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在求解扩散光学断层成像中的正向问题时, 目前普遍采用有限元法, 但是随着实际模型规模的增大, 有限元法的计算量问题日益显著, 而边界元法则由于可以降低计算维度使计算量减少而备受关注. 本文以均匀的高散射介质为模型, 研究了将快速多极边界元法用于扩散光学断层成像的正向问题. 快速多极边界元法利用核函数的多极展开, 将常规边界元法中系数矩阵和迭代矢量的乘积项等价为相应四叉树结构的一次递归, 再结合广义最小残量法进行迭代求解. 将计算结果和蒙特卡罗法的模拟结果进行了比较, 表明利用快速多极边界元法的模拟结果和蒙特卡罗法的结果有很好的一致性. 研究结果验证了快速多极边界元法可以用于扩散光学断层成像, 为其大规模和实时成像带来可观的前景.The forward problem of diffuse optical tomography (DOT) is commonly solved by the finite element method (FEM) currently. However, with the increase of the model scale, the computational complexity of FEM increases significantly; while the boundary element method (BEM) attracts much attention because of its reduction in calculated dimensions. In this paper, the fast multipole boundary element method (FMBEM) for DOT is studied using a model of highly scattering homogenous medium. In FMBEM, by the multipole expansions of kernel functions, the product of matrix coefficient and iterative vector can be equivalent to the recursion of a quadtree; and then a generalized minimal residual method is used to solve the BEM equation iteratively. The calculations of the FMBEM are compared with Monte Carlo simulations. The results show that the calculations of the FMBEM are in good agreement with Monte Carlo simulations. This demonstrates the feasibility of FMBEM in DOT and indicates that the FMBEM has a bright future for large-scale and real-time imaging.
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Keywords:
- diffuse optical tomography /
- boundary element method /
- fast multipole boundary element method
[1] Stephanie M W Y, Van de ven S M, Elias S G 2009 Eur. Radiol. 19 1108
[2] Hebden J C, Austin T 2007 Eur. Radiol. 17 2926
[3] Deng Y, Zhang X X, Luo Z Y, Xu J, Yang X Q, Meng Y Z, Gong H, Luo Q M 2012 Acta Phys. Sin. 62 014202 (in Chinese) [邓勇, 张喧轩, 罗召洋, 许军, 杨孝全, 孟远征, 龚辉, 骆清铭 2012 物理学报 62 014202]
[4] Srinivasan S, Ghadyani H R, Pogue B W, Paulsen K D 2009 Biomed. Opt. Express 1 398
[5] Elisee J P, Gibson A, Arridge S 2010 IEEE T. Bio-Med. Eng. 57 2737
[6] Elisee J P, Bonnet M, Arridge S 2011 Opt. Lett. 36 4101
[7] Li S D, Huang Q B, Li T Y 2012 Acta Phys. Sin. 61 064301 (in Chinese) [李善德, 黄其柏, 李天匀 2012 物理学报 61 064301]
[8] Xu K X, Gao F, Zhao H J 2007 Biomedical Photonics (Beijing: Science Press) p53 (in Chinese) [徐可欣, 高峰, 赵会娟 2007 生物医学光子学 (北京: 科学出版社)第53页]
[9] Sikora J, Zacharopoulos A, Douiri A, Schweiger M, Horesh L, Arridge S, Ripoll J 2006 Phys. Med. Biol. 51 497
[10] LiuY J 2009 Fast Multipole Boundary Element Method (1st Ed.) (New York: Cambridge University Press) p157
[11] Yaoi Z H, Wang H T 2009 Boundary Element Methods (Beijing: Higher Education Press) p226 (in Chinese) [姚振汉, 王海涛 2009 边界元法(北京: 高等教育出版社)第226页]
[12] Wang H T, Yao Z H, Wang P B 2005 Eng. Anal. Bound. Elem. 29 673
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[1] Stephanie M W Y, Van de ven S M, Elias S G 2009 Eur. Radiol. 19 1108
[2] Hebden J C, Austin T 2007 Eur. Radiol. 17 2926
[3] Deng Y, Zhang X X, Luo Z Y, Xu J, Yang X Q, Meng Y Z, Gong H, Luo Q M 2012 Acta Phys. Sin. 62 014202 (in Chinese) [邓勇, 张喧轩, 罗召洋, 许军, 杨孝全, 孟远征, 龚辉, 骆清铭 2012 物理学报 62 014202]
[4] Srinivasan S, Ghadyani H R, Pogue B W, Paulsen K D 2009 Biomed. Opt. Express 1 398
[5] Elisee J P, Gibson A, Arridge S 2010 IEEE T. Bio-Med. Eng. 57 2737
[6] Elisee J P, Bonnet M, Arridge S 2011 Opt. Lett. 36 4101
[7] Li S D, Huang Q B, Li T Y 2012 Acta Phys. Sin. 61 064301 (in Chinese) [李善德, 黄其柏, 李天匀 2012 物理学报 61 064301]
[8] Xu K X, Gao F, Zhao H J 2007 Biomedical Photonics (Beijing: Science Press) p53 (in Chinese) [徐可欣, 高峰, 赵会娟 2007 生物医学光子学 (北京: 科学出版社)第53页]
[9] Sikora J, Zacharopoulos A, Douiri A, Schweiger M, Horesh L, Arridge S, Ripoll J 2006 Phys. Med. Biol. 51 497
[10] LiuY J 2009 Fast Multipole Boundary Element Method (1st Ed.) (New York: Cambridge University Press) p157
[11] Yaoi Z H, Wang H T 2009 Boundary Element Methods (Beijing: Higher Education Press) p226 (in Chinese) [姚振汉, 王海涛 2009 边界元法(北京: 高等教育出版社)第226页]
[12] Wang H T, Yao Z H, Wang P B 2005 Eng. Anal. Bound. Elem. 29 673
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