微波断层成象重建算法研究
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摘要
微波断层成象是一种无损探测技术，它采用微波照射被测物体，利用置于
被测物体外部的检测器得到的散射数据，重建被测物体内部的复介电常数图象。
复介电常数包含了丰富的信息，其实部即为介电率，虚部与导电率成比例；并
且介电常数与很多生理参数有关，例如温度、含水量、血量、血氧浓度等。因
此，微波断层成象不仅能够得到人体内部的形态结构信息，还能得到相应的生
理信息，是一种非常有潜力的诊断成象技术。它既可以独立应用于诊断，也可
以作为其他成象系统的一种补充。
微波断层成象隶属于电磁逆散射问题，这里的散射是一个广义的概念，包
含了透射、反射、折射、衍射和散射等等。这样的问题具有非线性和非适应性，
一般难于求解。利用Born或Rytov近似，可以将问题线性化，然后应用与X-CT
类似的算法利用傅立叶变换对被测物体成象。但是此方法只适用于弱散射体，
而在生物医学领域中，大多数介质或组织介电常数的对比度较高，这种方法就
无能为力了。采用空域重建算法，可以将此问题视为优化问题迭代求解，并且
通过引入某些先验知识，如物体的外部形状、介电常数的上下限等，改善问题
的非适应性，因而可以应用于生物医学领域，进一步发展了微波断层成象技术。
在简要介绍微波断层成象的原理和基本重建算法之后，本文提出了几种不
同的方法应用于微波断层成象。扩展局部搜索重建算法不仅考虑了散射数据的
误差，同时还考虑了系数矩阵的计算误差，求解问题的总体最小二乘解，并利
用Tikhonov正则化改善问题的非适应性；神经网络重建算法引入了Markov随
机场模型，并且由于保边界正则化项中存在二进制变量，因而求解的混合变量
问题。耦合Hopfield神经网络和增广Hopfield神经网络都由两个子网络组成，
分别处理连续变量和二进制变量，并且这两个子网络之间也存在相互作用。这
两种神经网络的区别在于处理二进制变量的子网络，前者将二进制变量扩展为
0、1之间的连续变量，在计算之后再将其惩罚或强制为二进制变量；而后者可
以直接处理二进制变量，不需要扩展或强制变量；信赖域重建算法是将微波断
层成象问题视为约束最小二乘问题，其约束条件来自于先验知识。在迭代时，
根据一阶必要条件，可以将此问题转化为互补问题求解。实际上，互补问题所
求解的是两个拉格朗日乘子，由他们可以得到下次迭代所需要新的介电常数值。
这几种重建算法都可以方便地引入先验知识，并且应用Tikhonov或者其他
正则化过程来改善逆散射问题的非适应性。并且扩展局部搜索重建算法和神经
网络重建算法都可以搜索全局最优解，从而得到较好的重建结果。然而，这些
算法重建过程都较长，这是由迭代算法本身和散射问题的复杂性引起的，这也
是我们下一步研究的重点。
Microwave tomography is a promising nondestructive evaluation method, which exploit microwave as incidence to irradiate the object and reconstruct the internal complex permittivity image of the object by applying the scattered data from receivers settled outside of the object. Complex permittivity includes abundant information, since its real part is the dielectric constant, and the imaginary part is proportional to conductivity. Moreover, permittivity has close relationship with some physiological parameters, such as temperature, water content, blood content, blood oxygenation, and so on. Therefore microwave tomography can produce the internal information of not only morphology but also physiology. Thus it will be a potential imaging technique for medical diagnosis, as either an independent system competitive with other sophisticated imaging methods or merely a complementary method.
