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This paper mainly reviews the progress of methods, and research development in the field of computed tomography(CT) with incomplete projection data, at home and abroad, on limited angle projection data reconstruction with the detector fully covered, and truncated data reconstruction with the detector partially covered. Firstly, the discrete model iterative reconstruction algorithm and the compressed sensing(CS) sampling reconstruction algorithm are discussed for the sparsely uniform and limited-view angle sampling in the case that the detector fully covers the detected object. Secondly, the reconstruction algorithm of back-projection filtration(BPF) for helical cone beam and improved ones for cone beam FDK are discussed in the case that the detector could not fully cover the detected object. This paper could provide the researchers in CT reconstruction field the criticism of methods and summaries. Furthermore, it also points out current focus of the study and the research direction in the future.
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Keywords:
- incomplete projection date /
- compressed sensing /
- limited angle /
- truncated data
[1] Zhuang T G 1992 Principle and Algorithm of CT (Shanghai: Shanghai Jiao Tong University press) pp77-97 (in Chinese) [庄天戈 1992 CT原理与算法(上海: 上海交通大学出版社)第77–97页]
[2] Deng Z Y, Duan Z W, Liu J J, Xie Q G 2011 Nuclear Electronics & Detection Technology 31 53 (in Chinese) [邓贞宙, 段志文, 刘晶晶, 谢庆国 2011 核电子学与探测技术31 53]
[3] Tuy H K 1983 SIAM J. Applied Mathematics 43 546
[4] Smith B D 1985 IEEE Trans. Med. Imaging 4 14
[5] Grangeat P 1991 Mathematical Framework of Cone Beam 3D Reconstruction Via The First Derivative of The Radon Transform (Berlin: Springer-Verlag) pp66-97
[6] Yu H Y, Wang G 2009 Phys Med Biol. 54 2791
[7] Li J, Lu X Q, Sun Y 2011 Aeronautical Manufacturing Tech. 21 19 (in Chinese) [李镜, 卢孝强, 孙怡 2011 航空制造技术 21 19]
[8] Zou Y, Pan X C 2004 Phys. Med. Biol. 49 941
[9] Pack J D, Noo F, Clackdoyle R 2005 IEEE Trans. Med. Imaging 24 70
[10] Zou Y, Sidky E Y, Pelizzari C A, Munro P, Pan X C 2006 IEEE Trans. Med. Imaging 25 869
[11] Zhang H T, Chen M, Zhang P 2007 Progress in Natural Science 17 1589 (in Chinese) [张慧滔, 陈明, 张朋 2007 自然科学进展 17 1589]
[12] Ye Y, Wang G 2005 Med. Phys. 32 42
[13] Zhao S, Yu H, Wang G 2004 The SPIE 4th Conference on Developments in X-Ray Tomography Denver, CO, Aug 04-06, 2004 p318
[14] Guo J Q, Zeng L, Zou X B 2011 J. X-ray Sci. Technol. 19 293
[15] Li L, Chen Z Q, Zhang L, Xing Y X, Kang K J 2007 Appl. Radiat. Isot. 65 1041
[16] Zhang F, Yan B, Wang X C, Jiang H, Wei X 2013 Acta Phys. Sin. 62 168702 (in Chinese) [张峰, 闫镔, 汪先超, 江桦, 魏星 2013 物理学报 62 168702]
[17] Grunbaum F A 1980 Numerical Functional Analysis Opt. 2 31
[18] Tikhonov A N, Arsemin V Y 1979 Bull. Amer. Math. Soc. (N.S.) 1 521
[19] Rangayyan R, Dhawan A P, Gordon R 1985 Applied Optics. 24 4000
[20] Kak A C, Slaney M 1999 Principles of Computerized Tomographic Imaging (New York: IEEE Press) pp275-294
