脑磁源的定位计算的最优化

Best Minimization to Locate the Brain Sources of Magnetic Waves

脑磁源的定位是脑磁图（Ｍａｇｎｅｔｏｅｎｃｅｐｈａｌｏｇｒａｐｈｙ，ＭＥＧ）研究的一个基本问题。该问题属不定问题，即根据探头测量的微小磁场所建立的方程组求得的未知脑磁源有无穷组解。因此需要通过合理的数学模型补充适当信息。目前的两种主要的数学模型为偶极子源搜索法和Ｌｐ 最小模法。对前者若采用全局搜索费时太多，若采用梯度类搜索法又容易收敛到局部极值点。故一般得用其他方法引导，定出初值，再使用梯度类搜索法。后者算得脑磁源的深度依赖于该方法中权重函数的选择，而该权重函数又有不同的几种，因此不能准确的估计脑磁源的深度。为了克服上述两种方法的困难，采用多目标函数最优化法。该方法的目标函数为，（１）未知源的Ｌｐ 模，（２）由未知源计算出的磁场与测量的磁场之间的偏差，（３）源的数目。该方法与最小模方法相似也引入权重矩阵。但权重矩阵含多个参数，这些参数同未知源一样也要通过三目标最优化法来确定。适当选用权重函数及其参数，该方法可以准确的估计源的深度。

Locating the sources of brain magnetic fields is a basic question in magnetoencephalography (MEG). This question belongs to the indeterminate problem, that is, infinite solutions can be obtained from the equations of the measured small magnetic fields on the detector. Therefore a model of mathematics is needed to give more information. The major models of mathematics are the search method with many dipoles and the Lp minimum norm method. For the first model, if a global search is used, too much calculation time will be spent; if the gradient search is used, the searches usually converge to a local minimun. So other methods are introduced to find out the initial values, and then the gradient search can be used. For the second model, the depth of the brain sources will dependend on the choice of the weight functions which have several different types, and we do not know in advance which type is correct, so that it is difficult to have a good estimation of the depth of the brain sources. To overcome the difficulties, we use the method with three object functions which are the minimum Lp norm, the deviation between the calculated magnetic fields and the measured magnetic fields, and the number of the sources. This method is similar to the minimum norm method, a weight function is also introduced. But this weight function contains many reference variables which have to be determined through the minimization with three object functions. By choosing suitable weight function and its reference variables, the correct estimation of the depth of the brain sources can be achieved.