

A Relaxed Wishart Model for Polarimetric SAR Data
Anfinsen Stian Normann^{(1)}, Doulgeris Anthony Paul^{(1)} and Eltoft Torbjørn^{(1)}^{(1)} University of Tromsø, Auroral Observatory, N9037 Tromsø, Norway
Abstract
In this paper we demonstrate that simple yet flexible modelling of multilook polarimetric SAR (PolSAR) data can be obtained through a relaxation of the Wishart model. The degrees of freedom of the Wishart distribution is treated as a spatially nonstationary parameter, which is allowed to vary between thematic classes and segments of the PolSAR scene.
The Wishart distribution is the de facto statistical model for multilook PolSAR data. It is based on the assumption that the complex scattering coefficients are jointly circular Gaussian. However, this is only satisfied for homogeneous areas with fully developed speckle and no texture, which renders the model inadequate in many cases. Improved modelling is achieved by using more complex models that account for texture, such as the polarimetric Gdistribution family (Freitas et al., 2003), with the polarimetric Kdistribution (Lee et al., 1993) as a special case. These models allow for better adaption to data whose distribution is heavytailed and nonGaussian, but this comes at the cost of higher mathematical complexity.
The comparatively higher mathematical tractability of the Wishart distribution motivates us to pursue a relaxed Wishart model as an alternative. In the context of multilook PolSAR data, the degrees of freedom of the Wishart distribution is interpreted as the equivalent number of looks  a constant, global value that quantifies the effective number of data samples averaged in the multilooking process. In contrast, we treat it as a free parameter, which varies between  and possibly also within  classes and segments of the PolSAR scene. This reflects the highly variable degree of smoothing imposed on the data by nonlinear speckle filters. The choice can also be justified by looking at the degrees of freedom as a shape parameter of the distribution, which is determined not only by the degree of averaging, but also by texture. Thus, the influence of multilooking, speckle filtering, and texture is assimilated into one parameter, which can be estimated efficiently with a recently proposed estimator (Anfinsen et al., 2008).
Modelling results are shown for airborne NASA/JPL AIRSAR data. Application to classification is discussed.
References:
Freitas, C.C., A.C. Frery, and A.H. Correia. "The polarimetric G distribution for SAR data analysis," Environmetrics, vol. 16, no. 1, pp. 1331, 2005.
Lee, J.S., D.L. Schuler, L.H. Lang, and K.J. Ranson. "K distribution for multilook processed polarimetric SAR imagery," IEEE Int. Geosci. and Rem. Sensing Symp., Piscataway, USA, 1993, pp. 21792181.
Anfinsen, S.N., A.P. Doulgeris, and T. Eltoft, "Estimation of the equivalent number of looks in polarimetric SAR data," IEEE Int. Geosci. and Rem. Sensing symp., Boston, USA, 2008, in press.
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