Characterizing the Statistical Properties of SAR Clutter by Using an Empirical Distribution

e performances on the applications of synthetic aperture radar (SAR) data strongly depend on the statistical characteristics of the pixel amplitudes or intensities. In this paper, a new empirical model, called simply HGG, has been proposed to characterize the statistical properties of SAR clutter data over the wide range of homogeneous, heterogeneous, and extremely heterogeneous returns of terrain classes. A particular case of theHGGdistribution is the well-knownGG distributions. We also derived analytically the estimators of the presentedHGG model by applying the “method of log cumulants” (MoLCs).e performance of the proposed model is veri�ed by using some measured SAR images.


Introduction
Nowadays, synthetic aperture radar (SAR) has become an advanced tool as compared to optical sensors for monitoring land or sea surfaces because of its advantages regardless of weather conditions [1,2].e interpretation of statistical properties of SAR clutter with various terrain classes is a crucial task for designing �ltering [3], detection [4], segmentation [5], and classi�cation [6][7][8] of algorithms to optimally exploit the contents of the processed SAR images.ese analyses result in a growing interest in developing precise models for the statistics of the pixel amplitudes or intensities.
It is known that SAR images are strongly affected by speckle noise due to its coherent imaging mechanism [1,2,9,10].is noise effect degrades the information of SAR images and limits the surveillance performances.To overcome this drawback, much work has been done to match amplitude or intensity statistics [3,5,6,8,11].Among the existing statistical models, the parametric distributions have been intensively investigated because of their high accuracy and �exibility, which are usually obtained by two approaches.
e �rst is to consider the physical mechanism of the backscattering from the land surfaces.e multiplicative model [1,2] by combining the speckle noise and the terrain backscatter is commonly used in this class.Generally, multilook SAR speckle noise intensity is assumed to obey a gamma distribution.erefore, a central task for establishing this type of statistical models is to aim at the backscatter modeling of different terrain classes [1,2,11].Two important distributions, consisting of  as well as  0 , which, in turn, in relation to the Gamma and inverse Gamma distribution for the intensity backscatter, have received a great deal of attention [2, 5-7, 9, 11, 12].As compared to ,  0 agrees reasonably better with the heavy tail behavior coming from the extremely heterogeneous clutter like the cases of urban areas or other man-made structures [11].
e second is to conduct the distributions from a purely mathematics view irrespective with physical property of radar clutter backscatter.Some known examples are the lognormal, weibull, and more recently the Fisher [6] (completely identical to the  0 ) distributions.e lognormal and weibull �t sometimes well with the SAR histogram of some heterogeneous and ocean regions [9].However, they tend to occur a large deviation estimating the histograms with the heavy tail behavior.
is paper is devoted to report an empirical model (denoted simply as ℋ 0 ) for characterizing the statistical properties of SAR clutter data to obtain the modeling ability of more heterogeneous clutter.e proposed model has the  0 distribution as a special case.Furthermore, using the second-kind statistics theory developed by Nicolas [13], International Journal of Antennas and Propagation which relies on the Mellin transform [14], that is, "methodof-log-cumulants" (MoLC), we derive the parameter estimators of the new distribution model.
In the rest of this paper, the proposed distribution is �rst given in Section 2. Section 3 derives the corresponding parameter estimators based on the MoLC.We provide the experimental results of the ℋ 0 model using measured SAR data in Section 4, the comparisons with that  0 �ts are also discussed in this section.e last section concludes this paper and give a perspective in the future work.

The Proposed Distribution
e proposed amplitude distribution is de�ned as where , , and  are the power, shape, scale parameters, respectively. indicates the stretching parameter.Γ(⋅) represents the Gamma function. is the number of looks.e corresponding intensity expression of ( 1) is further given by ×  −    −  (−)/ , −, −, , , ,   0 (2) We refer to this distribution characterized by (1) or (2) as the ℋ 0 distribution.Speci�cally, we call the ℋ   distribution and the ℋ   distribution, correspond to (2) and (1), respectively, to distinct the intensity statistic as well as the amplitude statistic.Figure 1 gives some examples of the ℋ   distribution with respect to the various parameters.From this �gure, it can be seen clearly that the parameter  re�ects the degree of homogeneity for the tested returns, which implies that the smaller value of || obtains, the more in-homogeneous (i.e., larger tails) they are.Meanwhile,  corresponds to a stretching of the SAR image amplitude or intensity, showing a strong effect for low values of the return.Moreover, as an independent parameter,  indicates the whole �uctuation (contains magnifying or shrinking) of power of densities along the vertical axis.e parameter  controls the peak value of the density.
Although the probability density functions (PDFs) characterized by (1) or (2) are shown as the empirical models, the ℋ 0 model also can be derived within the structure of the multiplicative model [1,2,11].Herein, we account for the intensity distribution as an example, the following theorem is established.eorem 1. Letting   and   indicates the backscattering RCS component and speckle noise one, respectively,   denotes the observed intensity of SAR data.erefore, the relationship of this three variables is expressed by the multiplicative model as and the PDF of   is en the distribution of the intensity return is characterized by the density shown in (2), that is,   ∼ ℋ   (, , , , ).
Furthermore, ℋ 0 has the charming property that the well-known  0 , presented by Frery et al. [11] to model homogeneous, heterogeneous, and extremely heterogeneous terrains, is a special case of this proposed model when  =  and  = −.us, the proposed model exhibits higher �tting ability as compared to  0 .
As derived in Appendix A, the th order moments of the ℋ e th order moments of the corresponding amplitude random variable can easily be given by    ) =    ).

