基于软场的图像成像算法研究
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摘要
电学层析成像(ET)是重建内部电学特性(电导率或介电常数)分布的技术。ET中,阵列电极附于物体表面并交替加电压/电流激励,测量电极上的电容/电压。运用边界上的测量值重建内部场域的电导率或介电常数的近似空间分布。
ET技术具有无辐射、非侵入、快速响应和低成本等优点,它主要应用于三个领域:医学成像、工业过程成像和地球物理学测量。
电容层析成像(ECT)和电阻成像成像(ERT)是基于不同原理的两种电学层析成像。ECT适于测量以非导电物质为连续相的介质,ERT适于测量导电物质为连续相的介质。基于(ERT/ECT)的双模态技术可以应用于多相流中同时测量导电和非导电物质。
数学上,电学层析成像的物理模型符合拉普拉斯椭圆偏微分方程。对于场域中任意电导率或介电常数的分布,无法求取解析解,因此,采用有限元(FEM)方法求解正向问题。电学层析成像的图像重建问题是非线性的逆问题,需要采用特殊的方法才能获得较精确解。
本论文中,电学层析成像的正向问题采用COMSOL Multiphysics?和Matlab?软件进行求解,COMSOL具有建模方便简单和求解精确等优点。论文中分析了两个模态的相互影响并给出最优的电极配置和电极尺寸。
另外,文中讨论了逆问题求解的基本理论,提出了几种新的图像重建算法。
(1)提出求解逆问题的组合算法,该算法结合Krylov子空间和Tikhonov正则化算法对病态逆问题进行两重正则化。对组合算法、截断奇异值分解算法和Tikhonov正则化算法进行比较,实验表明,组合算法能够极大节约计算时间,通过运用L-曲线法选取正则化参数,该算法能够提高图像重建的质量。
(2)引入两重预迭代算法,该算法通过将原问题分解成两个子空间求解逆问题。实验表明,通过与非迭代算法和一般的迭代算法比较,该算法具有高效和稳定等特点。同时,文中讨论了参数的选择、滤波因素和收敛特性。
(3)通过结合先验知识,提出基于自适应剖分的图像重建算法。不但引入了自适应剖分,而且分析了后验误差、网格优化和网格质量。通过自适应剖分,能够获得研究区域更加精确的电势分布。自适应剖分结合正则化的Newton-Raphson算法,不但能够提高重建图像的质量,同时减少了计算时间。
(4)建立肺部模型,提出一种运用先验知识的新算法。结果表明该算法能够极大提高图像重建质量,并展示了这种结合先验知识算法的巨大发展潜力。
Electrical Tomography (ET) is a technique for reconstructing electrical properties (conductivity or permittivity) of the interior structures. In ET, an array of electrodes is attached around the object and alternating voltages/currents are injected via these electrodes and the resulting capacitances/voltages are measured. Using measurements on the boundary, an approximation for the spatial distribution of conductivity or permittivity within the object can be reconstructed.
Due to the advantages of non-radiant, non-intrusive, rapid response and low cost, ET technique is of great potential in three principal areas: medical imaging, industrial process imaging and geophysical surveying.
Electrical Capacitance Tomography (ECT) and Electrical Resistance Tomography (ERT) based on different principle are two important categories of ET. ECT is suitable to detect dielectric material, while ERT is capable to measure conducting medium. The dual-mode electrical tomography (ERT/ECT) technique is proposed to measure conducting and dielectric materials in multi-phase application.
Mathematically, the physical model for ET meets the Laplace elliptic partial differential equation. Since there is no analytical solution for an arbitrary conductivity or permittivity distribution in the volume, Finite Element Method (FEM) is a nice choice to solve the forward problem in ET. The ET reconstruction problem is a nonlinear ill-posed inverse problem which requires the use of particular methods to obtain accurate solution.
In this thesis, the forward problem in ET is solved using COMSOL Multiphysics? with Matlab? which can be modeled flexibly and solved accurately. The interactions of these two modes are analyzed and optimum design of the electrodes configuration and size for them are obtained respectively.
Besides, basic theories of inverse problem are discussed, and several new approaches of ET image reconstruction are proposed.