[21] Mueller K, Yagel R, Wheller J J 1999 IEEE Trans. Med. Imaging 18 519
[22] Herman G T, Meyer L B 1993 IEEE Trans. Med. Image 12 600
[23] Guan H, Gordon R 1994 Phys. Med. Biol. 39 2005
[24] Mueller K, Yagel R, Cornhill J F 1997 IEEE Trans. Med. Image 16 223
[25] Wang H J, Lu H N, Fu J 2006 Optical Technique 32 389 (in Chinese) [王宏钧, 路宏年, 傅健 2006 光学技术 32 389]
[26] Schölkopf B, Smola A, Mller K R 1998 Neural computation 10 1299
[27] Zhai Z, Zou Y G, Shen K, Wang Y, Xu W Z 2005 J. Computer Research and Development 42 1882 (in Chinese) [瞿中, 邹永贵, 沈宽, 王珏, 徐问之 2005 计算机研究与发展 42 1882]
[28] Qin Z Y, Mou X Q, Wang P, Cai Y L 2003 Chinese Journal of Electronics 33 1327 (in Chinese) [秦中元, 牟轩沁, 王平, 蔡元龙 2003 电子学报 33 1327]
[29] Wang X, Chen Z Q, Xiong H, Zhang L 2005 Nuclear Electronics & Detection Technology 25 785 (in Chinese) [王旭, 陈志强, 熊华, 张丽 2005 核电子学与探测技术 25 785]
[30] Siddon R 1985 Med. Phys. 12 252
[31] Guo W 2011 Ph.D. Dissertation (Jilin: Jilin University) (in Chinese) [郭威 2011 博士学位论文(吉林: 吉林大学)]
[32] Guo W, Chen H X, Geng W J, Lei L Y 2009 Proceedings of the 2nd International Conference on BioMedical Engineering and Informatics (China: Tianjin) p29
[33] Andersen A H, Kak A C 1984 Ultrason Imaging 6 81
[34] Li C F, Zhang X F, Pan J H, Shi D F 2002 J. Optoelectronics Laser 13 726 (in Chinese) [李春芳, 张新峰, 潘金虎, 是度芳 2002 光电子激光 13 726]
[35] Fang X, Mueller K 2005 IEEE Trans. Nuclear Science 52 654
[36] Kong H H, Pan J X 2008 Computerized Tomography Theory and Applications 17 40 (in Chinese) [孔慧华, 潘晋孝 2008 CT理论与应用研究 17 40]
[37] Gao X 2004 Ph.D. Dissertation (Zhejiang: Zhejiang University) (in Chinese) [高欣 2004 博士学位论文(浙江: 浙江大学)]
[38] Wang H 2008 MS Thesis (Liaoning: Dalian University of Technology) (in Chinese) [王浩 2008 硕士学位论文(辽宁: 大连理工大学)]
[39] Candes E J, Romberg J, Tao T 2006 IEEE Trans. Info. Theo. 52 489
[40] Donoho D L 2006 IEEE Trans. Info. Theo. 52 1289
[41] Kashin B 1977 Izvestiya Rossiiskoi Akademii Nauk Seriya Matematicheskaya 41 334
[42] Sidky E Y, Chartrand R, Pan X C 2007 IEEE Nuclear Science Symposium Conference Record (New York: IEEE Press) p3526
[43] Li Y 2011 MS Thesis (Taiyuan: North University of China) (in Chinese) [李毅 2011 硕士学位论文(太原: 中北大学)]
[44] Wang L Y, Wei Z H, Li X X 2012 J. Image Graphics 17 20 (in Chinese) [王丽艳, 韦志辉, 李星秀 2012 中国图象图形学报 17 20]
[45] Gong Z P 2012 MS Thesis (Guangzhou: South China University of Technology) (in Chinese) [龚紫平 2012 硕士学位论文(广东: 华南理工大学)]
[46] Yi X N 2011 Ph. D. Dissertation (Hubei: Huazhong University of Science and Technology) (in Chinese) [易学能 2011 博士学位论文(湖北: 华中科技大学)]
[47] Choi K, Wang J, Zhu L, Suk T, Boyd S, Xing L 2010 Med. Phys. 37 5113
[48] Fan J, Peng H 2004 The Annals of Statistics 32 928
[49] Candes E J, Tao T 2005 IEEE Trans. Info. Theo. 51 4203
[50] Zou Y, Pan X C, Sidky E Y 2005 Phys. Med. Biol. 50 13
[51] Sidky E Y, Pan X 2006 IEEE Nuclear Science Symposium Conference Record (New York: IEEE Press) p2904
