A Modification of Rician Distribution for SAR Image Modelling

This paper presents a novel statistical model i.e. the Laplace-Rician distribution, for the characterisation of synthetic aperture radar (SAR) images. Since accurate statistical models lead to better results in applications such as target tracking, classification, or despeckling, characterising SAR images of various scenes including urban, sea surface, or agricultural, is essential. The proposed Laplace-Rician model is investigated for SAR images of several frequency bands and various scenes in comparison to state-of-the-art statistical models that include K, Weibull, and Lognormal. The results demonstrate the superior performance and flexibility of the proposed model for all frequency bands and scenes.


Introduction
Statistical modelling of SAR images plays an essential role in characterising various scenes and underpins applications such as classification [1, 2] or denoising [3,4]. The literature abounds with numerous statistical models for different SAR scenes, and selecting the most accurate for SAR imagery is essential.
The standard SAR signal model defines the back-scattered SAR signal received by a SAR sensor as a complex signal R = x+iy, where x and y are the real and imaginary parts, respectively and follows several assumptions [5,6]: 1. The number of scatterers is large, 2. The scatterers are statistically independent, 3. The instantaneous scattering phases are statistically independent of the amplitudes, 4. The phase is uniformly distributed, 5. The reflectors are relatively small when compared to the illuminated scene, 6. There is no dominating scatterer in the the scene.
In particular, the first two assumptions recall the central limit theorem whereby the real and imaginary parts are jointly Gaussian. Combining with the assumption 6, this leads to the case where x and y are independent and identically distributed (i.i.d.) zero-mean Gaussian random variables. Thence, the amplitude distribution becomes the Rayleigh distribution, the probability density function (pdf) of which is given in (1).
Rician → r σ 2 e − r 2 +δ 2 2σ 2 I 0 rδ σ 2 , SR [5] → r ∞ 0 se −γs α J 0 (sr)ds, 2π 0 e − |r cos θ| α +|r sin θ| α γ α dθ. (7) However, in most cases, SAR reflections are non-Gaussian/Rayleigh, thus there are numerous statistical models in the literature which were developed to account for non-Rayleigh cases. In cases when the illuminated scene includes a dominating scatterer, where the assumption 6 is no longer valid, x and y become identically and independent non-zero-mean Gaussian random variables, which determines the amplitude distribution to follow the Rician distribution with pdf given in (2). Furthermore, K [7], Weibull [8] and Lognormal [9] distributions dominate the literature for modelling several different scenes from urban to agricultural, from land cover to sea surface in SAR imagery. The pdfs of K, Weibull and Lognormal models are given in (3), (4) and (5), respectively.
In a previous study, following the non-Gaussian reflections idea, Kuruoglu & Zerubia [5] have proposed a generalised central limit theorem based statistical model which extends the standard scattering model discussed above by addressing the real and the imaginary parts of the complex backscattered SAR signal as jointly symmetric-α-Stable random variables. This model, called the generalised Rayleigh distribution (will be denoted as Stable-Rayleigh, or shortly SR for the rest of the paper) for amplitude SAR images, the pdf of which is given in (6). SR has been shown to be a good choice for urban SAR image modelling in [5,10] and successfully applied to despeckling studies in [3]. Moser et, al, [6] have proposed another generalised statistical model for amplitude SAR modelling, which is similar to SR [5], by assuming the real and imaginary parts of the back-scattered signal to be independent zero-mean generalised Gaussian (GG) random variables, which leads to the generalised Gaussian Rayleigh (GGR) distribution, with the pdf given in (7).
In this paper, we present a novel statistical model, namely the Laplace-Rician distribution for modelling amplitude SAR images. The proposed model is based on the Rician distribution, whereby we assume that the real and imaginary parts of the back-scattered SAR signal are Laplace distributed. Notwithstanding its seldom usage [11,12], the Rician distribution occurs in cases when a scatterer dominates the SAR scene [13,14], and would characterise several SAR scenes. Combining the Rician idea with the non-Gaussian case via Laplace distribution in the proposed Laplace-Rician model addresses both the non-Rayleigh and heavy-tailed characteristics of amplitude SAR images. The rest of the paper is organised as follows: we present the proposed Laplace Rician model in Section 2. Then, in Section 3, we demonstrate the experimental analysis for various real SAR data, followed by concluding remarks and future work in Section 4.

