Scale Mixture of Gaussian Modelling of Polarimetric SAR Data
© A. P. Doulgeris and T. Eltoft. 2010
Received: 1 June 2009
Accepted: 28 September 2009
Published: 16 November 2009
This paper describes a flexible non-Gaussian statistical method used to model polarimetric synthetic aperture radar (POLSAR) data. We outline the theoretical basis of the well-know product model as described by the class of Scale Mixture models and discuss their appropriateness for modelling radar data. The statistical distributions of several Scale mixture models are then described, including the commonly used Gaussian model, and techniques for model parameter estimation are given. Real data evaluations are made using airborne fully polarimetric SAR studies for several distinct land cover types. Generic scale mixture of Gaussian features is extracted from the model parameters and a simple clustering example presented.
It is well known that POLSAR data can be non-Gaussian in nature and that various non-Gaussian models have been used to fit SAR images—firstly with single channel amplitude distributions [1–3] and later extended into the polarimetric realm where the multivariate K-distributions [4, 5] and G-distributions  have been successful. These polarimetric models are derived as stochastic product models [7, 8] of a non-Gaussian texture term and a multivariate Gaussian-based speckle term, and can be described by the class of models known as Scale Mixture of Gaussian (SMoG) models. The assumed distribution of the texture term gives rise to different product distributions and the parameters used to describe them.
In this paper we only investigate the semisymmetric zero-mean case, which is expected for scattering in the natural terrain, and the more general scale mixture model includes a skewness term to account for a dominant or coherent scatterer and a mean value vector. Extension to the non-symmetric case or expanding to a multitextural/nonscalar product will be addressed in the future. It is worth noting that these methods are general multivariate statistical techniques for covariate product model analysis and can be generally applied to single, dual, quad, and combined (stacked) dual frequency SAR images, or any type of coherent imaging system. The significance and interpretation of the parameters, however, may be different in each case.
The scale mixture models essentially describe the probability density function giving rise to the measured complex scattering coefficients. They therefore model at the scattering vector level, that is, Single-Look Complex (SLC) data sets, which contain 4-dimensional complex values. These complex vectors represent both magnitude and phase for the four combinations of both transmitted and received signals for both horizontal and vertical polarisation. Statistical modelling is achieved by looking at a small neighbourhood of pixels around each point and the model parameters are estimated from this collection of data vectors. Parameter estimation, particularly of higher-order statistical terms, is improved by using a larger neighbourhood size, but at the expense of image resolution and the introduction of class mixture effects at the boundaries. So a compromise must be made between a small neighbourhood to avoid mixtures and blurring and a large neighbourhood to improve parameter estimation.
The model fitting procedure generates the model parameters at each image pixel location which gives rise to a new feature space description of the image and can be used for subsequent classification or image interpretation. Although many different models have been used to describe non-Gaussian data, with quite different orders of complexity and parametric descriptions, the parameters are usually estimated from measurable sample moments. Since the parameters are simply nonlinear relations of measured moments, one can say that the moments themselves represent the rawest form, and additionally they are independent of the particular model in question. We therefore see two quite different avenues to take regarding analysis: firstly, one can choose a specific non-Gaussian model with an explicit probability density function (pdf) and use Bayesian statistical techniques to analyse the data, or alternatively, one can extract general scale mixture of Gaussian features (that are independent of any explicit model pdf) and work solely in a two-moment generic SMoG feature space. In this sense, the Gaussian-based analysis is a single moment method.
Speckle variation may be reduced by multilook averaging, either in the frequency domain during processing or in the spatial domain postimaging, and produces Multilook Complex (MLC) matrix data. Such multi-look averaging modifies the intensity distribution of the data and subsequent statistical modelling must take this into account for parameter estimation or statistical inference. The multi-looked matrix-variate distribution derived from purely Gaussian data is the complex Wishart distribution and for the Scale Mixture case is the generalised Wishart distribution, for example, the K-Wishart . Statistical clustering using these multi-look matrix-variate models has been demonstrated elsewhere [6, 10, 11], and here we only describe multi-look data for model parameter estimation.
The plan for this paper is to describe the modelling in Section 2, with general properties and suitability discussed in Section 3. Intercomparison and parametric feature results are shown for several data sets in Section 4, followed by our conclusions in Section 5.
We denote scalar values by either lower or upper case standard weight characters, vectors as lower case bold characters and matrices as bold uppercase characters. For simplicity, we have not distinguished between random variables and instances of random variables, as such can be ascertained through context.