[52] Sidky E Y, Kao C M, Pan X C 2006 J. X-ray Sci. Tech. 14 119
[53] Sidky E Y, Pan X C 2008 Phys. Med. Biol. 53 4777
[54] Niu T, Zhu L 2012 Med. Phys. 39 4588
[55] Lian Q S, Hao P P 2009 Opt. Tech. 35 422 (in Chinese) [练秋生, 郝鹏鹏 2009 光学技术 35 422]
[56] Donoho D L, Maleki A, Montanari A 2009 Proceedings of the National Academy of Sciences 106 18914
[57] Danielsson P E, Edholm P, Eriksson J, Magnusso M 1997 Proceedings of the 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Pittsburgh, PA) p141
[58] Zou Y, Pan X C 2004 Phys. Med. Biol. 49 941
[59] Zou Y, Pan X C 2004 Phys. Med. Biol. 49 2717
[60] Pan X C, Xia D, Zou Y, Yu L 2004 Phys. Med. Biol. 49 4349
[61] Katsevich A 2002 SIAM J. Applied Math. 62 2012
[62] Katsevich A 2003 Int. J. Math. Sci. 21 1305
[63] Sidky E Y, Zou Y, Pan X C 2005 Phys. Med. Biol. 50 1643
[64] Zhang F 2013 Ph. D. Dissertation (Zhengzhou: The PLA Information Engineering University) (in Chinese) [张峰 2013 博士学位论文 (郑州: 中国人民解放军信息工程大学)]
[65] Zeng L, Liu B D, Ji D J 2011 CN Patent 101387611 B [曾里, 刘宝东, 冀东江 2011 CN 101387611 B]
[66] Liu B D 2010 Ph. D. Dissertation (Chongqing: Chongqing University) (in Chinese) [刘宝东 2010 博士学位论文(重庆: 重庆大学)]
[67] Guo J Q 2011 Ph. D. Dissertation (Chongqing: Chongqing University) (in Chinese) [郭吉强 2011 博士学位论文(重庆: 重庆大学)]
[68] Hao J, Li L, Zhang L, Kang K J 2013 J. Tsinghua Univ: Sci. &Tech. 53 145 (in Chinese) [郝佳, 李亮, 张丽, 康克军 2013 清华大学学报: 自然科学版 53 145]
[69] Wang G, Ye Y, Yu H 2007 Phys. Med. Biol. 52 R1
[70] Pan X C, Siewerdsen J L, Riviere P J, Kalender W A 2008 Med. Phys. 35 3728
[71] Liu D H 2009PhD. Dissertation (Xi'an: Xidian University) (in Chinese) [刘丹华 2009 博士学位论文(西安: 西安电子科技大学)]
[72] Donoho D L, Maleki A, Montanari A 2009 Proc. Natl. Acad. Sci. 106 18914
[73] Yonina E, Gitta K 2012 Compressed Sensing: Theory and Applications (Oxford: Cambridge University Press) pp405-447
[74] Kamilov U S, Goyal V K, Rangan S 2012 IEEE Transactions On Signal Processing 60 6270
[75] Zhou W 2012 MS Thesis (Shanghai: Shanghai Jiao Tong University) (in Chinese) [皱伟 2012 硕士学位论文(上海: 上海交通大学)]
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[1] Zhuang T G 1992 Principle and Algorithm of CT (Shanghai: Shanghai Jiao Tong University press) pp77-97 (in Chinese) [庄天戈 1992 CT原理与算法(上海: 上海交通大学出版社)第77–97页]
[2] Deng Z Y, Duan Z W, Liu J J, Xie Q G 2011 Nuclear Electronics & Detection Technology 31 53 (in Chinese) [邓贞宙, 段志文, 刘晶晶, 谢庆国 2011 核电子学与探测技术31 53]
[3] Tuy H K 1983 SIAM J. Applied Mathematics 43 546
[4] Smith B D 1985 IEEE Trans. Med. Imaging 4 14
[5] Grangeat P 1991 Mathematical Framework of Cone Beam 3D Reconstruction Via The First Derivative of The Radon Transform (Berlin: Springer-Verlag) pp66-97
[6] Yu H Y, Wang G 2009 Phys Med Biol. 54 2791
[7] Li J, Lu X Q, Sun Y 2011 Aeronautical Manufacturing Tech. 21 19 (in Chinese) [李镜, 卢孝强, 孙怡 2011 航空制造技术 21 19]
[8] Zou Y, Pan X C 2004 Phys. Med. Biol. 49 941
[9] Pack J D, Noo F, Clackdoyle R 2005 IEEE Trans. Med. Imaging 24 70
