A Joint Test Statistic Considering Complex Wishart Distribution: Characterization of Temporal Polarimetric Data

Abstract

Polarimetric data of distributed scatterers can be fully characterized by the $(3 imes 3)$ Hermitian positive definite matrix which follows a complex Wishart distribution under Gaussian assumption. A second observation in time will also follow Wishart distribution. Then, these observations are correlated or uncorrelated process over time related to the monitored objects. To not to make any assumption concerning their independence, the $(6 imes 6)$ matrix which is also modeled as a complex Wishart distribution is used in this study to characterize the behavior of the temporal polarimetric data. According to the complex density function of $(6 imes 6)$ matrix, the joint statistics of two polarimetric observation is extracted. The results obtained in terms of the joint and the marginal distributions of Wishart process are based on the explicit closed-form expressions that can be used in pdf (probability density function) based statistical analysis. Especially, these statistical analysis can be a key parameter in target detection, change detection and SAR sequence tracking problem. As demonstrated the bias of the joint distribution can decrease with noise free signal and with increasing the canonical correlation parameter, number of looks and number of acquired SAR images. The results of this work are analyzed by means of simulated data.