm for Spiral Cone-beam CT, Proceedings of SPIE 5535 (2004).
[21] J.Yang, Q.Kong, T.Zhou and M.Jiang, Cone beam cover method: An approach to performing backprojection in Katsevich’s exact algorithm for spiral cone beam CT, Journal of X-ray Science and Technology 12: 199-214, 2004.
[22] Michael Sandborg, Computed Tomography physical principles and biohazards, Ph.D.Thesis, Department of Mathematical Modelling Section for Digital Signal Processing Technical University of Denmark.
[23] G. T. Herman. Image reconstruction from projections,. Academic Press,1980, pp. 63.
[24] C.H.Harris, “CT: It’s not just for diagnosis anymore,” Med.Imag., 16, (8),pp. 54-61(2001).
[25] P.M.Silverman, Helical(Spiral) Computed Tomography-A Practical Approach to Clinical Protocols, Lippincott Williams&Wilkins, Philadelphia(1998).
[26] H.C.Yoon, L.E.Greaser, III, R.Mather, S.Sinha, M.F.NcNitt-Gray, and J.G.Goldin, “Coronary artery calcium: alternate methods for accurate and reproduciblequantitation,” Academic Radiol., 4.pp.666-673(1997).
[27] Henrik Turbell Cone-beam reconstruction using filtered backprojection[D]. Sweden:Department of Electrical Engineering Linko pings universitet,2001.
[28] 陈立成,层析成像的数学方法与应用[M],成都,西南交通大学出版社,1994.11.
[29] A.H.Andersen and A.C.Kak.Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm. Ultrasonic Imaging, 6:81-94,1984.
[30] 潘晋孝,工业 CT 重建算法及其误差分析[D],华北工学院博士学位论文,2002.6.
[31] 高欣.新型迭代图像重建算法的理论研究与实现[D].浙江大学博士学位论文:浙江大学生物医学工程与仪器科学学院,2004.
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