Firstly, a hybrid method is proposed for solving the inverse problem for ET, which combines the Krylov subspace and the Tikhonov regularisation for double levels of regularisation to the ill-posed problem. Numerical simulation results using the hybrid method are presented and compared to those by Truncated singular value decomposition (TSVD) regularisation and the Tikhonov regularisation. Experimental results with the hybrid method are also presented, indicating that the hybrid method can reduce the computation time, and improve the resolution of reconstructed images with the regularisation parameter automatically chosen by the L-curve method. Then, to solve the inverse problem of ET, a two-level pre-iteration reconstruction algorithm is introduced which splits the solution space into two subspaces.
Experimental results indicate that the algorithm is much more stable and efficient comparing to direct methods and normal iterative methods. At the same time, the parameter choosing, filter factor and convergence property are discussed.
Thirdly, Reconstruction algorithm for electrical tomography based on adaptive mesh refinement which combines with prior information is discussed. The adaptive finite element method is introduced and the posteriori error, mesh refinement, mesh quality are analyzed. More accurate potential distribution in the investigated field domain is derived. The regularized Newton-Raphson algorithm is adopted based on adaptive mesh refinement to reconstruct the interior conductivity distribution. Experimental results show that this algorithm can not only improve the resolution of the recovered image but also reduce the reconstruction time.
Finally, the model of pulmo is built, and a novel reconstruction method which exploits available prior information is proposed. Results show that this method can improve the image quality greatly and exhibit the attractive developing potential of this kind of methods tremendously.
ET技术具有无辐射、非侵入、快速响应和低成本等优点,它主要应用于三个领域:医学成像、工业过程成像和地球物理学测量。
电容层析成像(ECT)和电阻成像成像(ERT)是基于不同原理的两种电学层析成像。ECT适于测量以非导电物质为连续相的介质,ERT适于测量导电物质为连续相的介质。基于(ERT/ECT)的双模态技术可以应用于多相流中同时测量导电和非导电物质。
数学上,电学层析成像的物理模型符合拉普拉斯椭圆偏微分方程。对于场域中任意电导率或介电常数的分布,无法求取解析解,因此,采用有限元(FEM)方法求解正向问题。电学层析成像的图像重建问题是非线性的逆问题,需要采用特殊的方法才能获得较精确解。
本论文中,电学层析成像的正向问题采用COMSOL Multiphysics?和Matlab?软件进行求解,COMSOL具有建模方便简单和求解精确等优点。论文中分析了两个模态的相互影响并给出最优的电极配置和电极尺寸。
另外,文中讨论了逆问题求解的基本理论,提出了几种新的图像重建算法。
(1)提出求解逆问题的组合算法,该算法结合Krylov子空间和Tikhonov正则化算法对病态逆问题进行两重正则化。对组合算法、截断奇异值分解算法和Tikhonov正则化算法进行比较,实验表明,组合算法能够极大节约计算时间,通过运用L-曲线法选取正则化参数,该算法能够提高图像重建的质量。
(2)引入两重预迭代算法,该算法通过将原问题分解成两个子空间求解逆问题。实验表明,通过与非迭代算法和一般的迭代算法比较,该算法具有高效和稳定等特点。同时,文中讨论了参数的选择、滤波因素和收敛特性。
(3)通过结合先验知识,提出基于自适应剖分的图像重建算法。不但引入了自适应剖分,而且分析了后验误差、网格优化和网格质量。通过自适应剖分,能够获得研究区域更加精确的电势分布。自适应剖分结合正则化的Newton-Raphson算法,不但能够提高重建图像的质量,同时减少了计算时间。
(4)建立肺部模型,提出一种运用先验知识的新算法。结果表明该算法能够极大提高图像重建质量,并展示了这种结合先验知识算法的巨大发展潜力。
Electrical Tomography (ET) is a technique for reconstructing electrical properties (conductivity or permittivity) of the interior structures. In ET, an array of electrodes is attached around the object and alternating voltages/currents are injected via these electrodes and the resulting capacitances/voltages are measured. Using measurements on the boundary, an approximation for the spatial distribution of conductivity or permittivity within the object can be reconstructed.
