A robust estimator of parameters for \( {\mathcal{G}}_I^0 \)modeled SAR imagery based on random weighting method
 CuiHuan Wang^{1, 2}Email authorView ORCID ID profile,
 XianBin Wen^{1, 2} and
 HaiXia Xu^{1, 2}
DOI: 10.1186/s1363401704525
© The Author(s). 2017
Received: 9 October 2016
Accepted: 6 February 2017
Published: 27 February 2017
Abstract
In monopolarized synthetic aperture radar (SAR) imagery, \( {\mathcal{G}}_I^0 \) distribution often is assumed as the universal model to characterize a large number of targets, which is indexed by three parameters: the number of looks, the scale parameter, and the roughness parameter. The latter is closely related to the number of elementary backscatters in each pixel, and it is the reason why so many researchers focus on it. Although many efforts have been paid on providing many estimates, numerical problems often exist in dependable estimation, such as ‘outlier’ and small samples and so on. Thus, a robust estimation scheme of two unknown parameters in \( {\mathcal{G}}_I^0 \) distribution based on random weighting method is proposed in this paper where the relationship between moments and parameters are utilized. Experimental results on SAR computational simulations data and real SAR images show that the particular scheme outperforms alternative forms of bias reduction mechanisms, and we can obtain more accurate estimation than that of other stateoftheart algorithms.
Keywords
Synthetic aperture radar (SAR) Random weighting method Parameter estimation Robust1 Introduction
In order to interpret synthetic aperture radar (SAR) images, the statistical modeling often be employed. In fact, the multiplicative model using \( \mathcal{G} \) family of distributions often is utilized to describe the speckled data, which is characterized by three parameters: the number of looks, the scale parameter, and the roughness parameter. By this distribution, it is able to describe rough and extremely rough areas, which is better than other distributions [1, 2]. Under the \( \mathcal{G} \) model, regions with different degree of roughness can be characterized by the parameters, thus, the accuracy of the estimation of these parameters becomes very important.
Several kinds of approaches for estimating roughness parameters with different number of looks have been proposed. For example, Gambini et al. gave an analogy estimator based on moments of order 1/2 and 1 [3, 4]. Vasconcellos et al. proposed an analytic change for improving performance with respect to bias and meansquared error, and the bias in the estimation of the roughness parameter of the \( {\mathcal{G}}_I^0 \) distribution by maximum likelihood (ML) was quantified [5]. In fact, the processing and understanding of SAR image is the problem of small samples, for instance, image filtering where with a few observations within a window a new value is computed. The parameter estimation with small samples is subjected to many problems, mainly including bias, large variance, and sensitivity to deviations from the hypothesized model. On the one hand, \( {\mathcal{G}}_I^0 \) distribution is heavytailed distributions. Hence, dealing with SAR data is essentially difficult because samples from the tail of the distribution will have a strong influence on parameter estimation, and bias will be introduced if we decline the weights of them [6]. Therefore, a common issue in all the aforementioned estimation procedures, including ML, and those based on fractional moments and logcumulants [7] is the need of iterative algorithms for which there is no granted convergence to global solutions. Frery et al. [8] and Pianto and CribariNeto [9] proposed techniques to aim at alleviating such issue at the cost of additional computational load.
Random weighting, proposed by Zheng (1987) [10], is an emerging computational method in statistics, and has been used to solve different problems [11]. The random weighting method has following advantages: (1) it is sample in computation; (2) it does not require the previous knowledge on the distribution, and the estimation results are unbiased; (3) the estimation error of the random weighting method is smaller than that of Bootstrap in the case of small samples; (4) it is independent and identically distributed, and robust; (5) statistic determined by the random weighting method has the density function, so it is particularly suitable for the problem described in the density function. This paper is to develop a robust estimation method of combining analogy with random weighting method for the \( {\mathcal{G}}_I^0 \) model, which has the good properties of unbiased, the small meansquared error and its ability to resist contamination. Even in small samples and low computational cost, its performance is still very robust.
The paper is organized as follows: Section II presents the random weighting estimation for parameters of \( {\mathcal{G}}_I^0 \) distribution. In Section III, we will present and discuss the main numerical results. Finally, Conclusions and future work are presented in Section IV.
Random weighting estimation for \( {\mathcal{G}}_I^0 \) distribution
1.1 The \( {\mathcal{G}}_I^0 \) model
where − α > 0 is the roughness parameter, γ > 0 is the scale parameter and L ≥ 1 is the number of looks [1].
The \( {\mathcal{G}}_I^0 \) distribution is very attractive for modeling data with speckle noise, due to its mathematical tractability and ability to describe information from most types of areas, for given α < − 1 and L. These densities are presented in semilogarithmic scale, showing that they have heavy (linear) tails with respect to the Gaussian distribution which displays quadratic behavior. It is noticeable that the larger values of α, the larger the variances have; in fact, the variance is not finite when α ≥ − 1.
1.2 The random weighting estimator of parameter
where ξ _{1}, …, ξ _{ n } are independent and identically distributed random variable with distribution function of random variable ξ _{1} = η − 2, where η is from gamma distribution function G(4, 2), i.e., the density function of η is \( \varGamma (4)=\frac{2}{3!}{\left(2 x\right)}^3 \exp \left\{2 x\right\} I\left\{ x>0\right\} \), I(A) is the indicator function of set A, and X _{1}, …, X _{ n } and ξ _{1}, …, ξ _{ n } are mutually independent.
2 Performance evaluations
In this section, the random weighting estimation of \( {\mathcal{G}}_I^0 \) distribution parameters is comprehensively evaluated on SAR computational simulations data and real SAR images.
2.1 Simulations data and analysis
The results of double bounce in SAR system is that a high return value exists in some pixels. The presence of such outliers may provoke big errors in the estimation. Since robustness is the ability to perform well when the data obey the assumed model and to not provide completely useless results when the observations do not exactly follow it, moreover, estimators in SAR signal and image processing are always used in various robustness algorithms, thus, the robustness is of highest importance.

