锥束CT有限角度三维重建算法研究
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摘要
计算机断层成像(Computed Tomography, CT)技术在医学和工业无损检测中具有广泛的应用,其核心技术之一的图像重建算法一直是CT成像技术研究的一大热点。然而,实际应用中由于考虑到剂量和对比度的因素,很多数据只在有限角度范围内扫描,这种情况称为有限角度问题。
本文针对有限角度三维图像重建算法进行研究,主要研究了利用迭代方式解决有限角度图像重建问题的一种新算法—重建参考差(Reconstruction-Reference Difference, RRD)算法。这种方法的主要思想是构造待重建对象的稀疏表示而后通过正则化方法求解。文中着重讨论了基于l_1-范数的正则化函数极小元的存在唯一性,证明了一类CT投影矩阵的“限制正交假设”(Restricted Orthonormality Hypothesis, ROH),在此基础上结合压缩感知(Compressive Sensing, CS)理论和正则化方法提出了重建参考差算法,并对此算法的高性能计算问题进行了研究。主要研究成果如下:
在压缩感知理论基础上,在正则化理论框架下研究了稀疏图像重建理论。讨论了基于l_1-范数的正则化泛函极小元的存在唯一性,对一类CT投影矩阵证明了极小元唯一性成立所需的“限制正交假设”。
在稀疏图像重建理论基础上,针对有限角度问题,提出了基于l_1-范数正则化泛函的有限角度稀疏重建算法。又在此基础上,根据实际重建问题的需要对该算法进行了改进,提出了重建参考差(RRD)算法。仿真实验和实际数据重建的结果表明,RRD算法针对有限角度问题有很好的重建效果。
针对RRD算法的实用化问题,研究了RRD算法的高性能计算。通过分析RRD算法的计算-存储-通信属性,给出了该算法的并行计算方案。同时结合对计算-存储-通信属性的分析,为硬件设计和体系结构调整提供了有意义的参考。
最后总结了全文,并对CT图像重建算法值得继续深入研究的几个方向进行了展望。
Computed Tomography (CT) is widely used in medicine and industrial non-destructive testing, the core technology of which—the image reconstruction—is always the research focus. However, in practical applications, the projection data often are acquired only in a limited angle range, because of the factors of the dose, contrast and so on. This situation is referred to as the limited-view problem.
The research of this dissertation is the three dimensions image reconstruction algorithm for limited-view problem. In this dissertation, a new algorithm for limited-view problem using iterative method, named reconstruction-reference difference (RRD), is introduced and discussed. The most basic idea of this algorithm is to build a sparse representation for the object and then solving it with the regularization method. The existence and uniqueness of minimal element of the regularization function based on l_1-norm is discussed, and the restricted orthonormality hypothesis to a kind of CT projection matrix is proved. Based on the conclusion above and compressive sensing theory, we develop the RRD algorithm using regularization. Moreover, high performance computing to RRD algorithm is studied. Generally the work consists of the following three parts.
Firstly, based on compressive sensing theory, the theory for reconstruction of sparse image is developed in the frame of regularization. The existence and uniqueness of minimal element of the regularization function based on l_1-norm is discussed, and the restricted orthonormality hypothesis to a kind of CT projection matrix is proved. Secondly, making use of the theory for reconstruction of sparse image, a reconstruction algorithm based on regularization function discussed above is developed in order to solve limited-view problem. In order to use the algorithm in real data reconstruction, the RRD algorithm is proposed with some improvements. The algorithm has been tested with a thin planar phantom and a real object in limited-view projection data. Moreover, all the studies showed the satisfactory results in accuracy.
In the third part, high performance computing to RRD algorithm is studied to use the algorithm in real more easily. We analyze the Processing-Memory-Communication (PMC) properties of the algorithm, and give a project for parallel computing. Taking into account PMC analyses, some advice is proposed for hardware design and architecture.
In the last part the dissertation is summarized, and prospected directions for the research of CT image reconstruction are proposed and discussed.
