基于全变差正则化的PD-IPM算法与对向驱动的等位线反投影算法的仿真对比研究
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摘要
电阻抗断层成像(Electrical Impedance Tomography, EIT)是一种兴起于20世纪80年代的成像技术,在医学领域具有巨大的应用潜力。相对于传统医学影像手段,EIT操作简便、对人无创无害、成本低廉,能够进行长时间的实时动态床旁监护,有望在未来成为一种与传统医学影像手段互补的辅助临床监护方式。等位线反投影算法以其快速的成像速度和一定的抗噪能力而成为应用于临床监护的动态EIT算法的代表,但是其本身的缺陷使得重建图像存在较大伪影,对扰动目标的边界保留情况较差,不能很好地满足医学图像对于空间分辨率和锐度的要求。
因此,为了解决上述问题,本文讨论了一种具有良好边界保留特性的算法——基于全变差(Total Variation, TV)正则化的PD-IPM(Primal Dual-Interior Point Method)算法。目前临床监护所采用算法的效果优于等位线反投影算法,但是需要调节重构参数,而等位线反投影算法无需如此。为了更方便的对算法进行评估,本文采用最为简单的等位线反投影算法作为对比,首先实现了对向驱动的等位线反投影算法,对其边界保留的情况做了初步评估;进而根据不足,开展了具有良好边界保留性质的PD-IPM算法的研究;最后通过仿真实验,以定量的评价指标作为标准,对两种算法的边界保留性质进行了全面的评估。全文大体可以分为三个部分:
(1) EIT基本问题的阐述
首先介绍了EIT的生理基础、基本原理和过程,随后介绍了EIT的关键组成部分——图像重构算法,最后从数学的角度对EIT的数学模型以及正问题和逆问题进行了阐述。
(2)等位线反投影算法的回顾和PD-IPM算法的介绍
对于经典的等位线反投影(Back Projection, BP)算法,从数学物理基础出发,介绍了反投影过程;随后介绍了改进驱动方式为对向驱动后的BP算法并给出了关键部分——反投影矩阵的计算方法,最后给出了计算公式并对其边界保留情况进行了初步的评估。对于基于TV正则化的PD-IPM算法,分别阐述了TV正则化的边界保留特性和PD-IPM算法的推导过程,最后给出了基于该算法的动态EIT的迭代公式并讨论了迭代次数和算法参数。
(3)仿真对比研究
以MATLAB为仿真平台对两种算法进行了对比研究。为了定量评价重建图像的质量,引入了重建质量函数D、图像结构偏离度函数SSIM和边界坡度函数G三个指标。首先在无噪声条件下对单扰动目标和双扰动目标两种情况分别进行了仿真对比研究。仿真结果表明,PD-IPM算法对正则化参数的选择敏感。无论是单扰动目标(6个成像位置)还是双扰动目标(4个成像位置),PD-IPM算法的重建结果均更加清晰,对于扰动目标的边界保留效果明显更优,而评价指标的比较也支持了这一结论。其次以单扰动目标(1个成像位置)为例,对比研究了两种算法的抗噪性能。结果表明,PD-IPM算法具有一定的抗噪能力,且在相同噪声水平下其重建结果仍然优于BP算法,但是对于噪声更加敏感。
通过仿真对比可以得出结论,PD-IPM算法具有良好的边界保留性质,同时具备一定的抗噪能力,是一种较优的算法。有必要进行进一步的物理模型和人体实验,从而为将该算法最终应用于临床监护打下基础。
Electrical Impedance Tomography (EIT) is an imaging technique which started in the 1980s and has great practical potential in the medical field. Compared to the conventional imaging approaches, EIT is convenient, harmless, non-invasive and has low cost. It is able to perform real-time dynamic bedside monitoring for a long period of time. Thus EIT is likely to become a clinical monitoring method that is complementary to the traditional medical imaging approaches. The Back Projection Algorithm (BP) is a dynamic EIT algorithm applied to clinical monitoring due to its fast imaging rate and fine noise immunity. However, the inherent defectiveness of the algorithm leads to the fact that the reconstructed image has relatively large artifacts and poor edge preservation of the target, which cannot perfectly meet the requirements of medical image in terms of spatial resolution and sharpness.