Due to the advantages of non-radiant, non-intrusive, rapid response and low cost, ET technique is of great potential in three principal areas: medical imaging, industrial process imaging and geophysical surveying.
Electrical Capacitance Tomography (ECT) and Electrical Resistance Tomography (ERT) based on different principle are two important categories of ET. ECT is suitable to detect dielectric material, while ERT is capable to measure conducting medium. The dual-mode electrical tomography (ERT/ECT) technique is proposed to measure conducting and dielectric materials in multi-phase application.
Mathematically, the physical model for ET meets the Laplace elliptic partial differential equation. Since there is no analytical solution for an arbitrary conductivity or permittivity distribution in the volume, Finite Element Method (FEM) is a nice choice to solve the forward problem in ET. The ET reconstruction problem is a nonlinear ill-posed inverse problem which requires the use of particular methods to obtain accurate solution.
In this thesis, the forward problem in ET is solved using COMSOL Multiphysics? with Matlab? which can be modeled flexibly and solved accurately. The interactions of these two modes are analyzed and optimum design of the electrodes configuration and size for them are obtained respectively.
Besides, basic theories of inverse problem are discussed, and several new approaches of ET image reconstruction are proposed.
Firstly, a hybrid method is proposed for solving the inverse problem for ET, which combines the Krylov subspace and the Tikhonov regularisation for double levels of regularisation to the ill-posed problem. Numerical simulation results using the hybrid method are presented and compared to those by Truncated singular value decomposition (TSVD) regularisation and the Tikhonov regularisation. Experimental results with the hybrid method are also presented, indicating that the hybrid method can reduce the computation time, and improve the resolution of reconstructed images with the regularisation parameter automatically chosen by the L-curve method. Then, to solve the inverse problem of ET, a two-level pre-iteration reconstruction algorithm is introduced which splits the solution space into two subspaces.
Experimental results indicate that the algorithm is much more stable and efficient comparing to direct methods and normal iterative methods. At the same time, the parameter choosing, filter factor and convergence property are discussed.
Thirdly, Reconstruction algorithm for electrical tomography based on adaptive mesh refinement which combines with prior information is discussed. The adaptive finite element method is introduced and the posteriori error, mesh refinement, mesh quality are analyzed. More accurate potential distribution in the investigated field domain is derived. The regularized Newton-Raphson algorithm is adopted based on adaptive mesh refinement to reconstruct the interior conductivity distribution. Experimental results show that this algorithm can not only improve the resolution of the recovered image but also reduce the reconstruction time.
Finally, the model of pulmo is built, and a novel reconstruction method which exploits available prior information is proposed. Results show that this method can improve the image quality greatly and exhibit the attractive developing potential of this kind of methods tremendously.
