基于贝叶斯理论的电容层析成像算法研究
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摘要
电容层析成像(ECT)是一种可获取封闭区域内介质分布图像的过程成像技术,它具有非侵入性、适用范围广、价格低廉等优点,因此在多种工业生产过程中得到广泛运用。
传统的确定性算法获得的是介质分布的近似最优解,而基于贝叶斯的抽样方法可以得到场域内介质分布的全面描述。
本文从贝叶斯理论出发,构造了一种新的ECT图像重建算法。首先,在正问题部分,通过自己编写的有限元仿真软件,进行有限元的计算,得到场域内介质电位分布情况。其次,在反问题部分,以标准正态分布作为先验信息,根据贝叶斯理论,获得了ECT介电常数的后验分伽模型。最后,引入线性反投影的结果作为马尔可夫链的初状态,应用Metropolis-Hastings抽样算法对后验分布进行抽样,根据抽样获得的样本期望值可重建图像,并获得了不同建议分布方差的成像结果。
Electrical Capacitance Tomography(ECT)is a promising technique of process tomography,which can obtain the distribution of the media image in an enclosed area. It has the advantage of being non-intrusive, broad application, and low price, etc, so it has been widely used in a variety of industrial processes.
Traditional ECT image reconstruction algorithm can only obtain the approximate optimal solution of the dielectric constant, but the sampling method based on Bayesian games can be distributed within a comprehensive description of the dielectric constant.
In this paper, based on Bayesian theory, we construct a new image reconstruction algorithm for ECT. Firstly, in the forword problem, we obtain the distribution of the potential field within the media by finite element calculations.Secondly, in the inverse problem, based on Bayesian theory and priori information of standard normal distribution, we obtain the posterior distribution model of the dielectric constant of ECT.Finally, we introduce the linear back-projection as the initial state of Markov chain,sample the posterior distribution with the application of Metropolis-Hastings sampling algorithm,and reconstruct the image by expectations based on samples.Then we obtain variance imaging results of different proposal distribution.
传统的确定性算法获得的是介质分布的近似最优解,而基于贝叶斯的抽样方法可以得到场域内介质分布的全面描述。
本文从贝叶斯理论出发,构造了一种新的ECT图像重建算法。首先,在正问题部分,通过自己编写的有限元仿真软件,进行有限元的计算,得到场域内介质电位分布情况。其次,在反问题部分,以标准正态分布作为先验信息,根据贝叶斯理论,获得了ECT介电常数的后验分伽模型。最后,引入线性反投影的结果作为马尔可夫链的初状态,应用Metropolis-Hastings抽样算法对后验分布进行抽样,根据抽样获得的样本期望值可重建图像,并获得了不同建议分布方差的成像结果。
Electrical Capacitance Tomography(ECT)is a promising technique of process tomography,which can obtain the distribution of the media image in an enclosed area. It has the advantage of being non-intrusive, broad application, and low price, etc, so it has been widely used in a variety of industrial processes.
Traditional ECT image reconstruction algorithm can only obtain the approximate optimal solution of the dielectric constant, but the sampling method based on Bayesian games can be distributed within a comprehensive description of the dielectric constant.
In this paper, based on Bayesian theory, we construct a new image reconstruction algorithm for ECT. Firstly, in the forword problem, we obtain the distribution of the potential field within the media by finite element calculations.Secondly, in the inverse problem, based on Bayesian theory and priori information of standard normal distribution, we obtain the posterior distribution model of the dielectric constant of ECT.Finally, we introduce the linear back-projection as the initial state of Markov chain,sample the posterior distribution with the application of Metropolis-Hastings sampling algorithm,and reconstruct the image by expectations based on samples.Then we obtain variance imaging results of different proposal distribution.
引文
[1]马军海,郑万明.电阻抗层析成像(EIT)的算法设计(Ⅲ)[J].天津：天津商学院学报,1996,(01)：16-19.
[2]宫莲,张克潜.用三维各向异性电阻抗成像做人脑活动的研究[J].清华大学学报自然科学版,1999,39(05)：1-4.
[3]江鹏,彭黎辉,萧德云.Gaussian窗函数在电容成像图像重建中的应用[J].清华大学学报(自然科学版),2007,47(01)：150-153.
[4]徐明.基于统计方法的电容层析成像重建算法研究：(硕士学位论文).天津：天津大学,2007.
[5]江鹏,彭黎辉,陆耿,萧德云.基于贝叶斯理论的电容层析成像图像重建迭代算法[J].中国电机工程学报,2008,28(11)：65-71.