In order to solve the problem mentioned above, an algorithm with fine ability of edge preservation is discussed in this thesis----the PD-IPM Algorithm (PD-IPM) based on Total Variation Regularization. Currently the algorithm applied in clinical monitoring has better performance than BP but has the inconvenience of adjusting reconstruction parameters, whereas there is no need for BP to do so. For more convenient evaluation of algorithms, this thesis adopts the simplest BP as the contrast. In the first place, BP with polar driven pattern is realized and preliminary evaluation of its edge preservation is carried out. Second, according to the shortages the study of PD-IPM which has fine edge preservation is conducted. Finally, comprehensive evaluation of edge preservation of the two algorithms with quantitative indexes as the standards is made through simulation. The thesis can be generally divided into three parts:
(1) Elaboration of basic EIT problems
Firstly the physiological basis, the fundamental principle and process of EIT are introduced. Subsequently the key components of EIT----image reconstruction algorithms are introduced. Finally the mathematic model of EIT as well as the forward problem and the inverse problem is described from the mathematic point of view.
(2) Retrospection of BP and introduction of PD-IPM
The retrospection of classical BP starts from its mathematical and physical basis and the back projection process is formulated. Subsequently the improved BP with polar driven pattern is introduced and the key part of the algorithm----the calculation of back projection matrix is given. At last the formula is presented and preliminary evaluation of its edge preservation is made. For the other algorithm, the edge preservation characteristic of total variation regularization and the derivation of PD-IPM algorithm are formulated. Finally the iterative formula of dynamic EIT based on the algorithm is given and iterations as well as the parameters are discussed.
(3) Comparative study of simulation
Comparative study is carried out between the two algorithms on the platform of MATLAB. For the purpose of evaluating the quality of the reconstruction quantificationally, three indexes are introduced, which are: reconstruction quality function D , image structure deviation function SSIM and edge gradient function G . First, one-target and two-target reconstructions without noise are simulated and compared. The results indicate that PD-IPM is sensitive to the selection of regularization parameter. In the cases of both one-target reconstruction (6 imaging positions) and two-target reconstruction (4 imaging positions), the reconstructions of PD-IPM are more clear and have evidently better edge preservation of the target, which are also supported by the indexes. Second, the anti-noise performance of the two algorithms is compared in the case of one-target reconstruction (1 imaging position). The results show that PD-IPM has moderate noise immunity and has better reconstructions than BP with same noise levels. However PD-IPM is more sensitive to noise.
Conclusions are made through the comparative study of simulation that PD-IPM has fine edge preservation property and moderate noise immunity, thus it is a relatively better algorithm. It is necessary to carry out further experiments on physical models and humans so that the foundation of future clinical application of the algorithm is laid.
因此,为了解决上述问题,本文讨论了一种具有良好边界保留特性的算法——基于全变差(Total Variation, TV)正则化的PD-IPM(Primal Dual-Interior Point Method)算法。目前临床监护所采用算法的效果优于等位线反投影算法,但是需要调节重构参数,而等位线反投影算法无需如此。为了更方便的对算法进行评估,本文采用最为简单的等位线反投影算法作为对比,首先实现了对向驱动的等位线反投影算法,对其边界保留的情况做了初步评估;进而根据不足,开展了具有良好边界保留性质的PD-IPM算法的研究;最后通过仿真实验,以定量的评价指标作为标准,对两种算法的边界保留性质进行了全面的评估。全文大体可以分为三个部分:
(1) EIT基本问题的阐述
首先介绍了EIT的生理基础、基本原理和过程,随后介绍了EIT的关键组成部分——图像重构算法,最后从数学的角度对EIT的数学模型以及正问题和逆问题进行了阐述。
(2)等位线反投影算法的回顾和PD-IPM算法的介绍
对于经典的等位线反投影(Back Projection, BP)算法,从数学物理基础出发,介绍了反投影过程;随后介绍了改进驱动方式为对向驱动后的BP算法并给出了关键部分——反投影矩阵的计算方法,最后给出了计算公式并对其边界保留情况进行了初步的评估。对于基于TV正则化的PD-IPM算法,分别阐述了TV正则化的边界保留特性和PD-IPM算法的推导过程,最后给出了基于该算法的动态EIT的迭代公式并讨论了迭代次数和算法参数。
(3)仿真对比研究
以MATLAB为仿真平台对两种算法进行了对比研究。为了定量评价重建图像的质量,引入了重建质量函数D、图像结构偏离度函数SSIM和边界坡度函数G三个指标。首先在无噪声条件下对单扰动目标和双扰动目标两种情况分别进行了仿真对比研究。仿真结果表明,PD-IPM算法对正则化参数的选择敏感。无论是单扰动目标(6个成像位置)还是双扰动目标(4个成像位置),PD-IPM算法的重建结果均更加清晰,对于扰动目标的边界保留效果明显更优,而评价指标的比较也支持了这一结论。其次以单扰动目标(1个成像位置)为例,对比研究了两种算法的抗噪性能。结果表明,PD-IPM算法具有一定的抗噪能力,且在相同噪声水平下其重建结果仍然优于BP算法,但是对于噪声更加敏感。
通过仿真对比可以得出结论,PD-IPM算法具有良好的边界保留性质,同时具备一定的抗噪能力,是一种较优的算法。有必要进行进一步的物理模型和人体实验,从而为将该算法最终应用于临床监护打下基础。
Electrical Impedance Tomography (EIT) is an imaging technique which started in the 1980s and has great practical potential in the medical field. Compared to the conventional imaging approaches, EIT is convenient, harmless, non-invasive and has low cost. It is able to perform real-time dynamic bedside monitoring for a long period of time. Thus EIT is likely to become a clinical monitoring method that is complementary to the traditional medical imaging approaches. The Back Projection Algorithm (BP) is a dynamic EIT algorithm applied to clinical monitoring due to its fast imaging rate and fine noise immunity. However, the inherent defectiveness of the algorithm leads to the fact that the reconstructed image has relatively large artifacts and poor edge preservation of the target, which cannot perfectly meet the requirements of medical image in terms of spatial resolution and sharpness.