$$ \mathrm{Case}\ 1:\left(1\varepsilon \right){\mathcal{G}}_I^0\left({\alpha}_1,{\gamma}_1, L\right)+\varepsilon {\mathcal{G}}_I^0\left({\alpha}_2,{\gamma}_2, L\right) $$

$$ \mathrm{Case}\ 2:\left(1\varepsilon \right){\mathcal{G}}_I^0\left({\alpha}_1,{\gamma}_1, L\right)+\varepsilon C $$

$$ \mathrm{Case}\ 3:\left(1\varepsilon \right){\mathcal{G}}_I^0\left({\alpha}_1,{\gamma}_1, L\right)+\varepsilon {\mathcal{G}}_I^0\left({\alpha}_1,{10}^{\mathrm{k}}{\gamma}_1, L\right) $$
From Figs. 1 and 2, we can observe that only three methods of five estimators are very close to the true value in mean, namely, \( {\widehat{\alpha}}_{\mathrm{ML}} \), \( {\widehat{\alpha}}_{\mathrm{ANA}} \) and \( {\widehat{\alpha}}_{\mathrm{RW}} \), and the farthest estimators from the true value are \( {\widehat{\alpha}}_{LC\mathrm{um}} \) and \( {\widehat{\alpha}}_{M\mathrm{om}12} \), when α = − 5. In most cases, all methods have very similar meansquared error in the lager sample data, thus, we cannot decide which one is the best. However, our proposed method is better than other four methods especially when L = 3 and L = 8.
Figures 3 and 4 show the influence of the perturbation in contaminated cases. As expected, it is more obvious. With the increase of the number of sample, the proposed estimation is further away from the contamination, and the meansquared errors of \( {\widehat{\alpha}}_{\mathrm{RW}} \) is smaller than that of \( {\widehat{\alpha}}_{\mathrm{ANA}} \), \( {\widehat{\alpha}}_{\mathrm{ML}} \), \( {\widehat{\alpha}}_{LC\mathrm{um}} \), \( {\widehat{\alpha}}_{M\mathrm{om}12} \) for L = 3, 8. But all methods do not have clear distinction for L = 1 except that \( {\widehat{\alpha}}_{\mathrm{RW}} \) is at least very competitive in the case of α = − 3.
From Figs. 5 and 6, our proposed estimator is approximately close to the true mean value, and its meansquare error is smaller than that of other methods, except for L = 1 and α = − 3. \( {\widehat{\alpha}}_{\mathrm{RW}} \) produces the closest estimates to the true value with reduced meansquared error from Figs. 7 and 8.
2.2 Real SAR image and analysis
Estimations of the α parameter using the samples shown in Fig. 9
Color  Size  \( {\widehat{\alpha}}_{RW} \)  \( {\widehat{\alpha}}_{ANA} \)  \( {\widehat{\alpha}}_{ML} \)  \( {\widehat{\alpha}}_{LC\mathrm{um}} \)  \( {\widehat{\alpha}}_{M\mathrm{om}12} \) 