Microwave tomography belongs to electromagnetic inverse scattering problems, where the word "scattering" is a generalized concept, which includes transmission, reflection, refraction, diffraction, and scattering. This kind of problems is quite difficult to develop to its full potential because of its nonlinearity and ill-posedness. Fortunately, the problem of a weakly scattering object can be linearized by Born or Rytov approximation and thus resolved in Fourier domain with methods similar to those in X-CT. However, these methods would not be valid in biomedical field, in which most tissues or media have high contrast of permittivity. A possible solution is to apply spatial methods, which discretize the integrate equations into matrix equations and solve them iteratively in the framework of optimization problem. The conspicuous advantage of these methods is the capability to improve the inherent ill-posedness of the inverse scattering problem by introducing some a priori knowledge of the object, such as the external structure, the lower and upper limit of permittivity, etc. It is these spatial methods that encourage the development of microwave tomography and make a lot of achievements.
After the brief explanation of the principle of microwave tomography and introduction of some basic reconstruction algorithms several spatial methods are proposed, including the extended local search reconstruction (ELSR), the neural network reconstruction (NNR), and the trust region reconstruction (TRR). ELSR focuses on not only the errors of scattered data, but also the errors of coefficient matrix coming from computing. Hence, ELSR is to search the global optimum solution of the total least squres problem and apply Tikhonov regularization to improve the ill-posedness of the problem. NNR is to resolve a mixed-variable problem, since a Morkov random field model is introduced and there are binary line processes, together with the continuous permittivity, in the edge-preserving regularization. Here, NNR includes two neural networks, coupled Hopfield network (CHN) and augmented Hopfield network (AHN), both of which are composed with two sub-networks to deal with continuous and binary variables, respectively. In
addition, these two sub-networks interact with each other. The sub-networks to continuous variables of CHN and AHN are based on the continuous Hopfield network and have the same structure and neurons. The difference between CHN and AHN exists in the other sub-network of handling the binary variables. For CHN, the subnetwork is also a continuous Hopfield network, for the line processes have been expanded into continuous one with the range from 0 to 1. A specific penalty term is incorporated into the energy function to penalize the continuous line processes back to binary variables. Situation is quite different in AHN. The sub-network of binary variable is composed of binary neurons with binary input and output, and thus can deal with binary variable directly. TRR resolve the problem with constrained least squares criterion, where the constrained condition comes from a priori knowledge. In each iteration the problem can be transformed as
被测物体外部的检测器得到的散射数据，重建被测物体内部的复介电常数图象。
复介电常数包含了丰富的信息，其实部即为介电率，虚部与导电率成比例；并
且介电常数与很多生理参数有关，例如温度、含水量、血量、血氧浓度等。因
此，微波断层成象不仅能够得到人体内部的形态结构信息，还能得到相应的生
理信息，是一种非常有潜力的诊断成象技术。它既可以独立应用于诊断，也可
以作为其他成象系统的一种补充。
微波断层成象隶属于电磁逆散射问题，这里的散射是一个广义的概念，包
含了透射、反射、折射、衍射和散射等等。这样的问题具有非线性和非适应性，
一般难于求解。利用Born或Rytov近似，可以将问题线性化，然后应用与X-CT
类似的算法利用傅立叶变换对被测物体成象。但是此方法只适用于弱散射体，
而在生物医学领域中，大多数介质或组织介电常数的对比度较高，这种方法就
无能为力了。采用空域重建算法，可以将此问题视为优化问题迭代求解，并且
通过引入某些先验知识，如物体的外部形状、介电常数的上下限等，改善问题
的非适应性，因而可以应用于生物医学领域，进一步发展了微波断层成象技术。
在简要介绍微波断层成象的原理和基本重建算法之后，本文提出了几种不
同的方法应用于微波断层成象。扩展局部搜索重建算法不仅考虑了散射数据的
误差，同时还考虑了系数矩阵的计算误差，求解问题的总体最小二乘解，并利
用Tikhonov正则化改善问题的非适应性；神经网络重建算法引入了Markov随
机场模型，并且由于保边界正则化项中存在二进制变量，因而求解的混合变量
问题。耦合Hopfield神经网络和增广Hopfield神经网络都由两个子网络组成，
分别处理连续变量和二进制变量，并且这两个子网络之间也存在相互作用。这
两种神经网络的区别在于处理二进制变量的子网络，前者将二进制变量扩展为
0、1之间的连续变量，在计算之后再将其惩罚或强制为二进制变量；而后者可
以直接处理二进制变量，不需要扩展或强制变量；信赖域重建算法是将微波断
层成象问题视为约束最小二乘问题，其约束条件来自于先验知识。在迭代时，
根据一阶必要条件，可以将此问题转化为互补问题求解。实际上，互补问题所
求解的是两个拉格朗日乘子，由他们可以得到下次迭代所需要新的介电常数值。
这几种重建算法都可以方便地引入先验知识，并且应用Tikhonov或者其他
正则化过程来改善逆散射问题的非适应性。并且扩展局部搜索重建算法和神经
