脑磁逆问题中的磁源重建算法研究
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摘要
脑磁图(MEG，magnetoencephalogram)通过非侵入性地测量微弱的脑磁场信号来研究人脑功能。其核心为所谓的MEG逆问题，即从头外测得的磁场数据，重建出产生该磁场的脑内神经活动的电流源分布。本论文是关于脑磁逆问题中的磁源重建算法研究，主要工作在于：
首先根据Maxwell方程组和Biot-Savat定律，分析了脑磁研究的相关生物电磁学的基本理论，使这些已成熟的理论能够完整和统一，便于开展以后的工作。
由于本文的工作是脑磁逆问题的研究，而考虑到目前尚没有较完整全面的脑磁逆问题求解方法的文献综述，因此本文对现有求逆方法进行了整理和总结，主要从基于点源模型的参数型偶极子定位和基于分布源模型的非参数型磁源成象这两大类求逆方法来进行详细的阐述和归纳，并对今后的发展趋势作了预测。
接下来，我们开展了逆问题的求解工作。首先从数学物理中关于不定问题的定义来解释脑磁逆问题的不适定性，脑磁文献中没有这方面的详细讨论。
在参数型的偶极子定位方法中，具体有非线性最小二乘法、优化算法以及阵列信号处理方法。本文提出了一种综合的优化方法，它不同于普遍采用的单一的优化方法，而是根据测量数据和计算数据的残差，来决定采用的非线性优化方法，这里是把Levenberg-Marquardt(LM)法和Quasi-Newton(QN)法结合起来考虑。此方法能提高迭代的速度，确保其收敛性，最主要的是降低了对初值的要求。
我们后期的工作重点主要是在非参数型的磁源分布图象重建上，它构架在图象重建基础上，从测量数据中重建出神经活动的电流源分布图象。它的不适定性决定了引入正则理论的必要性。实质就是在求解过程中将观测数据和先验知识综合在一起考虑，可解释为对观察数据的置信度和对先验知识的置信度之间的某种折衷。综观各种提出的脑磁源成象方法，我们均把它们纳入正则理论的框架内，划分为确定性和随机性正则，从这两方面来诠释磁源重建问题。从目前现有的文献资料来看，并没有一个磁源成象正则理论的完整解释，因此我们从正则化理论上把各种方法建立一个统一的磁源成象理论基础，这也是本文工作的重点。
由于脑磁逆问题的研究在国内才刚起步，开展的一些工作也仅是停留在点源模型的偶极子定位上。因此在这种背景下，本文开展了分布源模型下的磁源分布图象重建工作，分两大部分内容进行。首先，在基于Tikhonov正则的最小模估计算法中提出了一个区域加权因子，重建结果表明该方法能在一定程度上改善最小模估计等方法的图象平滑性，使结果能反映出神经生理学意义下的局部聚集总体稀疏的解分布。其次，在基于Markov随机场和Bayesian理论的磁
浙江大学博士学位论义
源成象中，构架了Bayesian重建框架后，把先验的神经电流活动分布图象视为
一Markov随机场模型，提出了一种把重建图象数据扩展为图象矢量和图象边
界的集合方法，这样可使重建问题转变为一个即有连续变量又有二进量的混合
优化问题，于是采用一种耦合梯度神经网络方法来求代价函数的最优解。仿真
实验表明算法不仅能反映重建图象的局部平滑性又能刻画局部之间的边界信
息。
文中最后对己经开展的以及下一步的后续工作作了总结和展望。
Magnetoencephalography(MEG), measurement of neuromagnetic fields by superconducting quantum interference devices (SQUIDs), is a useful noninvasive method for investigating human brain functions. The core of MEG research is its inverse problem obtaining the underlying neural activity from the magnetic field measured outside a human head. The scope of this paper is focus on the inverse problem solution. The author has done the following research work:
Firstly, the basic mathematical theory and electromagnetic concepts are thoroughly discussed, including general Maxwell equations and the quasistatic approximation on bioelectromagnetic studies. Meanwhile, MEG forward problem is analysis in general ways.