[10] Zou Y, Sidky E Y, Pelizzari C A, Munro P, Pan X C 2006 IEEE Trans. Med. Imaging 25 869
[11] Zhang H T, Chen M, Zhang P 2007 Progress in Natural Science 17 1589 (in Chinese) [张慧滔, 陈明, 张朋 2007 自然科学进展 17 1589]
[12] Ye Y, Wang G 2005 Med. Phys. 32 42
[13] Zhao S, Yu H, Wang G 2004 The SPIE 4th Conference on Developments in X-Ray Tomography Denver, CO, Aug 04-06, 2004 p318
[14] Guo J Q, Zeng L, Zou X B 2011 J. X-ray Sci. Technol. 19 293
[15] Li L, Chen Z Q, Zhang L, Xing Y X, Kang K J 2007 Appl. Radiat. Isot. 65 1041
[16] Zhang F, Yan B, Wang X C, Jiang H, Wei X 2013 Acta Phys. Sin. 62 168702 (in Chinese) [张峰, 闫镔, 汪先超, 江桦, 魏星 2013 物理学报 62 168702]
[17] Grunbaum F A 1980 Numerical Functional Analysis Opt. 2 31
[18] Tikhonov A N, Arsemin V Y 1979 Bull. Amer. Math. Soc. (N.S.) 1 521
[19] Rangayyan R, Dhawan A P, Gordon R 1985 Applied Optics. 24 4000
[20] Kak A C, Slaney M 1999 Principles of Computerized Tomographic Imaging (New York: IEEE Press) pp275-294
[21] Mueller K, Yagel R, Wheller J J 1999 IEEE Trans. Med. Imaging 18 519
[22] Herman G T, Meyer L B 1993 IEEE Trans. Med. Image 12 600
[23] Guan H, Gordon R 1994 Phys. Med. Biol. 39 2005
[24] Mueller K, Yagel R, Cornhill J F 1997 IEEE Trans. Med. Image 16 223
[25] Wang H J, Lu H N, Fu J 2006 Optical Technique 32 389 (in Chinese) [王宏钧, 路宏年, 傅健 2006 光学技术 32 389]
[26] Schölkopf B, Smola A, Mller K R 1998 Neural computation 10 1299
[27] Zhai Z, Zou Y G, Shen K, Wang Y, Xu W Z 2005 J. Computer Research and Development 42 1882 (in Chinese) [瞿中, 邹永贵, 沈宽, 王珏, 徐问之 2005 计算机研究与发展 42 1882]
[28] Qin Z Y, Mou X Q, Wang P, Cai Y L 2003 Chinese Journal of Electronics 33 1327 (in Chinese) [秦中元, 牟轩沁, 王平, 蔡元龙 2003 电子学报 33 1327]
[29] Wang X, Chen Z Q, Xiong H, Zhang L 2005 Nuclear Electronics & Detection Technology 25 785 (in Chinese) [王旭, 陈志强, 熊华, 张丽 2005 核电子学与探测技术 25 785]
[30] Siddon R 1985 Med. Phys. 12 252
[31] Guo W 2011 Ph.D. Dissertation (Jilin: Jilin University) (in Chinese) [郭威 2011 博士学位论文(吉林: 吉林大学)]
[32] Guo W, Chen H X, Geng W J, Lei L Y 2009 Proceedings of the 2nd International Conference on BioMedical Engineering and Informatics (China: Tianjin) p29
[33] Andersen A H, Kak A C 1984 Ultrason Imaging 6 81
[34] Li C F, Zhang X F, Pan J H, Shi D F 2002 J. Optoelectronics Laser 13 726 (in Chinese) [李春芳, 张新峰, 潘金虎, 是度芳 2002 光电子激光 13 726]
[35] Fang X, Mueller K 2005 IEEE Trans. Nuclear Science 52 654
[36] Kong H H, Pan J X 2008 Computerized Tomography Theory and Applications 17 40 (in Chinese) [孔慧华, 潘晋孝 2008 CT理论与应用研究 17 40]
[37] Gao X 2004 Ph.D. Dissertation (Zhejiang: Zhejiang University) (in Chinese) [高欣 2004 博士学位论文(浙江: 浙江大学)]
[38] Wang H 2008 MS Thesis (Liaoning: Dalian University of Technology) (in Chinese) [王浩 2008 硕士学位论文(辽宁: 大连理工大学)]
[39] Candes E J, Romberg J, Tao T 2006 IEEE Trans. Info. Theo. 52 489
[40] Donoho D L 2006 IEEE Trans. Info. Theo. 52 1289
[41] Kashin B 1977 Izvestiya Rossiiskoi Akademii Nauk Seriya Matematicheskaya 41 334
[42] Sidky E Y, Chartrand R, Pan X C 2007 IEEE Nuclear Science Symposium Conference Record (New York: IEEE Press) p3526
