Derivation of separability measures based on central complex Gaussian and Wishart distributions

In this paper the Bhattacharyya distance and the divergence are derived as two different measures of target class separability based on the central complex multivariate Gaussian and Wishart distributions with unequal covariance matrices. The derived Bhattacharyya distances for the two distributions, respectively, differ only in terms of a simple multiplier, i.e. the number of degrees of freedom (also known as number of looks in polarimetric synthetic aperture radar data). The same aspect was observed for the divergence. Furthermore, the Bhattacharyya distance was found to be proportional to the Bartlett distance, while the divergence is proportional to the symmetrized normalized log-likelihood distance. The use of the Bhattacharyya distance and the divergence as separability measures of target classes was demonstrated by using NASA/JPL AIRSAR POLSAR data and was benchmarked against the Euclidean distance. From the results, both the Bhattacharyya distance and the divergence were found to perform consistently in measuring the separability of target classes.