网络重建算法都可以搜索全局最优解，从而得到较好的重建结果。然而，这些
算法重建过程都较长，这是由迭代算法本身和散射问题的复杂性引起的，这也
是我们下一步研究的重点。
Microwave tomography is a promising nondestructive evaluation method, which exploit microwave as incidence to irradiate the object and reconstruct the internal complex permittivity image of the object by applying the scattered data from receivers settled outside of the object. Complex permittivity includes abundant information, since its real part is the dielectric constant, and the imaginary part is proportional to conductivity. Moreover, permittivity has close relationship with some physiological parameters, such as temperature, water content, blood content, blood oxygenation, and so on. Therefore microwave tomography can produce the internal information of not only morphology but also physiology. Thus it will be a potential imaging technique for medical diagnosis, as either an independent system competitive with other sophisticated imaging methods or merely a complementary method.
Microwave tomography belongs to electromagnetic inverse scattering problems, where the word "scattering" is a generalized concept, which includes transmission, reflection, refraction, diffraction, and scattering. This kind of problems is quite difficult to develop to its full potential because of its nonlinearity and ill-posedness. Fortunately, the problem of a weakly scattering object can be linearized by Born or Rytov approximation and thus resolved in Fourier domain with methods similar to those in X-CT. However, these methods would not be valid in biomedical field, in which most tissues or media have high contrast of permittivity. A possible solution is to apply spatial methods, which discretize the integrate equations into matrix equations and solve them iteratively in the framework of optimization problem. The conspicuous advantage of these methods is the capability to improve the inherent ill-posedness of the inverse scattering problem by introducing some a priori knowledge of the object, such as the external structure, the lower and upper limit of permittivity, etc. It is these spatial methods that encourage the development of microwave tomography and make a lot of achievements.
After the brief explanation of the principle of microwave tomography and introduction of some basic reconstruction algorithms several spatial methods are proposed, including the extended local search reconstruction (ELSR), the neural network reconstruction (NNR), and the trust region reconstruction (TRR). ELSR focuses on not only the errors of scattered data, but also the errors of coefficient matrix coming from computing. Hence, ELSR is to search the global optimum solution of the total least squres problem and apply Tikhonov regularization to improve the ill-posedness of the problem. NNR is to resolve a mixed-variable problem, since a Morkov random field model is introduced and there are binary line processes, together with the continuous permittivity, in the edge-preserving regularization. Here, NNR includes two neural networks, coupled Hopfield network (CHN) and augmented Hopfield network (AHN), both of which are composed with two sub-networks to deal with continuous and binary variables, respectively. In
addition, these two sub-networks interact with each other. The sub-networks to continuous variables of CHN and AHN are based on the continuous Hopfield network and have the same structure and neurons. The difference between CHN and AHN exists in the other sub-network of handling the binary variables. For CHN, the subnetwork is also a continuous Hopfield network, for the line processes have been expanded into continuous one with the range from 0 to 1. A specific penalty term is incorporated into the energy function to penalize the continuous line processes back to binary variables. Situation is quite different in AHN. The sub-network of binary variable is composed of binary neurons with binary input and output, and thus can deal with binary variable directly. TRR resolve the problem with constrained least squares criterion, where the constrained condition comes from a priori knowledge. In each iteration the problem can be transformed as
引文
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