本文针对有限角度三维图像重建算法进行研究,主要研究了利用迭代方式解决有限角度图像重建问题的一种新算法—重建参考差(Reconstruction-Reference Difference, RRD)算法。这种方法的主要思想是构造待重建对象的稀疏表示而后通过正则化方法求解。文中着重讨论了基于l_1-范数的正则化函数极小元的存在唯一性,证明了一类CT投影矩阵的“限制正交假设”(Restricted Orthonormality Hypothesis, ROH),在此基础上结合压缩感知(Compressive Sensing, CS)理论和正则化方法提出了重建参考差算法,并对此算法的高性能计算问题进行了研究。主要研究成果如下:
在压缩感知理论基础上,在正则化理论框架下研究了稀疏图像重建理论。讨论了基于l_1-范数的正则化泛函极小元的存在唯一性,对一类CT投影矩阵证明了极小元唯一性成立所需的“限制正交假设”。
在稀疏图像重建理论基础上,针对有限角度问题,提出了基于l_1-范数正则化泛函的有限角度稀疏重建算法。又在此基础上,根据实际重建问题的需要对该算法进行了改进,提出了重建参考差(RRD)算法。仿真实验和实际数据重建的结果表明,RRD算法针对有限角度问题有很好的重建效果。
针对RRD算法的实用化问题,研究了RRD算法的高性能计算。通过分析RRD算法的计算-存储-通信属性,给出了该算法的并行计算方案。同时结合对计算-存储-通信属性的分析,为硬件设计和体系结构调整提供了有意义的参考。
最后总结了全文,并对CT图像重建算法值得继续深入研究的几个方向进行了展望。
Computed Tomography (CT) is widely used in medicine and industrial non-destructive testing, the core technology of which—the image reconstruction—is always the research focus. However, in practical applications, the projection data often are acquired only in a limited angle range, because of the factors of the dose, contrast and so on. This situation is referred to as the limited-view problem.
The research of this dissertation is the three dimensions image reconstruction algorithm for limited-view problem. In this dissertation, a new algorithm for limited-view problem using iterative method, named reconstruction-reference difference (RRD), is introduced and discussed. The most basic idea of this algorithm is to build a sparse representation for the object and then solving it with the regularization method. The existence and uniqueness of minimal element of the regularization function based on l_1-norm is discussed, and the restricted orthonormality hypothesis to a kind of CT projection matrix is proved. Based on the conclusion above and compressive sensing theory, we develop the RRD algorithm using regularization. Moreover, high performance computing to RRD algorithm is studied. Generally the work consists of the following three parts.
Firstly, based on compressive sensing theory, the theory for reconstruction of sparse image is developed in the frame of regularization. The existence and uniqueness of minimal element of the regularization function based on l_1-norm is discussed, and the restricted orthonormality hypothesis to a kind of CT projection matrix is proved. Secondly, making use of the theory for reconstruction of sparse image, a reconstruction algorithm based on regularization function discussed above is developed in order to solve limited-view problem. In order to use the algorithm in real data reconstruction, the RRD algorithm is proposed with some improvements. The algorithm has been tested with a thin planar phantom and a real object in limited-view projection data. Moreover, all the studies showed the satisfactory results in accuracy.
In the third part, high performance computing to RRD algorithm is studied to use the algorithm in real more easily. We analyze the Processing-Memory-Communication (PMC) properties of the algorithm, and give a project for parallel computing. Taking into account PMC analyses, some advice is proposed for hardware design and architecture.
In the last part the dissertation is summarized, and prospected directions for the research of CT image reconstruction are proposed and discussed.
引文
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[2]田捷.医学影像处理与分析[M].北京:电子工业出版社,2003:10-12.
[3] Feldkamp L A, Davis L C, Kress J W. Practical cone-beam algorithm [J]. Opt Soc Am, 1984, 1: 612-619.
[4] Wang G, Lin T, Cheng P, et al. A general cone-beam reconstruction algorithm [J]. IEEE Trans on Medical Imaging, 1993, 12(3): 486-495.
[5] Grangeat P. Mathematical framework of cone-beam 3D reconstruction via the first derivative of the radon transform Mathematical Methods in Tomography [J]. In G T Herman, A K Louis and F Natterer (Eds.), Mathematical Methods in Tomography, Number 1497 in Lecture Notes in Mathematics, Springer Verlag, 1991, 66-97.
[6] Defrise M, and Clack R. A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection [J]. IEEE Transactions on Medical Imaging, 1994, 13(1): 186-195.
[7] Tam K C. Three-dimensional computerized tomography scanning method and system for large objects with smaller area detectors [J]. US Patent, 1995, 5(390): 112.
[8] Kudo H, Noo F, and Defrise M. cone-beam filtered-backprojection algorithm for truncated helical data [J]. Physics in Medicine and Biology, 1998, 43: 2885-2909.
[9] Tam K C, Lauritsch G, Sourbelle K, et al. Exact (spiral+circles) scan region-of-interest cone beam reconstruction via backprojection [J]. IEEE Transactions on Medical Imaging, 2000, 19(5): 376-383.
[10] Schaller S, Sauer F, Tam K C, et al. Exact radon rebinning algorithms using local region-of-interest for helical cone-beam CT [J]. International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, 1999: 23-26.
[11] Kudo H, Park S, Noo F, et al. Performance of quasi-exact cone-beam filtered backprojection algorithm for axially truncated data [J].IEEE Transactions on Nuclear Science,1999,46(3):608-617.
[12] Defrise M, Noo F, and Kudo H. A solution to the long-object problem in helical cone-beam tomography [J]. Phys. Med. Bio., 2000, 45: 623-643.
[13] Sourbelle K, Lauritsch G, Tam K C, et al. Performance evaluation of local ROI algorithms for exact ROI reconstruction in spriral cone-beam computed tomography [J]. IEEE Medical Imaging, 2000: 16-20.