In order to solve the problem mentioned above, an algorithm with fine ability of edge preservation is discussed in this thesis----the PD-IPM Algorithm (PD-IPM) based on Total Variation Regularization. Currently the algorithm applied in clinical monitoring has better performance than BP but has the inconvenience of adjusting reconstruction parameters, whereas there is no need for BP to do so. For more convenient evaluation of algorithms, this thesis adopts the simplest BP as the contrast. In the first place, BP with polar driven pattern is realized and preliminary evaluation of its edge preservation is carried out. Second, according to the shortages the study of PD-IPM which has fine edge preservation is conducted. Finally, comprehensive evaluation of edge preservation of the two algorithms with quantitative indexes as the standards is made through simulation. The thesis can be generally divided into three parts:
(1) Elaboration of basic EIT problems
Firstly the physiological basis, the fundamental principle and process of EIT are introduced. Subsequently the key components of EIT----image reconstruction algorithms are introduced. Finally the mathematic model of EIT as well as the forward problem and the inverse problem is described from the mathematic point of view.
(2) Retrospection of BP and introduction of PD-IPM
The retrospection of classical BP starts from its mathematical and physical basis and the back projection process is formulated. Subsequently the improved BP with polar driven pattern is introduced and the key part of the algorithm----the calculation of back projection matrix is given. At last the formula is presented and preliminary evaluation of its edge preservation is made. For the other algorithm, the edge preservation characteristic of total variation regularization and the derivation of PD-IPM algorithm are formulated. Finally the iterative formula of dynamic EIT based on the algorithm is given and iterations as well as the parameters are discussed.
(3) Comparative study of simulation
Comparative study is carried out between the two algorithms on the platform of MATLAB. For the purpose of evaluating the quality of the reconstruction quantificationally, three indexes are introduced, which are: reconstruction quality function D , image structure deviation function SSIM and edge gradient function G . First, one-target and two-target reconstructions without noise are simulated and compared. The results indicate that PD-IPM is sensitive to the selection of regularization parameter. In the cases of both one-target reconstruction (6 imaging positions) and two-target reconstruction (4 imaging positions), the reconstructions of PD-IPM are more clear and have evidently better edge preservation of the target, which are also supported by the indexes. Second, the anti-noise performance of the two algorithms is compared in the case of one-target reconstruction (1 imaging position). The results show that PD-IPM has moderate noise immunity and has better reconstructions than BP with same noise levels. However PD-IPM is more sensitive to noise.
Conclusions are made through the comparative study of simulation that PD-IPM has fine edge preservation property and moderate noise immunity, thus it is a relatively better algorithm. It is necessary to carry out further experiments on physical models and humans so that the foundation of future clinical application of the algorithm is laid.
引文
[1]董秀珍.生物电阻抗成像研究的现状与挑战.中国生物医学工程年报, 2008,第27卷第5期: 641-643
[2] K.S. Cole. Electric impedance of suspensions of arbacia eggs. J Gen Physiol, 1928, 12: 37-54.
[3] K.S. Cole, R.S. Guttman. Electric impedance of frog egg. J Gen Physiol, 1942, 25: 765-775.
[4] K. Boone, A.M. Lewis, D.S. Holder. Imaging of spreading depression by EIT: Implications for localization of Epileptic Foci. Physiological Measurement, 1994, 15(Suppl2A): A189-A198
[5] D.S. Holder. Electrical impedance tomography of global cerebral ischaemia with cortical or scalp electrodes in the anaesthetized rat. Clinical Physics and Physiological Measurement, 1992, 13: 87-98
[6] D.S. Holder, Rao A, Hanquan Y. Imaging of physiologically evoked responses by electrical impedance tomography with electrodes in the anaesthetized rabbit. Physiological Measurement, 1996, 17 (Suppl4A): A179-A186.
