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低剂量计算机断层成像（computed tomography，CT）具有减少X射线对患者的伤害的优势. 本文主要针对从不完备投影数据重建出高质量低剂量CT图像的问题. 通常，这个问题可以通过统计图像重建方法来实现，而统计重建算法需要非常多的迭代次数，导致了巨大的计算时间压力，以至于很难应用在实践中. 为解决此问题，本文提出一种有序子集重建算法，该算法结合了全变分最小化和快速一阶方法以减少重建的迭代次数，采用Split Bregman交替方向法求解上述优化问题，利用投影到凸集合的方法加快迭代的收敛速率. 实验结果表明，在同样的迭代次数下，本文提出的方法与基于有序子集的一阶方法相比较，相对重建误差的下降速度更快.Low-dose computed tomography(CT) has an advantage to reduce X-rays that are harmful to the body. This paper considers the issue of reconstructing high-quality low-dose CT images from incomplete projection data. Generally, this can be done by statistical image reconstruction methods. However, the huge number of iterations of the statistical reconstruction algorithms leads to long computing time, making them difficult to be of practical value. To solve this problem, we propose a method to alleviate the issue by using total variation minimization and fast first-order method for the ordered subsets. We use Split Bregman alternating direction method to solve the optimization problem. Then, the projection onto convex sets method is used to speed up the convergence rate of the iterative method. Numerical experiments show that the relative reconstruction error of the proposed method can decrease faster than the first-order method of ordered subsets with the same iterative number.
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Keywords:
- total variation /
- projection onto convex sets /
- alternating direction method /
- ordered subsets
[1] Brenner D J, Hall E J 2007 New Engl. J. Med. 357 2277
[2] Candès E J, Romberg J, Tao T 2006 IEEE Trans. Info. Theory 52 489
[3] Candès E J, Tao T 2006 IEEE Trans. Info. Theory 52 5406
[4] Donoho D 2006 IEEE Trans. Info. Theory 52 1289
[5] Wang L Y, Li L, Yan B, Jiang C S, Wang H Y, Bao S L 2010 Chin. Phys.B 19 088106
[6] Yang F Q, Zhang D H, Huang K D, Wang K, Xu Z 2014 Acta Phys. Sin. 63 058701 (in Chinese)[杨富强, 张定华, 黄魁东, 王鹍, 徐哲 2014 63 058701]
[7] Rudin L, Osher S, Fatemi E 1992 Physica D 60 259
[8] Li S P, Wang L Y, Yan B, Li L, Liu Y J 2012 Chin. Phys. B 21 108703
[9] Gu Y F, Yan B, Li L, Wei F, Han Yu, Chen J 2014 Acta Phys. Sin. 63 018701 (in Chinese)[古宇飞, 闫镔, 李磊, 魏峰, 韩玉, 陈健 2014 63 018701]
[10] Boyd S, Parikh N, Chu E, Peleato B, Eckstein J 2010 Foundations and Trends?in Machine Learning 3 1
[11] Goldstein T, Osher S 2009 SIAM J. Imaging Sci. 2 323
[12] Wang L Y, Zhang H M, Cai A L, Yan B, Li L, Hu G E 2013 Acta Phys. Sin. 62 198701 (in Chinese)[王林元, 张瀚铭, 蔡爱龙, 闫镔, 李磊, 胡国恩 2013 62 198701]
[13] Ramani S, Fessler J A 2012 IEEE Trans. Med. Imag. 31 677
[14] Matakos A, Ramani S, Fessler J A 2013 IEEE Trans. Image Process. 22 2019
[15] Zhang H M, Wang L Y, Yan B, Li L, Xi X Q, Lu L Z 2013 Chin. Phys.B 22 078701
[16] Sidky E Y, Jørgensen J S, Pan X 2013 Med. Phys. 40 031115
[17] Nesterov Y E 1983 Dokl. Akad. Nauk SSSR 269 543 (in Russian)