[14] Danielsson P E, Edholm P, Eriksson J, et al. Towards exact 3D-reconstruction for helical cone-beam scanning of long objects: a new arrangement and a new completeness condition [J].International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, 1997: 25-28.
[15] Tam K C, Samarasekera S and Sauer F. Exact cone-beam ct with a spiral scan [J]. Phys. Med. Bio., 1998, 43: 1015-24.
[16] Katsevich A. An improved exact filtered backprojection algorithm for spiral computed tomography [J]. Adv. Appl. Math., 2004, 32: 681-697.
[17] Zou Y and Pan X. Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT [J]. Phys. Med. Bio., 2004, 49: 941-959.
[18] Zou Y and Pan X. Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT [J]. Phys. Med. Bio., 2004, 49: 2717-31.
[19] Pan X, Zou Y, Xia D. Image reconstruction in peripheral and central regions-of-interest and data recundancy [J]. Med. Phys., 2005, 32: 673-684.
[20] Zou Y, Pan X, Sidky E Y. Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan [J]. Phys. Med. Bio., 2005, 50: 13-27.
[21] Zou Y, Pan X, Sidky E Y. Theory and algorithms for image reconstruction on chords and within regions-of-interest [J]. J. Opt. Soc. Am. A, 2005, 22: 2372-84.
[22] Zou Y, Sidky E Y, Pan X. Partical volume and aliasing artifacts in helical cone-beam CT [J]. Phys. Med. Bio., 2004, 49: 2365-75.
[23] Pan X, Yu L, Kao C M. Spatial-resolution enhancement in computed tomography [J]. IEEE Med. Imag., 2005, 24: 248-253.
[24] Andersen A H. Algebraic reconstruction in CT from limited views [J]. IEEE Trans. Med. Imag., 1989, 8: 50-55.
[25] Lange K and Carson R. EM reconstruction algorithms for emission and transmission tomography [J]. Comput. Assist. Tomogr., 1984, 8: 306-16.
[26] Sauer K and Bouman C. A local update strategy for iterative reconstruction from projections [J]. IEEE Trans. Signal Process, 1993, 41: 534-48.
[27] Manglos S H, Gagne G M, Krol A, et al. Transmission maximum-likelihood reconstruction with ordered subsets for cone-beam CT [J]. Phys. Med. Biol., 1995, 40: 1225-41.
[28] Elbakri I A and Fessler J A. Segmentation-free statistical image reconstruction for polyenergetic x-ray computed tomography with experimental validation [J]. Phys. Med. Biol., 2003, 48: 2453-77.
[29] Kole J S. Statistical image reconstruction for transmission tomography using relaxed ordered subset algorithms [J]. Phys. Med. Biol., 2005, 50: 1533-45.
[30] Zbijewski W, Defrise M, Viergever M A, et al. Statistical reconstruction for x-ray CT systems with non-continuous detectors [J]. Phys. Med. Biol., 2007, 52: 403-18.
[31] Andersen A H, Kak A C. Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm[J]. Ultrasonic Imaging, 1984, 6: 81-94.
[32] Dusaussory N J. Some new multiplicative algorithms for image reconstruction from projections [J]. Linear algebra and its application, 1990, 130: 111-132.
[33] Wang Y, Friedrich M W. Vector-entropy optimization based neural network approach to image reconstruction from projections [J]. IEEE Transactions on Neural Networks, 1997, 8(5): 1008-1014.
[34] Minerbo G. A maximum entropy algorithm for reconstructing a source from projection data [J]. Computer Graphics and Image Processing, 1979, 10(1): 48-68.
[35] Soller C, Wenskus R, Middendorf P, et al. Interferometric tomography for flow visualization of density field in supersonic jets and convective flow [J]. Appl. Opt., 1994, 30(14): 2921-32.
[36] Sato T, Norton S J, Linzer M, et al. Tomographic imaging reconstruction from limited projections using iterative revisions in image and transform space [J]. Opt Soc Amer, 1981, 71: 582-592.
[37] Nassi M, Brody W R, Medoff B P, et al. Iterative reconstruction-reprojection: An algorithm for limited data cardiac-computed tomography [J]. IEEE Trans Biomed Eng, 1982, 29: 331-341.
[38] Reeds J A, Shepp L A. Limited angle reconstruction in tomography via squashing [J]. IEEE Trans Med Imaging, 1987, 6: 89-97.
[39] Tuy H. An inversion formula for cone-beam reconstruction [J]. SIAMJ. Apply. Math, 1983, 43(3): 546-552.
[40] Smith B D. Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods [J]. IEEE Trans. Med. Imag., 1985, 4(1): 14-25.
[41]高河伟,张丽,陈志强等.有限角度CT图像重建算法综述[J].CT理论与应用研究,2006,15(1):46-50.
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