MODEL-BASED STATISTICAL ANALYSIS OF POLSAR DATA. Torbjørn Eltoft 1,2 , Anthony Doulgeris 1 and Stian N. Anfinsen 1 1 Department of Physics & Technology, University of Tromsø 2 Northern Research Institute (Norut), Tromsø, Email: torbjorn.eltoft, anthony.doulgeris, stian.normann.anfinsen all at @uit.no ABSTRACT In this paper, we consider statistical analysis of PolSAR data in the framework of the multivariate product model. The complex scattering vector is here considered as a double stochastic circular Gaussian variable, in which the variance is linearly scaled by a common stochastic scaling factor z. The scaling factor is associated with texture. We discuss various parametric probability density functions for z, and indicate how model parameters can be estimated from data by a simple moment based method. Experimental analysis shows that for some surface covers, certain texture distributions fit better than others. Then, polarimetric covariance matrix data analysis is addressed in the framework of product models, and we propose a processing scheme which perform image segmentation using a stochastic EM approach. I. INTRODUTRODUCTION Polarimetric SAR (PolSAR) data are complex multidi- mensional image data, which can be analyzed along several processing schemes. In the literature, we find that much emphasis has been put on analysis based on target decompo- sition theorems [1], [2]. Through this approach, information about scattering mechanisms can be gained. The knowledge of the exact statistical properties of PolSAR data founds the basis for another strategy of multidimensional image analysis, which in some cases is complementary to the target decomposition approach. Statistical properties can be used to discriminate different types of land cover (e.g. [3], [4]), or to device specialized filters for speckle noise reduction (e.g. [5]), to mention two areas of application. It is well known that SAR (and PolSAR) data can be non- Gaussian in nature. For this reason, various non-Gaussian models have been proposed to represent SAR data. These have later been extended into the polarimetric realm, where the multivariate K-distributions [6] and G-distributions [7] are successful examples. Both these distributions are mem- bers of the so-called product model, which states that, under certain conditions, the backscattered signal results from the product between a Gaussian speckle noise component and the terrain backscatter. Several distributions could be used for the terrain backscatter, in order to model different types of surface classes with their characteristic spatial correlation properties and degrees of homogeneity. Mathematically, the multivariate product model can be formulated as follows: Let x be a d-dimensional, zero mean Gaussian variable with covariance matrix equal to the identity matrix. Let furthermore, Γ ∈R d×d be a positive definite, Hermitian matrix with determinant det Γ =1, and let Z be a scalar random variable with pdf f z (z), which can attain only positive values. A new variable y is now generated as y = √ zΓ 1 2 x. (1) The matrix Γ defines the internal covariance structure of the component variables of y. For this reason we will refer to this matrix as the covariance structure matrix. The above modeling scheme constitutes flexible models, which have the capabilities to model data which ranges in Gaussianity from highly non-Gaussian to Gaussian data. The rich parameter space associated with the product models al- lows for the development of image segmentation algorithms, both in data space and in the parametric feature space. In this paper some characteristics related to product mod- els are briefly discussed, as well as certain approaches to the analysis of PolSAR data in this framework. Experimental results from various application areas will be presented. II. PARAMETRIC MODELS FOR VECTOR OBSERVATIONS The product model scheme describes different parametric families of distributions, depending on the scale parameter probability density function, f z (z). Given the pdf for the scale parameter, the marginal pdf for y can be obtained by integrating the conditional pdf of y|z, which is multivariate Gaussian, over the density of z. That is f y (y)=  ∞ 0 f y|z (y|z) f z (z) dz (2) As can be seen, the probability distribution of the resulting double stochastic variable is obtained as a continuous mix- ture of scaled Gaussians. For this reason we often refer to these models as scale mixture of Gaussian (SMoG) models, or normal variance mixture variates. III - 955 978-1-4244-3395-7/09/$25.00 ©2009 IEEE IGARSS 2009