Statistical Analysis of Multilook Polarimetric Radar Images with the Mellin Transform

Abstract

This thesis presents methods for statistical analysis of the probability distributions used to model multilook polarimetric radar images. The methods are based on a matrix-variate version of Mellin's integral transform.

The proposed theoretical framework is referred to as Mellin kind statistics. It is an extension of a theory recently developed for single polarisation amplitude and intensity data to the complex matrix-variate case describing multilook polarimetric images. This generalisation is made possible by the rediscovery of a generalised Mellin transform, which is defined for functions of positive definite Hermitian matrices. The domain makes it suited for application to the distributions used to model the polarimetric covariance and coherency matrix.

The analysis tools include the matrix-variate Mellin kind characteristic function, which is defined with the Mellin transform in place of the conventional Fourier transform. Matrix log-moments and matrix log-cumulants are retrieved from this function. The matrix log-cumulants are used in a moment based approach to parameter estimation of the distribution parameters. The estimators make efficient use of all the statistical information in the polarimetric covariance matrix, and are superior to all known alternatives. The matrix log-cumulants are also used to construct the first known goodness-of-fit test for matrix distributions based on the multilook polarimetric product model. The algorithms are interpreted by means of a highly informative graphical visualisation tool displaying a space spanned by certain matrix log-cumulants.

It is demonstrated that the matrix-variate Mellin transform is the natural tool for analysing multilook polarimetric radar images. This conclusion is based on the simple and elegant mathematical expressions obtained, the superb statistical properties of developed estimators, as well as the intuitive interpretations offered by the Mellin kind statistics.