[6]胡红利.用于两相流测量的图像重构技术研究工业仪表与自动化装置[J],2010,(2)：100-103.
[7]郑万松.EIT腹部电极系统设计研究：(硕士学位论文).西安：第四军医大学,2006.
[8]罗辞勇,陈民铀,王平,何为.电阻抗成像交义测量模式的抗噪声性能研究[J].仪器仪表学报,2009,30(01)：15-19.
[9]彭珍瑞,祁文哲,吴刊选,孟建军.电容层析成像技术的近期研究进展.自动化仪表,2008,29(9)：1-5.
[10]Xie C G, Huang S M, Hoyle B S. Electrica] capacitance tomography for flow imaging: system model for development of image reconstruction algorithm and design of primary sensors[J]. IEEE Proceedings 2G,1992,139(1):89-98.
[11]Isaksen O, Nordtvedt J E. A new reconstruction algorithm for process tomography [J]. Measurement Science and Technology,1993,4(12):1464-1475.
[12]Nooralahiyan A Y, Hoyle B S. Three2 component tomographic flow imaging using artificial neural network reconstruction[J]. Chemical Engineering Science,1997, 52(13):2139-2148.
[13]Yang W Q, Spink D M, York T A.An image reconstruction algorithm based on Landweber's iteration method for electrical capacitance tomography [J]. Measurement Science and Technology,1999,10(11):1065-1069.
[14]姜凡,刘靖,刘石,梁世强,王雪瑶.电容层析成像技术应用于冰水两相测试研究[J].中国 电机工程学报,2010,30(05)：49-53.
[15]高鹤明,许传龙,王式民.优化Tikhonov迭代法在电容层析成像中的应用[J].东南大学学报(自然科学版),2010,40(03)：527-532.
[16]杨道业,许传龙,周宾,王式民.基于单检测通道的电容层析成像系统[J].仪器仪表学报,2010,31(01)：132-136.
[17]王淑荣.电容层析成像中最优化算法的研究与应用[D]：(硕士学位论文).北京：华北电力大学,2010.
[18]Kaipio J P and Somersalo E. Statistical and computational inverse problems (New York:Applied Mathematical Sciences, Springer).2004
[19]Fox C and Nicholls G K. Exact map states and expectations from perfect sampling: Greig, Porteous and Seheult revisited, in Twentieth Int. Workshop on Bayesian Inference andMaximum Entropy Methods in Sci. and Eng, A. Mohammad-Djafari ed., American Institute of Physics.2001
[20]陈姝君.电阻抗断层成像技术的研究图像重建算法及实现[D]：(硕士学位论文).南京：南京理工大学,2008.
[21]Fox C and Nicholls G K. Sampling conductivity images via MCMC, The Art and Science of Bayesian Image Analysis-Leeds Annual Statistics Research Workshop.1997,14(2) 91-100.
[22]Higdon D, Lee H and Holloman C.Markov chain Monte Carlo-based approaches for inference in computationally intensive inverse problems. Oxford University Press:Bayesian Statistics 7.2003.
[23]Nicholls G K and Fox C. Prior modelling and posterior sampling in impedance imaging, Proc. of Bayesian Inference for Inverse Problems.1998, (3459):116-127.
[24]Kolehmainen V et al.A Bayesian approach and total variation priors in 3D electrical impedance tomography, Proceedings of the 20t" Annual International Conference of the IEEE Engineering in Medicine and Biology Society,1998.
[25]Wang M. Inverse solutions for electrical impedance tomography based on conjugate gradients methods. Meas. Sci. Technol,2002,13:101-117.
[26]Kaipio J P etal. Statistic inversion in Monte Carlo sampling methods in electrical Impedance tomography. Inverse Problems,2000,16:1487-1522.
[27]Vauhkonen M. Electrical impedance tomography and prior information. Ph. D Thesis, Kuopio University, Kuopio, Finland,1997.
[28]Martin T and Idier J. A FEM based nonlinear MAP estimator in electrical impedance tomography. Proc. IEEE ICIP'97,1997.
[29]Mosegaard K. Resolution analysis of general inverse problems through inverse monte carlo sampling. Inverse Problems,1998,14:405-426.
[30]Kolehmainen V and Somersalo E. A Bayesian approach and total variation priors in 3D electrical impedance tomography. Proc.20th Ann. Int. Conf. IEEE Eng. Med. Biol. Soc,1998:1028-31.
[31]Dobson D and Samosa F. An image enhancement technique for electrical impedance tomography. Inverse Problems,1994,10:317-334.
