Modeling Multilook Magnitude and Phase in Extremely Heterogeneous Clutter

This paper proposes a joint distribution of magnitude and phase for multilook SAR interferogram in extremely heterogeneous clutter. The presented theoretical distribution, called simply here J2 distribution and derived from the multiplicative model and the known joint distribution of the homogeneous clutter, is shown to be able to model the extremely heterogeneous clutter areas. Moreover, we estimate distribution parameters by means of the well-known method-of-log-cumulants (MoLC). The experimental results applied on actual dual-channel SAR images prove the good performance of the proposed distribution.


Introduction
Ground moving target identification (GMTI) using synthetic aperture radar (SAR) has been a growing interest over the last couple of decades in many applications, such as military surveillance and reconnaissance of ground vehicles and civilian ship monitoring of harbor [1][2][3].The recent works [3,4] reported in this field show that the multilook interferogram, defined as the product of the first channel and the complex conjugate of the second, is an important tool for detecting moving targets.However, precise knowledge of the interferogram's phase and magnitude statistics, that is, the joint probability density function (PDF), is a major issue currently under study in the development of statistically based detector tests for distinguishing the moving targets from clutter [3][4][5].
Some investigations for statistical modeling of multilook SAR interferogram have been presented in the past, for example, [3][4][5].Lee et al. [5] firstly proposed the joint distribution of interferometric magnitude and phase with the condition of a constant radar cross section (RCS) background based on the complex Wishart distribution, presented by Goodman [6].The analytical expression of this joint distribution is given as [5]  , (, ) where  and  are the normalized interferogram's magnitude and the multilook phase. represents number of looks, and  indicates the magnitude of the complex correlation coefficient. −1 (⋅) is the second type modified Bessel function with order ( − 1).Γ(⋅) indicates the Gamma function.In the analysis of lot of literatures, it is shown [3][4][5] that the PDF shown in (1) is valid for modeling homogeneous areas, whereas it also tends to deviate strongly in most cases whose scenes contain heterogeneous or extremely heterogeneous regions.
Additionally, as the phase statistic is highly invariant against changes of the clutter type [3], the marginal PDFs of interferometric magnitude for heterogeneous and extremely heterogeneous regions are derived by Gierull [4].Meanwhile, an original joint PDF of interferometric magnitude and phase for heterogeneous clutter is also given in afore-mentioned literature.Unfortunately, the joint PDFs of interferometric magnitude and phase for extremely heterogeneous regions are still a hard task by means of combining the marginal PDFs of magnitude and the ones of phase owing to that  and  are not statistically independent [3].
In this paper, our objective is to present a novel joint distribution of interferometric magnitude and phase for extremely heterogeneous clutter.We test the performance of the proposed distribution utilizing a representative dualchannel SAR image of urban area described as an extremely heterogeneous region.

The Known Joint Distribution for Heterogeneous Regions.
Considering an -look interferogram   , it is the average of  single-look interferograms.Assuming the energy of two channels is identical, it is well known that   can be modeled by the multiplicative model as [4] where  represents the backscattering RCS magnitude of each channel.As analyzed by Frery et al. [7], the random variable  obeys a reciprocal of the square root of a Gamma distribution to characterize highly heterogeneous situation, that is,  ∼ Γ −1/2 (−, ).Supposing  =  2 , the PDF of  is given by where  (− ∈ (0, ∞)) is a shape parameter, which essentially reflects the degrees of homogeneity for processed areas. is a scale parameter related to the mean energy of processed areas.Therefore, the modified interferometric magnitude  of heterogeneous clutter is given as  =  2  :=  and the joint distribution of  and  can be expressed by Applying (1), we get the right hand-side of the integral shown in (4) as ) . ( Combining ( 3) and ( 5) by ( 4), and utilizing the integral formula ∫ ∞ 0  −1  −   () = (√(2)  /( + ) + ) ⋅ ((Γ( + )Γ( − ))/Γ( + 1/2)) 2  1 (+, +1/2; +1/2; (− )/( + )) [8], the joint distribution of magnitude and phase in the heterogeneous clutter is finally derived as [4]  , (, ) where 2  1 is the Gauss hypergeometric function and

The New Joint Distribution for Extremely Heterogeneous
Regions.For extremely heterogeneous clutter like the urban areas, the histograms show the heavy trail.To solve this problem, Gierull [4] proposed a transformation of  into  =   ,  ∈ R + , where the interferometric magnitude  and phase of extremely heterogeneous clutter can be derived by (6) with where   1 (, ) and   2 (, ) are the following functions: Hereafter ( 8) is denoted simply by J 2 .The marginal PDF of interferometric magnitude  is further given by integrating (8) with respect to the phase  as  It is clear from the derivations of ( 5) and ( 8) that the J 2 distribution converges to the  , (, ) when −,  > 0,  → 1.Similarly,  , (, ) also converges to  , (, ) under the condition that −,  → ∞, (−/) → 1.The properties stated in Figure 1 show that either homogeneous, heterogeneous, or extremely heterogeneous interferogram's magnitude and phase statistics can be treated as the outcome of the proposed J 2 distribution.

