Wishart分布矩阵行列式值的统计特性及其在参数估计中的应用

摘要

The statistics of the covariance matrix with the Wishart distribution is fully developed to be used in the data analy-sis of multilook polarimetric radar images .The determinant of the covariance matrix describes the distributed degree of target scatter-ing ,which gives a good performance in parameter estimation .In this paper a comparison has been performed among the determinant of the normalized covariance matrix ,polarization entropy and polarization diversity ,which all characterize the distributed degree of the random scattering .The mathematical relations among them are been presented with the two variables :the degree of polarization and the degree of direction .The differences among three characterizing methods only lie on the different combination of the two variables .Then the statistics of the determinant of the Wishart distribution is analyzed via Mellin transform .The log-cumulants are obtained from the distribution of the determinant of a complex Wishart distributed matrix based on its characteristic function of the logarithm of the determinant of a complex covariance matrix .The probability density function of the determinant of the matrix is de-rived .The maximum likelihood estimation (MLE ) is put up to improve the precision of parameter estimation based on the distribu-tion of the determinant of the matrix .Finally the simulated data and experiment data are used to verify the correctness of the above theory ,and the results are according to our derivation .The statistics of the determinant and the novel ML estimator of the equivalent number of looks are helpful to target detection ,parameter estimation and segmentation .%极化协方差矩阵行列式值表征了Pol-SAR图像目标散射极化散布程度，它在参数估计中表现出突出的优势，研究行列式值的物理内涵及其统计特性对揭示行列式值的优势的本质有着重要的理论价值。在此背景下，将表征目标散射极化散布程度的标准行列式值与表征目标散射随机性的极化熵、极化散度进行了比较，给出了它们与平面极化度和方向极化度的数学关系，指出他们在描述目标散射随机特性方面具有部分一致性，存在差异的本质在于三个随机性描述参数对极化方向随机度和平面极化度的加权方法的不同。然后研究了极化协方差矩阵行列式值的统计分布特性，给出了极化协方差矩阵行列式值的概率密度函数，指出在视图数和样本数足够的情况下，其行列式值的测量均值是渐进无偏估计。针对样本数较少时等效视图数参数估计误差较大的问题提出了最大似然修正算法，提高了小样本数目时的等效视图数参数估计精度。最后用仿真数据对理论推导进行了验证，结果表明了理论结果的正确性。极化协方差矩阵行列式值的统计特性和目标散射随机性参数的研究为目标极化散射随机特性、参数估计和目标检测等的研究提供了新思路。