[32]Fox C and Nicholls G.Sampling conductivity images via MCMC. Proc. Leeds Ann. Statistics Research Workshop,1997:91-100.
[33]Glidewell M. Anatomically constrained electrical impedance tomography for anisotropic bodies. IEEE Trans. Med. Imaging,1995,14:498-503.
[34]王富耻,张朝晖.ANSYS 10.0有限元分析理论与工程应用.北京：电子工业出版社.(2005)
[35]Glidewell M. Anatomically constrained electrical impedance tomography for three dimensional anisotropic bodies. IEEE Trans. Med. Imaging,1997,16:572-580.
[36]Mosegaard K and Tarantola A.Monte Carlo sampling of solutions to inverse problems. J. Geophys,1995,12:431-447.
[37]Nicholls G K and Fox C. Prior modeling and posterior sampling in impedance imaging Bayesian Inference for Inverse Problems. A Mohammad,1998,59:116-127.
[38]PinheiroP A T. Smoothness-constrained inversion for two-dimensional electrical resistance tomography. Meas. Sci. Technol,1997,8:293-302.
[39]赵琪.MCMC方法研究：(硕十学位论文).济南：山东大学.2007.
[40]Somersalo E and Kaipio J. Impedance imaging and Markov chain Monte Carlo methods. Medical and Nonmedical Applications,1997,10:175-85.
[41]Tierney L. Markov chains for exploring posterior distributions Ann. Stat,1994,22: 1701-1762.
[42]Vauhkonen M and Kaipio J, Somersalo E and Karj alainen P. Electrical impedance tomography with basis constraints. Inverse Problems,1997,13:523-30.
[43]Isaksen G. A review of reconstruction techni ques for capacitance tomography. Meas. Sci. Technol,1996,7:325-337. [20] Andersen K E, Brooks S P and Hansen M B 2003 Bayesian inversion of geoelectrical resistivity data, Jnl. Of the Royal Stat. Soc.:Series B 65 619-642.
[44]Christen J A and Fox C. MCMC using an approximation. Jnl. of Comp. and Graphical Stat,2005,14,795-810.
[45]慈璋.电磁场(第二版).北京：高等教育出版社,1983.99-168.
[46]马艺馨.电阻层析成像技术及其在气/液两相泡状流检测中的应用：(博士学位论文).天津：天津大学,1999.
[47]王保良.电容层析成像技术及其在两相流参数检测中的应用研究：(博士学位论文).浙江：浙江大学,1998.
[48]冯康等编.数值计算方法.北京：国防工业出版社,1978.156-168.
[49]连汉雄.电磁场理论的数学方法.北京：北京理工大学出版社,1990.45-62.
[50]刘鹏程.电磁场解析方法.北京：电子工业出版社,1995.82-96.
[51]符果行.经典电磁理论方法.成都：电子科技大学出版社,1998.67-89.
[52]肖庭延,于慎根,王彦飞.反问题的数值解法.北京：科学出版社,2003.
[53]傅鹂,刘石,杨五强.两相流测量中电容层析成像图像重建的广义逆最小模解与线性反投影和迭代法的比较[J].仪器仪表学报,2001,22(01)：74-78.
[54]倪光正.电磁场数值计算.北京：高等教育出版社,1996.82-96.
[55]陈希儒.高等数理统计学.合肥：中国科学技术大学出版社,1999.444-457.
[56]南京大学数学系计算数学专业.最优化方法.北京：科学出版社,1978.143-148.
[57]王彦飞.反演问题的计算方法及其应用.北京：高等教育出版社.2007.
[58]刘军著.唐年胜等译.科学计算中的蒙特卡洛策略.北京：高等教育出版社.2009.
[59]徐明.基于统计方法的电容层析成像重建算法研究：(硕士学位论文).天津：天津大学,2007.
[60]雷兢.多相流的电容层析成像图像重建研究：(博士学位论文).北京：中国科学院工程热物理研究所,2008.
[61]李大心,沈博.近代物理检测原理与技术.武汉：中国地质大学出版社.2007.
[62]朱嵩.基于贝叶斯推理的环境水力学反问题研究：(博士学位论文).浙江：浙江大学,2008.
[2]宫莲,张克潜.用三维各向异性电阻抗成像做人脑活动的研究[J].清华大学学报自然科学版,1999,39(05)：1-4.
[3]江鹏,彭黎辉,萧德云.Gaussian窗函数在电容成像图像重建中的应用[J].清华大学学报(自然科学版),2007,47(01)：150-153.