2. Scale Mixture of Gaussian Scheme
The Scale Mixture of Gaussian models, also known as normal-variance mixtures [12, 13], are a statistical product model with a texture random variable times a speckle random variable. The pure speckle term has a standard complex multivariate Gaussian distribution and the texture term has any positive only scalar distribution. Since the textural random variable models the variance of the signal rather than its amplitude, it is introduced as a square root term in the data vector (described in ).
Mathematically, we model the vector of polarimetric scattering coefficients ( ) under the multidimensional SMoG scheme as
where is the mean vector, the scale parameter is a strictly positive random variable (scalar), is the internal covariance structure matrix, normalised such that the determinant , and is a standardised, complex multivariate Gaussian variable with zero mean and identity covariance matrix, that is, . We will hereafter assume that . This assumption is well justified for natural environments (i.e., distributed targets without dominant coherent scatterers), where the complex values of are theoretically expected to be, and generally are, zero mean. Theoretically, this is the case of distributed coherent imaging where the resolution cell size and roughness are large relative to the illuminating wavelength, leading to the absolute phase variation over all scatterers in the cell being uniformly randomly distributed and the integrated in-phase and quadrature signals are therefore expected to be zero. We have chosen to normalise the covariance structure matrix instead of the scale parameter in our work, because of the analogy between the average scale, , and the radar cross section, , of 1-dimensional data (also described in ), even though this interpretation is not straight forward for multidimensional data.
This scheme describes different parametric families of distributions, depending on the scale parameter probability density function, . Given the pdf for the scale parameter, the marginal pdf for can be obtained by integrating the conditional pdf of , which is multivariate Gaussian, over the density of . That is,
Scale mixture of Gaussian models.
Multivariate scale mixture distribution
Constant (Dirac delta)
Normal Inverse Gaussian,
Given such a general scheme as in (1), it can be readily shown that
Moment expressions and parameter solutions for each model.
Solution given and RK
Exponential ( )
Gamma ( )
Inverse Gaussian ( )
Estimation in the case of L-look MLC data is based upon the neighbourhood mean of the matrix-variate data, plus the variance of a mean squared Mahalanobis measure ( ) which is equivalent to trace . Assuming that for simplicity, it is easily shown that
Note that the expectation of equals because of the normalisation with respect to each local covariance matrix in . The parameters and are obtained from the mean matrix by applying the constraint that , and RK is obtained in terms of var( ) by rearranging (10). Subsequently, the texture parameters are solved for as in Table 2.
3. Properties and Suitability
All models are symmetric about the mean and although each dimension may have different relative widths, distributed by the covariance matrix , they will each have a similar (global) shape governed by the scalar parameters. All models are also sparse distributions, meaning that they are more pointed in the peak and heavier tailed than the Gaussian. The MG and ML distributions have a fixed shape and the scalar parameter varies the width. The MK's and MNIG's two scalar parameters lead to a range of shapes as well as overall width. The shapes range from more pointed than Laplacian, through to rounded like the Gaussian (see Figure 1). The effect of the shape parameter on the density function is highly nonlinear with value, with the clearly visible variation occurring for small parameter values (e.g., for the K-distribution) and converging rapidly towards the Gaussian in shape from only moderate values (e.g., ) up to infinity. Also note that both the ML and MK distribution's pdfs can go to infinity at the mean value, whereas the MNIG always has a finite peak.
If we take our assumption of scale mixture of Gaussians modelling and our theoretical radar scattering as a vector sum with uniformly random phase, then three main properties emerge: zero-mean, semisymmetric shape, and global shape. It seemed appropriate to investigate whether the real PolSAR data showed similar general features as a validation for using such a mixture model.
It is interesting to also note that the polarimetric information becomes visible in the form of the different widths of each dimension, which can vary distinctly as in the first set, showing very little cross-polarisation scattering, or be much more evenly scaled as in the other two locations. Also note the pairwise equality in the distributions, because the real and imaginary parts will have equal magnitudes, and the centre four dimensions being equally scaled due to reciprocity.
Clearly, the choice of semi-symmetric, zero-mean scale mixture of Gaussian models appears to be well suited for this type of PolSAR data.
4. Modelling Results
After obtaining four parametric descriptions of the data, we then compare a goodness-of-fit measure of each to determine which model fits best. Since we are comparing four different parametric descriptions to the same data set, it is sufficient to use a relative ranking measure only, and we do not require an absolute or normalised measure of fit. The log-likelihood measure is fast and efficient and simply requires summing the log of the model pdf value at each data point. The logarithmic nature of this measure also makes it sensitive to differences in the tails of the distributions and is therefore well suited for testing heavy-tailed distributions.
Uniform, smooth, or homogeneous areas are usually best fitted as Gaussian (white), as seen in the central lake area in (a), the large open snow areas in (b), the (presumably) snow covered old ice patches in (c), and the water inlet and several large fields in (d).
