Segmentation and Classification of Polarimetric SAR Data based on the KummerU Distribution
Olivier Harant(1), Lionel Bombrun(1) and Michel Gay(1)
(1) GIPSA-Lab, 961 rue de la Houille Blanche, 38400 Saint Martin d'Hères, France
Synthetic Aperture Radar (SAR) data are the result of a coherent imaging system that produces the speckle noise phenomenon.
For multi-look multichannel Polarimetric SAR (PolSAR) data, the covariance matrix is used to characterize the scatterers. For fully developed speckle, the covariance matrix follows the complex Wishart distribution . It is assumed that land cover backscatter characteristics are homogeneous (uniform or not textured) over the area. With the development of new sensors, the resolution of PolSAR data becomes more and more high. Consequently, homogeneous assumptions cannot be validated and texture must be used to model polarimetric SAR images. For textured scenes, the ”product model” has been proposed . The observed signal is the product of a positive scalar texture component μ with the speckle component.
For PolSAR data, this model assumes that the texture component is independent of the polarization. The texture term is generally modeled by a Gamma distribution. The observed signal then follows a K distribution. Recent works have proposed to model the texture parameter by a Fisher distribution . Under the scalar product model assumption, it follows that the covariance matrix distribution can be expressed by means of the confluent hypergeometric function of the second kind (KummerU).
This new distribution has been implemented in a hierarchical segmentation algorithm and has been successfully validated on PolSAR data. The use of an appropriate distribution to model the texture parameter is essential to interpret PolSAR data.
Like the Fisher distribution is equal by definition to the Mellin convolution of a Gamma distribution by an Inverse Gamma distribution, the Fisher distribution has an hybrid behaviour between those two pdf. In this paper, a study of the asymptotic cases of the Fisher pdf is carried out. We mathematically show that if the Fisher pdf tends to the Gamma pdf, the KummerU pdf tends to the well known K distribution.
Next, authors propose to adapt the Wishart classifier  to the KummerU distribution. Classification results are shown on synthetic and real data acquired by the ESAR sensor over the Oberpfaffenhofen test-site. Finally, an evaluation of the classification accuracy is proposed by computing detection and false alarm ratio for each iteration of the algorithm.
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 L. Bombrun and J.-M. Beaulieu, “Fisher Distribution for Texture Modeling of Polarimetric SAR Data,” IEEE Geoscience and Remote Sensing Letters, vol. 5, no. 3, Juillet 2008.
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