The Parameter Estimators of the Proposed Distribution
where ℳ is the Mellin transform operator.e th order derivative of   ) at  =  is the log-cumulants of order , International Journal of Antennas and Propagation that is, Hereaer, we take the intensity distribution ℋ   as an example to estimate its parameters , , and .e processing of the amplitude one is similar.Owing to   () = (  ), consequently, via ( 8) and ( 9), the second-kind second characteristic functions of the ℋ   distribution yields (13)

e Parameter
Estimators of ℋ 0 .We notice that ( 12) is independent of the parameter  and the th log-cumulants shown in (12) are irrespective with the parameter  on the condition that   2. In addition, we regard the parameter  as a known constant, which can be replaced by the equivalent number of looks (ENLs) [4,11] or obtained from some prior knowledge about processed SAR images, hence, allowing us to divide the parameter estimates to three stages.First, the estimates   and   of the parameters  as well as  are obtained by solving the following equations resorting to the numerical computation:

Experimental Results
In this section, we aim at verifying the performance of the proposed ℋ 0 .In order to assess how the ℋ 0 performs, several space-borne TerraSAR-X geocoded scenes with various land-over typologies as examples are reported.Figures 2(a)-2(d) show four selected typical patches from a large "Sanchagang" TerraSAR-X image with low resolution, which are the portions of water body, drying riverbed, mountain, and a town returns, respectively.e four types of scenes are related to the homogeneous and heterogeneous terrains.For simplicity, we denote them as "water-body", "drying riverbed", "mountain", and "town." Additionally, two urban areas shown in Figures 2(e)-2(f), extracted from a large "Beijing" TerraSAR-X scene with high resolution, are further carried out to demonstrate the effectiveness of the proposed model on the extremely heterogeneous terrains.Likewise, the two urban images are denoted by "urban1" and "urban2." e main parameters of TerraSAR-X systems for "Sanchagang" and "Beijing" data are listed in Table 1.
e estimated PDFs of the proposed ℋ 0 model for the histograms of six selected areas indicated in Figure 2 are shown in Figure 3.As the corresponding comparisons, the �tting results with the intensively used  0 distribution are also provided, where parameter estimates of this distribution are derived by Tison et al. [6] based on the MoLC.Herein, the estimate  in ℋ 0 is �rst replaced by the ENL, that is, [4] where   and   are the effective number of looks in the azimuth and range.For the previous "Sanchagang" and "Beijing" TerraSAR-X images used in this investigation,   and   are listed in the last two columns of Table 1 according   3 (see the yellow textbox).It can be clearly seen that the proposed ℋ   better agrees with the amplitude histograms of all six terrains than the    , as expected, because the  values are larger than 1 for all six areas.e same conclusion can be con�rmed from a visual point of view and implies the higher �tting precision, using ℋ  than using  0 , over homogeneous, heterogeneous, and extremely heterogeneous regions.

Conclusion and Perspective
We have developed an empirical model, ℋ 0 , to exploit the knowledge of statistical characteristics of SAR amplitude or intensity images over the wide terrain classes with homogeneous, heterogeneous, and extremely heterogeneous backscattering properties.e parameter estimators of this model based on the MoLC are also provided.Consequently, we report the performances of different land-over typologies with ℋ 0 distribution �ts.e experimental results show that the ℋ 0 distribution is a more advanced model compared with the known  0 distribution to characterize the multilook processed SAR data.
As we know, a preliminary statistical analysis of SAR clutter data is important for designing signal processing algorithms, such as speckle �ltering, target detection, building extraction, image segmentation, and classi�cation.In future, it is worth expecting to use the ℋ 0 distribution in these �elds.Herein, we �rstly attempt to give an analytical derivation for constructing a constant false alarm rate (CFAR) detector to promote the upcoming studies of target detection in SAR images. Given Considering  ℋ   ( is strictly monotonously increasing, the threshold  can be accurately calculated with the help of the numerical solution or a simple bisection method [16].
Our future work will focus on demonstrating the performances of the proposed CFAR detector using some measured SAR data and investigating how the ℋ 0 distribution performs when extending it to other application �elds.

F 3 :
Plots of a�plitude �istogra�s and of t�e esti�ated P�Fs: (a)-(f) are corresponded to t�e �tting results for t�e scenes s�o�n in Figures 2(a)-2(f), respectively.
We assume the observed amplitude PDF is     () from the actual data, which corresponds to the histogram of tested data.e theoretical amplitude PDF denotes as    ().As the previous analysis, since  indicates the whole proportion �uctuation of    (),    () can be calculated by using  ,  , and  .Given a sample amplitude set {  |  =   }, let the symbol    () be equal to    (), then the estimate   of the parameter  is simply given by

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International Journal of Antennas and Propagation T 1: e main parameters of TerraSAR-X space-borne systems for the data in this study.is a sample set,   (⋅ represents the compared theoretical PDF,   (⋅ is the basic theoretical PDF, and (⋅ indicates the actual PDF from the observed data.esymbol ‖ ⋅ ‖ denotes 2-norm.It is obvious that the numerator and denominator in (18) separately imply the total �tting errors with the   and   to approximate .ebetter the performances of   related to   �t, the smaller  is, and vice verse.Speci�cally,   , if both   and   have the identical capability for �tting the measured data.Let ℋ   represent the basic amplitude PDF, and let    be the compared amplitude one.e  values of the previous six areas in this study are given in Figure