The land in general, the visible icy crevasses, rocky outcrops, urban areas, and certainly anything with small scale details and high contrast are certainly non-Gaussian in nature and were poorly fitted by the Gaussian model.
All types of vegetated land appear to be best described by the normal inverse Gaussian distribution, whereas the sea ice image by the K-distribution, although the difference compared to the NIG was negligible.
The urban areas and coastlines are best fitted more often by the Laplacian; however this may be due to high contrast edge mixture effects because it appears at all water/land boundaries, around point sources like known huts within the forest, and along hedge/fence lines around fields.
The Gaussian model is usually a poor fit for significant parts of the image area, over 20%.
The Laplacian model is very good at detecting edges and point sources and is otherwise very poor at fitting to natural terrain types. Its seemingly good fit for urban areas is presumably because of the predominance of points and edges of mixed terrain in the urban landscape.
In all cases the two parameters of the MK and MNIG give a shape space that finds a "good" fit for the majority of the data points (over 90%), and mostly "fail", that is, are more poorly fitted, for the high contrast edges and point sources.
The normal inverse Gaussian model has the greatest "good" fitted area for all images and is usually the greatest best fit also.
Our results indicate that using a single, flexible two parameter model is sufficient to capture the majority of shapes seen in real PolSAR imagery. Our results indicate that the normal inverse Gaussian model is the best choice, and the K-distribution model for sea ice analysis, although both are flexible enough for all types of data.
It is important to remember that only the goodness-of-fit testing of each model has been depicted in the figures so far, and not an actual image segmentation based upon the modelled parameters. The modelled parameters consist of a brightness (or total intensity) value, a non-Gaussianity (shape or texture) value, and a polarimetric matrix. The main emphasis of our method is to include the non-Gaussianity measure, which gives additional information that is otherwise ignored in a purely Gaussian-based approach. Additionally, by working with the raw non-Gaussianity measure, these features are independent of the specific scale model and can be considered a general two moment SMoG model.
The polarimetry can be interpreted in the usual manner because our matrix is simply a normalised covariance matrix. For example, a Pauli RGB colouring scheme, or the Freeman-Durden decomposition , can be used for display and interpretation with respect to general scattering mechanisms. Simple polarimetric features, extracted from the covariance matrix, are cross-polarisation fraction, co-pol ratio, co-pol correlation magnitude, and correlation phase. In total we have six scalar features, and we found that a logarithmic transformation of the brightness, non-Gaussianity, cross-pol fraction, and co-pol ratio improved visualisation and linearity of those features.
Besides image segmentation, the features may be useful for physical parameter extraction or physical interpretation in terms of polarimetry or backscattering brightness or texture. This has not yet been rigourously studied, but the level of detail shown in these parameters is encouraging.
The scale mixture of Gaussians models indeed seems well suited to modelling PolSAR data which show inherently heavy-tailed distributions with zero-mean and a global shape for each dimension.
We have confirmed that many terrain types are clearly non-Gaussian in nature and that a flexible two-parameter model is able to capture the full shape range of PolSAR data distributions, whereas the Gaussian model cannot. Different terrain types can show quite different distribution shapes; therefore the non-Gaussianity/shape parameter should be of benefit to subsequent image segmentation.
It was demonstrated that the normal inverse Gaussian distribution is the better fitting model, out of those analysed, and usually better than the more commonly used K-distribution, with the exception of over sea ice. The MNIG model captures the greater proportion of distribution shape variations and has less trouble at boundary mixtures than the MK. The normal inverse Gaussian also has strong theoretical grounds derived from Brownian motion theory. A detailed study of why it is generally superior, and why not for sea ice, has not yet been undertaken.
We also described how a generic, two-moment, scale mixture of Gaussian analysis may be performed without the need for choosing a specific model. The feature space obtained from the modelling contains non-Gaussianity, Brightness and a Polarimetric matrix. Six features were extracted, displayed, and discussed, with a final simple image segmentation as an example application.
The methods described here can be considered the foundation for our statistical analysis of PolSAR data and future work will investigate some of the observations made here as well as address several important extensions to the model that were discussed in the introduction.
The authors would like to thank Professor Henning Skriver, Dr. Jorgen Dall, and the Danish Technical University for the Foulum data set. Thanks to Dr. Daniel Delisle, Dr. Sahebi Mahmod Reza, and the Canadian Space Agency for the sea ice data set. Both data-sets were downloaded from the European Space Agency website (http://earth.esa.int/polsarpro/datasets.html). Thanks to Norut Tromsø, Norway, for the Bleikvatnet and Okstinden data sets.
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