引文
[1]王化祥,杨五强电容过程成像技术的进展,仪器仪表学报,2000,21(1):4-7
[2] M. Schweiger, S. R. Arridge and D. T. Delpy, Application of the finite-element method for the forward and inverse models in optical tomography, J. Math. Imag. Vision 3, 263-283 (1993)
[3] Shepp LA, et al. Maximum Likelihood Reconstruction in Emission Tomography. IEEE Trans Med Imag, 1982, MI-1:113~122
[4]庄天戈,CT原理与算法,上海:上海交通大学出版社,1992, 1-8
[5] Kwong KK, Belliveau JW, Chesler DA, et al. Dynamic magnetic resonance imaging of human brain activity during primary sensory stimulation. Proc Nat Acad Sci USA, 1992, 89:5675-5679
[6]钱英.医学超声设备的新进展.医疗设备信息, 2002, (11) :25-27
[7] D. M. Otten, Cryosurgical monitoring using visible light imaging and electrical impedance tomography—A feasibility study, Ph.D dissertation, University of California, Berkeley, 2000
[8] A. M. Dijkstra, B. H. Brown, A. D. Leathard, et al, Review Clinical Application of Electrical Impedance Tomography, Journal of Medical Engineering & Technology, 1993, 17(3): 89~98
[9] Todd Eric Kerner, Electrical Impedance Tomography for Breast Imaging, Ph.D dissertation, Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire, 2001,7
[10]任超世, EIT—一种诱人的医学成像技术,电子科技导报, 1996, 5: 9~12
[11] H Griffiths, A phantom for electrical impedance tomography, Clin. Phys. Physiol. Meas., 1988, 9, Suppl. A, 15~20
[12] Xie C G et al, 8-electrode Capacitance System for Two-component Flow Identification. Part1: Tomography Flow Imaging, IEE Proc.A, 136, 1989, 184~190
[13] Xie C G et al, Electrical Capacitance Tomography for Imaging System Model for Development of Reconstruction Algorithms and Design of Primary Sensors, IEE Proc. G, 139, 1992, 89~98
[14] M.Tang,W.Wang,I.Wheeler,et al. Effects of incompatible boundary information in EIT on the convergence behavior of an iterative algorithm[J].IEEE Trans. Med. Imag, 2002, 21: 620-628
[15]汤孟兴,董秀珍,一个临床应用的实时电阻抗断层成像系统的设计与初步结果,国外医学生物医学工程分册,1995,8(5): 288~294
[16]贺兴柏,周守昌,阻抗CT图像重建算法的研究,重庆大学学报,1992,15(5): 38~43
[17]程吉宽,孙进平,杜岩等,电阻抗成像技术的原理及其发展,北京航空航天大学学报, 1998, 24(2): 137~140
[18]毕德显,电磁场理论,北京:电子工业出版社,1985,65~232
[19]冯慈璋,马西奎,工程电磁场导论,北京:高等教育出版社,2000,1~88
[20]王超,医学电阻抗断层成像的研究,[博士学位论文],天津大学,2001
[21]马艺馨,电阻抗成像技术及其在气/液两相泡状流检测中的应用,[博士学位论文],天津大学, 1999
[22] Vauhkonen M, Electrical impedance tomography and prior information, PhD thesis, Department of Physics, University of Kuopio, Finland. 1997
[23]何永勃,电阻抗(ECT/ERT)双模态层析成像技术研究, [博士学位论文],天津大学,2006
[24]问雪宁,基于ANSYS的ECT/ERT电磁场仿真及阵列电极优化设计,[硕士学位论文],天津大学,2006
[25]盛剑宏,电磁场数值分析,北京:科学出版社, 1984
[26]李开泰,有限元方法及应用,西安:西安交通大学出版社, 1992
[27] M. V. K.查利,电磁场问题的有限元解法,北京:科学技术出版社, 1985
[28] Vauhkonen M, Lionheart W R B, Heikkinen L M, Vauhkonen P J and Kaipio L P (2001), A MATLAB package for the EIDORS projects to reconstruct two-dimensional EIT images, Physiol . Meas., 22:107-111
[29]王超,王化祥,医学电阻抗成像系统电极结构优化设计,第四军医大学学报, 2001, 22(1): 78~80
[30] Y Li and W Q Yang. Virtual electrical capacitance tomography sensor. Journal of Physics: Conference Series 15 (2005)183–188
[31]彭黎辉,陆耿,杨五强,电容成像图象重建算法原理及评价,清华大学学报,2004,41(4):478-484
[32] R. Gadd, P. Record, P. Rolfe, A sensitivity region reconstruction algorithm using adjacent drive current injection strategy, Clin. Phys. Physiol. Meas., 1992, 13, Supp1. A, 101~105
[33] Bjorck A (1988), A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations, BIT, 28:659-670
[34] Borsic A (2002), Regularisation methods for imaging from electrical measurements, PhD thesis, School of Engineering, Oxford Brookes University, UK