[7] A. Rao. Electrical Impedance Tomography of brain activity: studies into its accuracy and physiological mechanisms. Ph.D. Thesis. University College London, London, 2000
[8] W.Q. Yang, D.M. Spink, T.A. York, et al. An image reconstruction algorithm based on Landweber iteration method for electrical capacitance tomography. Meas. Sci. Technol., 1999, 10:1065-1069
[9] A. Adler. Electrical impedance tomography: regularized imaging and contrast detection. IEEE Transactions on Medical Imaging, 1996, 15: 170-179
[10] D.C. Barber, B.H. Brown. Imaging spatial distributions of resistivity usingapplied potential tomography. Electronic Letters, 1983, 19: 933-935.
[11]刘国强.医学电磁成像.科学出版社, 2006
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[22] J.P. Morucci, P.M. Marsili. Bioelectrical impedance techniques in medicine. PartⅢ:Impedance imaging. Second section: reconstruction algorithms. Crit Rev Biomed Eng 1996, 24(4-6): 599-654
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[2] K.S. Cole. Electric impedance of suspensions of arbacia eggs. J Gen Physiol, 1928, 12: 37-54.
[3] K.S. Cole, R.S. Guttman. Electric impedance of frog egg. J Gen Physiol, 1942, 25: 765-775.
[4] K. Boone, A.M. Lewis, D.S. Holder. Imaging of spreading depression by EIT: Implications for localization of Epileptic Foci. Physiological Measurement, 1994, 15(Suppl2A): A189-A198
[5] D.S. Holder. Electrical impedance tomography of global cerebral ischaemia with cortical or scalp electrodes in the anaesthetized rat. Clinical Physics and Physiological Measurement, 1992, 13: 87-98
[6] D.S. Holder, Rao A, Hanquan Y. Imaging of physiologically evoked responses by electrical impedance tomography with electrodes in the anaesthetized rabbit. Physiological Measurement, 1996, 17 (Suppl4A): A179-A186.
[7] A. Rao. Electrical Impedance Tomography of brain activity: studies into its accuracy and physiological mechanisms. Ph.D. Thesis. University College London, London, 2000
[8] W.Q. Yang, D.M. Spink, T.A. York, et al. An image reconstruction algorithm based on Landweber iteration method for electrical capacitance tomography. Meas. Sci. Technol., 1999, 10:1065-1069
[9] A. Adler. Electrical impedance tomography: regularized imaging and contrast detection. IEEE Transactions on Medical Imaging, 1996, 15: 170-179
[10] D.C. Barber, B.H. Brown. Imaging spatial distributions of resistivity usingapplied potential tomography. Electronic Letters, 1983, 19: 933-935.
[11]刘国强.医学电磁成像.科学出版社, 2006
[12]付峰,蒋大宗.电阻抗断层成像重构算法.国外医学:生物医学工程分册, 1995, 18(5): 255-262.
[13]侯文生.电阻抗断层成像中的图像重建技术.生物医学工程学杂志, 2000, 17(2): 214-217
[14]金建铭,王建国,葛德彪.电磁场有限元方法.西安电子科技大学出版社, 2001
[15] N.O. Matthew. Numerical Techniques in Electromagnetics, Second Edition. CRC Press, 2000
[16]付峰,董秀珍,刘锐岗等.实时电阻抗成像系统中的成像算法及软件系统实现.医疗设备信息, 2003, 18: 2-4
[17]帅万钧.腹部内出血电阻抗图像监护关键技术及其实验研究.博士学位论文,第四军医大学,西安, 2008
[18]何为,罗辞勇,徐征,等.电阻抗成像原理.科学出版社.2009
[19] B.H. Brown, D.C.Barber.‘Applied Potential Tomography’--- a new in-vivo medical imaging technique. Proc. of the HPA Ann. Conf., Sheffield, 1982
[20] D.C. Barber, A.D. Seagar. Fast reconstruction of resistance images. Clin Phys Physiol Meas, 1987, 8(Suppl A): 47-54
[21]汤孟兴,董秀珍,秦明新,等.动态电阻抗断层成像反投影算法的仿真研究.第四军医大学学报, 1997, 1 (18): 42-44
[22] J.P. Morucci, P.M. Marsili. Bioelectrical impedance techniques in medicine. PartⅢ:Impedance imaging. Second section: reconstruction algorithms. Crit Rev Biomed Eng 1996, 24(4-6): 599-654
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