Secondly, an overview is given on the solution method for MEG inverse problem. During the development of inverse problem research, the source reconstruction methods are classified as dipole source localization and source imaging (or current density reconstruction). The former, applying the point source model, leads to a nonlinear inverse problem. And the latter, applying the distributed source model, can be cast a linear problem.
In dipole localization based on point source model, we proposed a synthetic algorithm, which vary the used optimization method during the source scanning procedure. We use quasi-Newton(QN) method for a high-speed coarse scan over a large area of the head, and Levenberg-Marquardt(LM) method for a fine scan near the source area. In comparison with those single optimization methods, this synthetic approach proved to be more efficient both in terms of computation time and sensitivity to the iterative initial value.
MEG source image reconstruction, an important and difficult problem in image processing applications, is formulated as ill-posed inverse problem, so regularization is necessary. Regularization can be classified into two common approaches, i.e., deterministic and stochastic. Their common feature is that they both make use of a priori constraints concerning the current density distribution. In this paper we review several aspects of the application of regularization theory in MEG source image reconstruction, where the minimum norm estimation with Tikhonov regularization and Bayesian framework based on MAP-MRF model are introduced in detail. The characteristics, difference of these methods are also discussed.
To our knowledge, there is no paper talking about the regularization for the MEG source image reconstruction from a unifying viewpoint. So we try to present a relative complete regularization viewpoint on this special imaging.
In deterministic regularization, i.e., adopting minimum-norm estimation with Tikhonov regularization, we proposed the concept of region weighing. In order to obtain unique and physiologically justified solution, an operator of region weighing is introduced, meanwhile incorporating the depth weighing in the reconstruction procedure. Computer experiments show the method presented here is promising. And limitations of the proposed method and future work are then discussed.
In stochastic regularization, we present a reconstruction method based on a Markov
prior image model in order to obtain a stable solution, in this image model, the neural current density distributed image is extended to include not only the image vector of the moment values but also include the quantities corresponding to the image edges that are binary line processed. Thus the reconstruction is defined as the maximum a posteriori estimate(MAP) based on Bayesian framework. To acquire a global minimum solution from the posterior energy function, we here incorporate the ideas of coupled gradient artificial neural networks into this mixed integer optimization task.
After the present study, some issues remain open. Next, we are going to achieve the improvements of our works. Finally, we discuss the major results of this paper and some comments on MEG future trends.
首先根据Maxwell方程组和Biot-Savat定律，分析了脑磁研究的相关生物电磁学的基本理论，使这些已成熟的理论能够完整和统一，便于开展以后的工作。
由于本文的工作是脑磁逆问题的研究，而考虑到目前尚没有较完整全面的脑磁逆问题求解方法的文献综述，因此本文对现有求逆方法进行了整理和总结，主要从基于点源模型的参数型偶极子定位和基于分布源模型的非参数型磁源成象这两大类求逆方法来进行详细的阐述和归纳，并对今后的发展趋势作了预测。
接下来，我们开展了逆问题的求解工作。首先从数学物理中关于不定问题的定义来解释脑磁逆问题的不适定性，脑磁文献中没有这方面的详细讨论。
在参数型的偶极子定位方法中，具体有非线性最小二乘法、优化算法以及阵列信号处理方法。本文提出了一种综合的优化方法，它不同于普遍采用的单一的优化方法，而是根据测量数据和计算数据的残差，来决定采用的非线性优化方法，这里是把Levenberg-Marquardt(LM)法和Quasi-Newton(QN)法结合起来考虑。此方法能提高迭代的速度，确保其收敛性，最主要的是降低了对初值的要求。
我们后期的工作重点主要是在非参数型的磁源分布图象重建上，它构架在图象重建基础上，从测量数据中重建出神经活动的电流源分布图象。它的不适定性决定了引入正则理论的必要性。实质就是在求解过程中将观测数据和先验知识综合在一起考虑，可解释为对观察数据的置信度和对先验知识的置信度之间的某种折衷。综观各种提出的脑磁源成象方法，我们均把它们纳入正则理论的框架内，划分为确定性和随机性正则，从这两方面来诠释磁源重建问题。从目前现有的文献资料来看，并没有一个磁源成象正则理论的完整解释，因此我们从正则化理论上把各种方法建立一个统一的磁源成象理论基础，这也是本文工作的重点。
由于脑磁逆问题的研究在国内才刚起步，开展的一些工作也仅是停留在点源模型的偶极子定位上。因此在这种背景下，本文开展了分布源模型下的磁源分布图象重建工作，分两大部分内容进行。首先，在基于Tikhonov正则的最小模估计算法中提出了一个区域加权因子，重建结果表明该方法能在一定程度上改善最小模估计等方法的图象平滑性，使结果能反映出神经生理学意义下的局部聚集总体稀疏的解分布。其次，在基于Markov随机场和Bayesian理论的磁
浙江大学博士学位论义