[35] Hanke M and Hansen P C (1993), Regularization methods for large-scale problems, Surveys on Mathematics for Industry, 3:253-315
[36] Hansen P C (2005), Discrete inverse problems, xxxx, Informatics and Mathematical Modelling, Technical University of Denmark, Denmark
[37] Holder D S (2004), electrical impedance tomography, Institute of Physics Publishing, Bristol and Philadelphia
[39] Jacobsen M (2004), Modular regularization algorithms, PhD thesis, Informatics and Mathematical Modelling, Technical University of Denmark, Denmark
[40] O’Leary D P and Simmons J A (1981), A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems, SIAM J. Sci. Stat. Comput., 2:474-488
[41] Vauhkonen M, Vadasz D, Kaipio J P, Somersalo E and Karjalainen P A, (1998), Tikhonov regularization and priori information in electrical impedance tomography, IEEE Trans. Medical Imaging, 17:285-293
[42] Webster J G (1990), Electrical impedance tomography, Adam Higler
[43] M. Molinari , B.H. Blott , S.J. Cox , et al. Optimal imaging with adaptive mesh refinement in electrical impedance tomography. Physiol .Meas. 2002 . 23:121-128
[44] M Monlinari , S. J. Cox , B.H. Blott, et al. Adaptive mesh refinement techniques for electrical impedance tomography . Physiol. Meas. , 2001. 22:91-96
[45] M Molinari, H Fangohr, J Generowicz, et al . Finite Element Optimizations for Efficient Non-Linear Electrical Tomography Reconstruction. 2nd World Congress on Industrial Process Tomography, 2001 .29-31
[46] P. Fernandes, P. Girdinio, P. Molfino, et al . LOCAL ERROR ESTIMATES FOR ADAPTIVE MESH REFINEMENT.IEEE TRANSACTIONS ON MAGNETICS, 1988 . VOL. 24, NO. 1
[47] T.J.Yorkey, J.G. Webster, and W.J.Tomkins. Comparing Reconstruction algorithm for Electrical impedance tomography . IEEE Trans Biomed Eng , 1987 . 34:843-852
[48] Michael Jacobsen(2000). Two-Grid Iterative Methods for Ill-Posed Problems. PhD thesis, Informatics and Mathematical Modelling, Technical University of Denmark, Denmark
[49]余学进, D. Redekop.自适应有限元的应用和发展.南昌航空工业学院学报,1998,4:78-83
[50]王超,王化祥.电阻抗断层图像重建算法研究.信号处理,2002, 18(6):547-550
[2] M. Schweiger, S. R. Arridge and D. T. Delpy, Application of the finite-element method for the forward and inverse models in optical tomography, J. Math. Imag. Vision 3, 263-283 (1993)
[3] Shepp LA, et al. Maximum Likelihood Reconstruction in Emission Tomography. IEEE Trans Med Imag, 1982, MI-1:113~122
[4]庄天戈,CT原理与算法,上海:上海交通大学出版社,1992, 1-8
[5] Kwong KK, Belliveau JW, Chesler DA, et al. Dynamic magnetic resonance imaging of human brain activity during primary sensory stimulation. Proc Nat Acad Sci USA, 1992, 89:5675-5679
[6]钱英.医学超声设备的新进展.医疗设备信息, 2002, (11) :25-27
[7] D. M. Otten, Cryosurgical monitoring using visible light imaging and electrical impedance tomography—A feasibility study, Ph.D dissertation, University of California, Berkeley, 2000
[8] A. M. Dijkstra, B. H. Brown, A. D. Leathard, et al, Review Clinical Application of Electrical Impedance Tomography, Journal of Medical Engineering & Technology, 1993, 17(3): 89~98
[9] Todd Eric Kerner, Electrical Impedance Tomography for Breast Imaging, Ph.D dissertation, Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire, 2001,7
[10]任超世, EIT—一种诱人的医学成像技术,电子科技导报, 1996, 5: 9~12
[11] H Griffiths, A phantom for electrical impedance tomography, Clin. Phys. Physiol. Meas., 1988, 9, Suppl. A, 15~20