Parameter Estimations
The MoLC, which relies on the Mellin transform [9], is a more feasible parametric PDF estimation technique for distributions defined on [0, +∞).Given  as a function defined over R + , the Mellin transform of  is defined as Thus, the second-kind first characteristic function and the second-kind second characteristic function are given, respectively, by The estimate ρ of parameter  has been obtained by Abdelfattah and Nicolas; the details can be found in [10].However, here, we are interested in estimates of the parameters , , , and  in the J 2 distribution.To make the deriving process be simplified, we notice that the parameters of the J 2 distribution and the corresponding marginal PDF of magnitude given by (10)  ( The kth-order derivative of Φ() at  = 1 is the kthorder second-kind cumulant also named "log-cumulant." Consequently, the th-order log-cumulants corresponding to the J 2 distribution are where Ψ(⋅) represents the digamma function and Ψ(, ⋅) is the kth-order polygamma function.Additionally, the logcumulants c can be directly estimated by  samples   as follows: The Ψ(, ⋅)functions with varying  are shown in Figure 2 Thus (15) is continuous and strictly monotonically for each parameter of , , and .To obtain the numerical solution quickly and simply, we set  as an even (i.e., letting  = 2,  = 4 and  = 6 in (15), resp.).On the other hand, we stress that (15) does not contain , thus allowing us to split the nonlinear solution procedure into two distinct stages.First, the estimates δ, n, and α of the parameters , , and  are accquired by solving (15) and (16), that is, International Journal of Antennas and Propagation  Second, via ( 14) and ( 16), the estimate γ of the parameter  is

Experimental Analysis
In this section, we present simulation results obtained by measured SAR data using the proposed J 2 distribution.Especially, we provide the fitting performance of urban area indicated as extremely heterogeneous clutter to support the prior theoretical analysis that the J 2 distribution is able to model the clutter areas with widely varying degrees of homogeneity.
As a representative example, the test dual-channel SAR data of urban used in this study were acquired by an airborne SAR system of China in Beijing operated in X band and HH polarization, with the spatial resolution 10 m × 2 m (azimuth  We apply the proposed parametric estimation algorithm in Section 3 to the above-mentioned urban area.The results of parametric estimation of the corresponding J 2 distribution are listed in Table 1.Based on these estimated parameters, the fitting histogram is given in Figure 5.It is clearly seen that the J 2 distribution performs very well in fitting the histogram of the urban area from the viewpoint of the visual comparison between the histogram and the estimated PDF.
Furthermore, in order to assess the effectiveness of the presented J 2 distribution, we test the developed model on the previous urban area by limiting various phase  values.Figure 6 yields the fitting result with the condition of  = 0.It is easy to observe that the J 2 distribution agrees well with the given urban area which implies the wide modeling ability of the proposed distribution, as expected.

Conclusion
In this paper, we have developed a joint distribution of magnitude and phase for multilook SAR interferogram in extremely heterogeneous clutter.We also provide the corresponding parameter estimation technique based on the MoLC.The theoretical analysis and experimental results of measured multilook SAR data both have shown the good performance modeling extremely heterogeneous clutter.Since either homogeneous, heterogeneous, or extremely heterogeneous interferogram's magnitude and phase statistics can be treated as the outcome of the proposed J 2 distribution (as Figure 1 stated), the presented distribution turns out to be suitable for the clutter with widely varying degrees of homogeneity.

Figure 1 :
Figure 1: The relationship of the joint distributions.

2. 3 .
Relationship between Distributions.The relations of the aforementioned joint distributions are summarized in Figure 1, where the symbol  ⇀ denotes the convergence in distribution.

Figure 3 :
Figure 3: The tested SAR data of urban.

Figure 4 :
Figure 4: The three-dimension histogram of the tested urban area.

Figure 5 :Figure 6 :
Figure 5: The estimated PDF of the previous area.

Table 1 :
Parameter estimations of noted clutter areas in Figure3.