[43] Li Y 2011 MS Thesis (Taiyuan: North University of China) (in Chinese) [李毅 2011 硕士学位论文(太原: 中北大学)]
[44] Wang L Y, Wei Z H, Li X X 2012 J. Image Graphics 17 20 (in Chinese) [王丽艳, 韦志辉, 李星秀 2012 中国图象图形学报 17 20]
[45] Gong Z P 2012 MS Thesis (Guangzhou: South China University of Technology) (in Chinese) [龚紫平 2012 硕士学位论文(广东: 华南理工大学)]
[46] Yi X N 2011 Ph. D. Dissertation (Hubei: Huazhong University of Science and Technology) (in Chinese) [易学能 2011 博士学位论文(湖北: 华中科技大学)]
[47] Choi K, Wang J, Zhu L, Suk T, Boyd S, Xing L 2010 Med. Phys. 37 5113
[48] Fan J, Peng H 2004 The Annals of Statistics 32 928
[49] Candes E J, Tao T 2005 IEEE Trans. Info. Theo. 51 4203
[50] Zou Y, Pan X C, Sidky E Y 2005 Phys. Med. Biol. 50 13
[51] Sidky E Y, Pan X 2006 IEEE Nuclear Science Symposium Conference Record (New York: IEEE Press) p2904
[52] Sidky E Y, Kao C M, Pan X C 2006 J. X-ray Sci. Tech. 14 119
[53] Sidky E Y, Pan X C 2008 Phys. Med. Biol. 53 4777
[54] Niu T, Zhu L 2012 Med. Phys. 39 4588
[55] Lian Q S, Hao P P 2009 Opt. Tech. 35 422 (in Chinese) [练秋生, 郝鹏鹏 2009 光学技术 35 422]
[56] Donoho D L, Maleki A, Montanari A 2009 Proceedings of the National Academy of Sciences 106 18914
[57] Danielsson P E, Edholm P, Eriksson J, Magnusso M 1997 Proceedings of the 1997 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Pittsburgh, PA) p141
[58] Zou Y, Pan X C 2004 Phys. Med. Biol. 49 941
[59] Zou Y, Pan X C 2004 Phys. Med. Biol. 49 2717
[60] Pan X C, Xia D, Zou Y, Yu L 2004 Phys. Med. Biol. 49 4349
[61] Katsevich A 2002 SIAM J. Applied Math. 62 2012
[62] Katsevich A 2003 Int. J. Math. Sci. 21 1305
[63] Sidky E Y, Zou Y, Pan X C 2005 Phys. Med. Biol. 50 1643
[64] Zhang F 2013 Ph. D. Dissertation (Zhengzhou: The PLA Information Engineering University) (in Chinese) [张峰 2013 博士学位论文 (郑州: 中国人民解放军信息工程大学)]
[65] Zeng L, Liu B D, Ji D J 2011 CN Patent 101387611 B [曾里, 刘宝东, 冀东江 2011 CN 101387611 B]
[66] Liu B D 2010 Ph. D. Dissertation (Chongqing: Chongqing University) (in Chinese) [刘宝东 2010 博士学位论文(重庆: 重庆大学)]
[67] Guo J Q 2011 Ph. D. Dissertation (Chongqing: Chongqing University) (in Chinese) [郭吉强 2011 博士学位论文(重庆: 重庆大学)]
[68] Hao J, Li L, Zhang L, Kang K J 2013 J. Tsinghua Univ: Sci. &Tech. 53 145 (in Chinese) [郝佳, 李亮, 张丽, 康克军 2013 清华大学学报: 自然科学版 53 145]
[69] Wang G, Ye Y, Yu H 2007 Phys. Med. Biol. 52 R1
[70] Pan X C, Siewerdsen J L, Riviere P J, Kalender W A 2008 Med. Phys. 35 3728
[71] Liu D H 2009PhD. Dissertation (Xi'an: Xidian University) (in Chinese) [刘丹华 2009 博士学位论文(西安: 西安电子科技大学)]
[72] Donoho D L, Maleki A, Montanari A 2009 Proc. Natl. Acad. Sci. 106 18914
[73] Yonina E, Gitta K 2012 Compressed Sensing: Theory and Applications (Oxford: Cambridge University Press) pp405-447
[74] Kamilov U S, Goyal V K, Rangan S 2012 IEEE Transactions On Signal Processing 60 6270
[75] Zhou W 2012 MS Thesis (Shanghai: Shanghai Jiao Tong University) (in Chinese) [皱伟 2012 硕士学位论文(上海: 上海交通大学)]
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