A Joint Density of Temporal Scattering Elements: Application to Change Detection
Esra Erten(1,2), Andreas Reigber(1) and Olaf Hellwich(2)
(1) German Aerospace Center, DLR, D-82234 Wessling, Germany
(2) Technical University Berlin, Sekr. FR 3-1, Franklinstr. 28/29, 10587 Berlin, Germany
Polarimetric data of distributed scatterer can be fully characterized by the (3x3) Hermitian positive definite covariance matrix which follows a complex Wishart distribution under Gaussian assumption. A second observation in time will also follow Wishart distribution. These observations are correlated or uncorrelated process over time depending on the monitored objects. To not to make any assumption concerning their independence, the (6x6) matrix, which is also modeled as a complex Wishart distribution, is used in this study to characterize the temporal behavior of polarimetric data. While statistical aspects concerning Wishart matrices are well developed, there seem to be little work on the eigenvalue statistics of correlated and/or uncorrelated Wishart process over time. Generally, to characterize the change temporal covariance matrices are compared with each other according to null hypothesis. If this hypothesis is accepted it can be concluded that all the scattering mechanism have the same variance over time and hence contribute equally to the total change. It means also that there is no need to perform target decomposition to model the natural change regarding physical properties of the object over time. However, it is reasonable to consider the null hypothesis which deals with eigenvalues regarding specific scattering mechanism rather than all eigenvalues (determinant of covariance matrix) at once. In this framework, change detection analysis regarding only covariance matrices is a restrictive assumption in the polarimetric case. In this paper, statistical model due to eigendecomposition of the covariance matrix describing the amount of the change of a scene are postulated. The joint distribution of the different scattering mechanism and their contribution to whole change phenomenon is derived. The theoretical performance of the change detection statistic is derived and shows a significant improvement over the null hypothesis considering only the equality of the covariance matrices (considering whole scattering mechanism). Finally, the change detection performance is demonstrated using experimental data collected using DLR E-SAR system, and the theoretical results of this work are analyzed by means of simulated data.