[4]徐明.基于统计方法的电容层析成像重建算法研究：(硕士学位论文).天津：天津大学,2007.
[5]江鹏,彭黎辉,陆耿,萧德云.基于贝叶斯理论的电容层析成像图像重建迭代算法[J].中国电机工程学报,2008,28(11)：65-71.
[6]胡红利.用于两相流测量的图像重构技术研究工业仪表与自动化装置[J],2010,(2)：100-103.
[7]郑万松.EIT腹部电极系统设计研究：(硕士学位论文).西安：第四军医大学,2006.
[8]罗辞勇,陈民铀,王平,何为.电阻抗成像交义测量模式的抗噪声性能研究[J].仪器仪表学报,2009,30(01)：15-19.
[9]彭珍瑞,祁文哲,吴刊选,孟建军.电容层析成像技术的近期研究进展.自动化仪表,2008,29(9)：1-5.
[10]Xie C G, Huang S M, Hoyle B S. Electrica] capacitance tomography for flow imaging: system model for development of image reconstruction algorithm and design of primary sensors[J]. IEEE Proceedings 2G,1992,139(1):89-98.
[11]Isaksen O, Nordtvedt J E. A new reconstruction algorithm for process tomography [J]. Measurement Science and Technology,1993,4(12):1464-1475.
[12]Nooralahiyan A Y, Hoyle B S. Three2 component tomographic flow imaging using artificial neural network reconstruction[J]. Chemical Engineering Science,1997, 52(13):2139-2148.
[13]Yang W Q, Spink D M, York T A.An image reconstruction algorithm based on Landweber's iteration method for electrical capacitance tomography [J]. Measurement Science and Technology,1999,10(11):1065-1069.
[14]姜凡,刘靖,刘石,梁世强,王雪瑶.电容层析成像技术应用于冰水两相测试研究[J].中国 电机工程学报,2010,30(05)：49-53.
[15]高鹤明,许传龙,王式民.优化Tikhonov迭代法在电容层析成像中的应用[J].东南大学学报(自然科学版),2010,40(03)：527-532.
[16]杨道业,许传龙,周宾,王式民.基于单检测通道的电容层析成像系统[J].仪器仪表学报,2010,31(01)：132-136.
[17]王淑荣.电容层析成像中最优化算法的研究与应用[D]：(硕士学位论文).北京：华北电力大学,2010.
[18]Kaipio J P and Somersalo E. Statistical and computational inverse problems (New York:Applied Mathematical Sciences, Springer).2004
[19]Fox C and Nicholls G K. Exact map states and expectations from perfect sampling: Greig, Porteous and Seheult revisited, in Twentieth Int. Workshop on Bayesian Inference andMaximum Entropy Methods in Sci. and Eng, A. Mohammad-Djafari ed., American Institute of Physics.2001
[20]陈姝君.电阻抗断层成像技术的研究图像重建算法及实现[D]：(硕士学位论文).南京：南京理工大学,2008.
[21]Fox C and Nicholls G K. Sampling conductivity images via MCMC, The Art and Science of Bayesian Image Analysis-Leeds Annual Statistics Research Workshop.1997,14(2) 91-100.
[22]Higdon D, Lee H and Holloman C.Markov chain Monte Carlo-based approaches for inference in computationally intensive inverse problems. Oxford University Press:Bayesian Statistics 7.2003.
[23]Nicholls G K and Fox C. Prior modelling and posterior sampling in impedance imaging, Proc. of Bayesian Inference for Inverse Problems.1998, (3459):116-127.
[24]Kolehmainen V et al.A Bayesian approach and total variation priors in 3D electrical impedance tomography, Proceedings of the 20t" Annual International Conference of the IEEE Engineering in Medicine and Biology Society,1998.
[25]Wang M. Inverse solutions for electrical impedance tomography based on conjugate gradients methods. Meas. Sci. Technol,2002,13:101-117.
[26]Kaipio J P etal. Statistic inversion in Monte Carlo sampling methods in electrical Impedance tomography. Inverse Problems,2000,16:1487-1522.
[27]Vauhkonen M. Electrical impedance tomography and prior information. Ph. D Thesis, Kuopio University, Kuopio, Finland,1997.
[28]Martin T and Idier J. A FEM based nonlinear MAP estimator in electrical impedance tomography. Proc. IEEE ICIP'97,1997.
[29]Mosegaard K. Resolution analysis of general inverse problems through inverse monte carlo sampling. Inverse Problems,1998,14:405-426.