[18] Choi K, Wang J, Zhu L, Suh T S, Boyd S, Xing L 2010 Med. Phys. 37 5113
[19] Jensen T L, Jørgensen J S, Hansen P C, Jensen S H 2012 BIT Numer. Math. 52 329
[20] Beck A, Teboulle M 2009 SIAM J. Imaging Sci. 2 183
[21] Daubechies I, Fornasier M, Loris I 2008 J. Fourier Anal. Appl. 14 764
[22] Daubechies I, Defrise M, Mol C D 2004 Comm. Pure Appl. Math. 57 1413
[23] Beck A, Teboulle M 2009 IEEE Trans. Image Process. 18 2419
[24] Erdogan H, Fessler J A 1999 Phys. Med. Biol. 44 2835
[25] Hudson H M, Larkin R S 1994 IEEE Trans. Med. Imag. 13 601
[26] Kim D, Ramani S, Fessler J A 2013 The 12th International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine California, USA, June 16-21, 2013 p22
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[1] Brenner D J, Hall E J 2007 New Engl. J. Med. 357 2277
[2] Candès E J, Romberg J, Tao T 2006 IEEE Trans. Info. Theory 52 489
[3] Candès E J, Tao T 2006 IEEE Trans. Info. Theory 52 5406
[4] Donoho D 2006 IEEE Trans. Info. Theory 52 1289
[5] Wang L Y, Li L, Yan B, Jiang C S, Wang H Y, Bao S L 2010 Chin. Phys.B 19 088106
[6] Yang F Q, Zhang D H, Huang K D, Wang K, Xu Z 2014 Acta Phys. Sin. 63 058701 (in Chinese)[杨富强, 张定华, 黄魁东, 王鹍, 徐哲 2014 63 058701]
[7] Rudin L, Osher S, Fatemi E 1992 Physica D 60 259
[8] Li S P, Wang L Y, Yan B, Li L, Liu Y J 2012 Chin. Phys. B 21 108703
[9] Gu Y F, Yan B, Li L, Wei F, Han Yu, Chen J 2014 Acta Phys. Sin. 63 018701 (in Chinese)[古宇飞, 闫镔, 李磊, 魏峰, 韩玉, 陈健 2014 63 018701]
[10] Boyd S, Parikh N, Chu E, Peleato B, Eckstein J 2010 Foundations and Trends?in Machine Learning 3 1
[11] Goldstein T, Osher S 2009 SIAM J. Imaging Sci. 2 323
[12] Wang L Y, Zhang H M, Cai A L, Yan B, Li L, Hu G E 2013 Acta Phys. Sin. 62 198701 (in Chinese)[王林元, 张瀚铭, 蔡爱龙, 闫镔, 李磊, 胡国恩 2013 62 198701]
[13] Ramani S, Fessler J A 2012 IEEE Trans. Med. Imag. 31 677
[14] Matakos A, Ramani S, Fessler J A 2013 IEEE Trans. Image Process. 22 2019
[15] Zhang H M, Wang L Y, Yan B, Li L, Xi X Q, Lu L Z 2013 Chin. Phys.B 22 078701
[16] Sidky E Y, Jørgensen J S, Pan X 2013 Med. Phys. 40 031115
[17] Nesterov Y E 1983 Dokl. Akad. Nauk SSSR 269 543 (in Russian)
[18] Choi K, Wang J, Zhu L, Suh T S, Boyd S, Xing L 2010 Med. Phys. 37 5113
[19] Jensen T L, Jørgensen J S, Hansen P C, Jensen S H 2012 BIT Numer. Math. 52 329
[20] Beck A, Teboulle M 2009 SIAM J. Imaging Sci. 2 183
[21] Daubechies I, Fornasier M, Loris I 2008 J. Fourier Anal. Appl. 14 764
[22] Daubechies I, Defrise M, Mol C D 2004 Comm. Pure Appl. Math. 57 1413
[23] Beck A, Teboulle M 2009 IEEE Trans. Image Process. 18 2419
[24] Erdogan H, Fessler J A 1999 Phys. Med. Biol. 44 2835
[25] Hudson H M, Larkin R S 1994 IEEE Trans. Med. Imag. 13 601
[26] Kim D, Ramani S, Fessler J A 2013 The 12th International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine California, USA, June 16-21, 2013 p22
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