Blue  25  −3.0821  −5.2631  −3.1776  −2.8362  −2.0477 
Red  30  −4.2572  −6.9932  −5.6990  −3.0953  −2.0067 
Estimations of the γ parameter using the samples shown in Fig. 9
Color  Size  \( {\widehat{\gamma}}_{RW} \)  \( {\widehat{\gamma}}_{ANA} \)  \( {\widehat{\gamma}}_{ML} \)  \( {\widehat{\gamma}}_{LC\mathrm{um}} \)  \( {\widehat{\gamma}}_{M\mathrm{om}12} \) 

Blue  25  2.0821  4.2631  2.1776  2.8362  1.0477 
Red  30  3.2572  5.9932  4.6990  2.0953  1.0067 
Sample p values of the K–S test with samples from the image in Fig. 9
p value  

Color  \( {\widehat{\alpha}}_{RW} \)  \( {\widehat{\alpha}}_{ANA} \)  \( {\widehat{\alpha}}_{ML} \)  \( {\widehat{\alpha}}_{LC\mathrm{um}} \)  \( {\widehat{\alpha}}_{M\mathrm{om}12} \) 
Blue  0.5298  0.3722  0.3649  0.2621  0.1541 
Red  0.7375  0.5315  0.1969  0.1290  0.1052 
3 Conclusions
A new estimator for the roughness parameter of the c = 100 distribution based on the random weighing method is proposed. Moreover, three models of contamination inspired in real situations are defined to assess the impact of outliers in the performance of the estimators. By the experiments and analysis under the contamination, we can observe that: (1) do not consider the intensity of the contamination, the bigger the number of looks, the smaller the percentage of no convergence. (2) Under the contamination cases, the convergence with the increase of the level of contamination or with the reduction of α. In conclusion, our proposed method is much closer to the real mean value, and its meansquared error is lower than that of other methods in the case of small samples.
Declarations
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (Grant No. 61472278 and 61102125) and the Natural Science Foundation of Tianjin (Grant No. 12JCYBJC10200).
Authors’ contributions
CHW contributed significantly to the conception of the study and performed the experiments. Moreover, the author analyzed the data and drafted the manuscript; XBW and HXX helped perform the analysis with constructive discussions and revised the manuscript; XBW and HXX approved the final version. All authors read and approved the final manuscript.
Authors’ information
CuiHuan Wang received the B.S. degree in Communication Engineering from North College of Beijing University of Chemical Technology, Hebei, China, in 2014.
Currently, she is pursuing her master’s degree in School of Computer and Communication Engineering at Tianjin University of Technology, Tianjin, China. Her research interests include synthetic aperture radar (SAR) image processing.
XianBin Wen received the master’s degree in Applied Mathematics and the PhD degree in Computer Application Technology both from Northwestern Polytechnical University, Xi’an, China, in 1992 and 2005, respectively.
Currently, he is a Professor with Tianjin University of Technology. His current research interests include remote sensing image understanding, pattern recognition, computer vision, intelligent computer, and statistical signal analysis.
HaiXia Xu received the B.S. and PhD degrees in Computer Application Technology from Northwestern Polytechnical University, Xi’an, China.
Currently, she is an associate professor with Tianjin University of Technology. Her research interests include remote sensing image understanding, pattern recognition, and computer vision.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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