源成象中，构架了Bayesian重建框架后，把先验的神经电流活动分布图象视为
一Markov随机场模型，提出了一种把重建图象数据扩展为图象矢量和图象边
界的集合方法，这样可使重建问题转变为一个即有连续变量又有二进量的混合
优化问题，于是采用一种耦合梯度神经网络方法来求代价函数的最优解。仿真
实验表明算法不仅能反映重建图象的局部平滑性又能刻画局部之间的边界信
息。
文中最后对己经开展的以及下一步的后续工作作了总结和展望。
Magnetoencephalography(MEG), measurement of neuromagnetic fields by superconducting quantum interference devices (SQUIDs), is a useful noninvasive method for investigating human brain functions. The core of MEG research is its inverse problem obtaining the underlying neural activity from the magnetic field measured outside a human head. The scope of this paper is focus on the inverse problem solution. The author has done the following research work:
Firstly, the basic mathematical theory and electromagnetic concepts are thoroughly discussed, including general Maxwell equations and the quasistatic approximation on bioelectromagnetic studies. Meanwhile, MEG forward problem is analysis in general ways.
Secondly, an overview is given on the solution method for MEG inverse problem. During the development of inverse problem research, the source reconstruction methods are classified as dipole source localization and source imaging (or current density reconstruction). The former, applying the point source model, leads to a nonlinear inverse problem. And the latter, applying the distributed source model, can be cast a linear problem.
In dipole localization based on point source model, we proposed a synthetic algorithm, which vary the used optimization method during the source scanning procedure. We use quasi-Newton(QN) method for a high-speed coarse scan over a large area of the head, and Levenberg-Marquardt(LM) method for a fine scan near the source area. In comparison with those single optimization methods, this synthetic approach proved to be more efficient both in terms of computation time and sensitivity to the iterative initial value.
MEG source image reconstruction, an important and difficult problem in image processing applications, is formulated as ill-posed inverse problem, so regularization is necessary. Regularization can be classified into two common approaches, i.e., deterministic and stochastic. Their common feature is that they both make use of a priori constraints concerning the current density distribution. In this paper we review several aspects of the application of regularization theory in MEG source image reconstruction, where the minimum norm estimation with Tikhonov regularization and Bayesian framework based on MAP-MRF model are introduced in detail. The characteristics, difference of these methods are also discussed.
To our knowledge, there is no paper talking about the regularization for the MEG source image reconstruction from a unifying viewpoint. So we try to present a relative complete regularization viewpoint on this special imaging.
In deterministic regularization, i.e., adopting minimum-norm estimation with Tikhonov regularization, we proposed the concept of region weighing. In order to obtain unique and physiologically justified solution, an operator of region weighing is introduced, meanwhile incorporating the depth weighing in the reconstruction procedure. Computer experiments show the method presented here is promising. And limitations of the proposed method and future work are then discussed.
In stochastic regularization, we present a reconstruction method based on a Markov
prior image model in order to obtain a stable solution, in this image model, the neural current density distributed image is extended to include not only the image vector of the moment values but also include the quantities corresponding to the image edges that are binary line processed. Thus the reconstruction is defined as the maximum a posteriori estimate(MAP) based on Bayesian framework. To acquire a global minimum solution from the posterior energy function, we here incorporate the ideas of coupled gradient artificial neural networks into this mixed integer optimization task.
After the present study, some issues remain open. Next, we are going to achieve the improvements of our works. Finally, we discuss the major results of this paper and some comments on MEG future trends.
引文
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