[30]Kolehmainen V and Somersalo E. A Bayesian approach and total variation priors in 3D electrical impedance tomography. Proc.20th Ann. Int. Conf. IEEE Eng. Med. Biol. Soc,1998:1028-31.
[31]Dobson D and Samosa F. An image enhancement technique for electrical impedance tomography. Inverse Problems,1994,10:317-334.
[32]Fox C and Nicholls G.Sampling conductivity images via MCMC. Proc. Leeds Ann. Statistics Research Workshop,1997:91-100.
[33]Glidewell M. Anatomically constrained electrical impedance tomography for anisotropic bodies. IEEE Trans. Med. Imaging,1995,14:498-503.
[34]王富耻,张朝晖.ANSYS 10.0有限元分析理论与工程应用.北京：电子工业出版社.(2005)
[35]Glidewell M. Anatomically constrained electrical impedance tomography for three dimensional anisotropic bodies. IEEE Trans. Med. Imaging,1997,16:572-580.
[36]Mosegaard K and Tarantola A.Monte Carlo sampling of solutions to inverse problems. J. Geophys,1995,12:431-447.
[37]Nicholls G K and Fox C. Prior modeling and posterior sampling in impedance imaging Bayesian Inference for Inverse Problems. A Mohammad,1998,59:116-127.
[38]PinheiroP A T. Smoothness-constrained inversion for two-dimensional electrical resistance tomography. Meas. Sci. Technol,1997,8:293-302.
[39]赵琪.MCMC方法研究：(硕十学位论文).济南：山东大学.2007.
[40]Somersalo E and Kaipio J. Impedance imaging and Markov chain Monte Carlo methods. Medical and Nonmedical Applications,1997,10:175-85.
[41]Tierney L. Markov chains for exploring posterior distributions Ann. Stat,1994,22: 1701-1762.
[42]Vauhkonen M and Kaipio J, Somersalo E and Karj alainen P. Electrical impedance tomography with basis constraints. Inverse Problems,1997,13:523-30.
[43]Isaksen G. A review of reconstruction techni ques for capacitance tomography. Meas. Sci. Technol,1996,7:325-337. [20] Andersen K E, Brooks S P and Hansen M B 2003 Bayesian inversion of geoelectrical resistivity data, Jnl. Of the Royal Stat. Soc.:Series B 65 619-642.
[44]Christen J A and Fox C. MCMC using an approximation. Jnl. of Comp. and Graphical Stat,2005,14,795-810.
[45]慈璋.电磁场(第二版).北京：高等教育出版社,1983.99-168.
[46]马艺馨.电阻层析成像技术及其在气/液两相泡状流检测中的应用：(博士学位论文).天津：天津大学,1999.
[47]王保良.电容层析成像技术及其在两相流参数检测中的应用研究：(博士学位论文).浙江：浙江大学,1998.
[48]冯康等编.数值计算方法.北京：国防工业出版社,1978.156-168.
[49]连汉雄.电磁场理论的数学方法.北京：北京理工大学出版社,1990.45-62.
[50]刘鹏程.电磁场解析方法.北京：电子工业出版社,1995.82-96.
[51]符果行.经典电磁理论方法.成都：电子科技大学出版社,1998.67-89.
[52]肖庭延,于慎根,王彦飞.反问题的数值解法.北京：科学出版社,2003.
[53]傅鹂,刘石,杨五强.两相流测量中电容层析成像图像重建的广义逆最小模解与线性反投影和迭代法的比较[J].仪器仪表学报,2001,22(01)：74-78.
[54]倪光正.电磁场数值计算.北京：高等教育出版社,1996.82-96.
[55]陈希儒.高等数理统计学.合肥：中国科学技术大学出版社,1999.444-457.
[56]南京大学数学系计算数学专业.最优化方法.北京：科学出版社,1978.143-148.
[57]王彦飞.反演问题的计算方法及其应用.北京：高等教育出版社.2007.
[58]刘军著.唐年胜等译.科学计算中的蒙特卡洛策略.北京：高等教育出版社.2009.
[59]徐明.基于统计方法的电容层析成像重建算法研究：(硕士学位论文).天津：天津大学,2007.
[60]雷兢.多相流的电容层析成像图像重建研究：(博士学位论文).北京：中国科学院工程热物理研究所,2008.
[61]李大心,沈博.近代物理检测原理与技术.武汉：中国地质大学出版社.2007.
[62]朱嵩.基于贝叶斯推理的环境水力学反问题研究：(博士学位论文).浙江：浙江大学,2008.