The Proposed Model
In this study, we follow the idea of generalising the Rayleigh distribution through heavy-tailed distributions proposed in [5,6], however, this time we apply this to the case when the there are dominant reflectors leading to nonzero mean reflections previously modelled with the Rician distribution.
In particular, we first assume that signal components, x and y are non-zero mean Laplace distributed: where µ is the location parameter, γ refers to the scale parameter, and u is either x or y. Since the SAR signal components x and y are independent [6], we write the joint pdf as Therefore, by using the identity f (r, θ|µ, γ) = rf (r cos θ, r sin θ|µ, γ) (11) we write the joint pdf for the amplitude (r) and phase (θ) of the back-scattered SAR signal as Hence, the corresponding marginal amplitude pdf can be obtained by averaging (12) over θ and boils down to [15]: which we call the Laplace-Rician distribution.

Results and Discussions
The proposed method was tested for 33 different SAR images coming from various platforms with frequency bands of X (TerraSAR-X, COSMO-SkyMed and ICEYE), L (ALOS-2) and C (Sentinel-1). Each SAR image corresponds to one type of scene investigate, i.e. urban, agricultural, mountain, land cover and sea surface with and without ships and their wakes. Since we have three sources for X band SAR imagery as mentioned above, we have more X band example images in this study, the exact distribution of images in terms of scenes and frequency bands is given in Table 2.   Figure 1 confirm the numerical results presented in Table 3, whereby the Laplace-Rician model outperforms all reference methods utilised in this study and follows the data histograms better than the others. Figure 2 depicts a visual representation to the statistical significance values given in Table 3. Dark pixels show lower values, which also refer to better modelling results. When examining both sub-figures in Figure 2, we can easily state that for over all images, the proposed Laplace-Rician model is obviously better than the state-of-the-art models.

Conclusions
In this paper, we presented a study on the statistical modelling of SAR images of various scenes. We proposed a novel statistical model, the Laplace-Rician distribution, which is based on a generalisation of the Rician distribution by replacing the Gaussian assumption with the Laplace distribution. The performance of the Laplace-Rician in modelling SAR images corresponding to urban, agricultural, mountain, land cover, and sea surface scenes from four satellite platforms, i.e. TerraSAR-X, ALOS2, COSMO-SkyMed and Sentinel-1, was assessed in comparison to state-of-theart statistical models that include K, Rayleigh, Rician, Weibull, lognormal and GGR [6] distributions. The results demonstrate that the proposed method achieves the best modelling results for most of the images, and outperforms state-of-the-art models for images from various frequency bands and sources. It is interesting to note that the results show us the need for combining the advantages of non-Gaussian heavy-tailed modelling and non-zero mean reflections modelling provided by Rician model. That is quite obvious when we compare Laplace-Rician with GGR, Rician and Rayleigh directly. Furthermore, using non-zero reflections along with the heavy-tailed modelling in Laplace-Rician model shows important success in all types of SAR scenes and frequency bands. There is one specific exemption (I 25 ) which contradict this contribution. Image I 25 shows the Rician is the best model, which states the need for nonzeros reflections but not heavy-tailed modelling. Moreover, There are three cases (I 5 , I 12 and I 18 ) that lognormal model outperforms the proposed Laplace-Rician for both of the statistical measures. These four specific examples will guide us for future works on further developments of the method on where our modelling gain is low or none.

Acknowledgments
The authors are grateful to the UK Satellite Applications Catapult for providing the COSMO-SkyMed datasets.