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In this paper, we propose to use a matrix-variate Mellin transform in the statistical analysis of multilook polarimetric radar data. The domain of the transform integral is the cone of complex positive definite matrices, which allows for transformation of the distributions used to model the polarimetric covariance and coherency matrix. Based on the matrix-variate Mellin transform, an alternative characteristic function is defined, from which we can retrieve a new kind of matrix log-moments and log-cumulants. It is demonstrated that the matrix log-cumulants are of great value to analysis of polarimetric radar data, and that they can be used to derive estimators for the distribution parameters with low bias and variance.