[12] Xie C G et al, 8-electrode Capacitance System for Two-component Flow Identification. Part1: Tomography Flow Imaging, IEE Proc.A, 136, 1989, 184~190
[13] Xie C G et al, Electrical Capacitance Tomography for Imaging System Model for Development of Reconstruction Algorithms and Design of Primary Sensors, IEE Proc. G, 139, 1992, 89~98
[14] M.Tang,W.Wang,I.Wheeler,et al. Effects of incompatible boundary information in EIT on the convergence behavior of an iterative algorithm[J].IEEE Trans. Med. Imag, 2002, 21: 620-628
[15]汤孟兴,董秀珍,一个临床应用的实时电阻抗断层成像系统的设计与初步结果,国外医学生物医学工程分册,1995,8(5): 288~294
[16]贺兴柏,周守昌,阻抗CT图像重建算法的研究,重庆大学学报,1992,15(5): 38~43
[17]程吉宽,孙进平,杜岩等,电阻抗成像技术的原理及其发展,北京航空航天大学学报, 1998, 24(2): 137~140
[18]毕德显,电磁场理论,北京:电子工业出版社,1985,65~232
[19]冯慈璋,马西奎,工程电磁场导论,北京:高等教育出版社,2000,1~88
[20]王超,医学电阻抗断层成像的研究,[博士学位论文],天津大学,2001
[21]马艺馨,电阻抗成像技术及其在气/液两相泡状流检测中的应用,[博士学位论文],天津大学, 1999
[22] Vauhkonen M, Electrical impedance tomography and prior information, PhD thesis, Department of Physics, University of Kuopio, Finland. 1997
[23]何永勃,电阻抗(ECT/ERT)双模态层析成像技术研究, [博士学位论文],天津大学,2006
[24]问雪宁,基于ANSYS的ECT/ERT电磁场仿真及阵列电极优化设计,[硕士学位论文],天津大学,2006
[25]盛剑宏,电磁场数值分析,北京:科学出版社, 1984
[26]李开泰,有限元方法及应用,西安:西安交通大学出版社, 1992
[27] M. V. K.查利,电磁场问题的有限元解法,北京:科学技术出版社, 1985
[28] Vauhkonen M, Lionheart W R B, Heikkinen L M, Vauhkonen P J and Kaipio L P (2001), A MATLAB package for the EIDORS projects to reconstruct two-dimensional EIT images, Physiol . Meas., 22:107-111
[29]王超,王化祥,医学电阻抗成像系统电极结构优化设计,第四军医大学学报, 2001, 22(1): 78~80
[30] Y Li and W Q Yang. Virtual electrical capacitance tomography sensor. Journal of Physics: Conference Series 15 (2005)183–188
[31]彭黎辉,陆耿,杨五强,电容成像图象重建算法原理及评价,清华大学学报,2004,41(4):478-484
[32] R. Gadd, P. Record, P. Rolfe, A sensitivity region reconstruction algorithm using adjacent drive current injection strategy, Clin. Phys. Physiol. Meas., 1992, 13, Supp1. A, 101~105
[33] Bjorck A (1988), A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations, BIT, 28:659-670
[34] Borsic A (2002), Regularisation methods for imaging from electrical measurements, PhD thesis, School of Engineering, Oxford Brookes University, UK
[35] Hanke M and Hansen P C (1993), Regularization methods for large-scale problems, Surveys on Mathematics for Industry, 3:253-315
[36] Hansen P C (2005), Discrete inverse problems, xxxx, Informatics and Mathematical Modelling, Technical University of Denmark, Denmark
[37] Holder D S (2004), electrical impedance tomography, Institute of Physics Publishing, Bristol and Philadelphia
[39] Jacobsen M (2004), Modular regularization algorithms, PhD thesis, Informatics and Mathematical Modelling, Technical University of Denmark, Denmark
[40] O’Leary D P and Simmons J A (1981), A bidiagonalization-regularization procedure for large scale discretizations of ill-posed problems, SIAM J. Sci. Stat. Comput., 2:474-488
[41] Vauhkonen M, Vadasz D, Kaipio J P, Somersalo E and Karjalainen P A, (1998), Tikhonov regularization and priori information in electrical impedance tomography, IEEE Trans. Medical Imaging, 17:285-293
[42] Webster J G (1990), Electrical impedance tomography, Adam Higler
[43] M. Molinari , B.H. Blott , S.J. Cox , et al. Optimal imaging with adaptive mesh refinement in electrical impedance tomography. Physiol .Meas. 2002 . 23:121-128
[44] M Monlinari , S. J. Cox , B.H. Blott, et al. Adaptive mesh refinement techniques for electrical impedance tomography . Physiol. Meas. , 2001. 22:91-96
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