Compound Gaussian Model Research Articles

Compound-Gaussian Model with Nakagami-Distributed Textures for High-Resolution Sea Clutter at Medium/High Grazing Angles

In this paper, a compound-Gaussian model (CGM) with the Nakagami-distributed textures (CGNG) is proposed to model sea clutter at medium/high grazing angles. The corresponding amplitude distributions are referred to as the CGNG distributions. The analysis of measured data shows that the CGNG distributions can provide better goodness-of-the-fit to sea clutter at medium/high grazing angles than the four types of commonly used biparametric distributions. As a new type of amplitude distribution, its parameter estimation is important for modelling sea clutter. The estimators from the method of moments (MoM) and the [zlog(z)] estimator from the method of generalized moments are first given for the CGNG distributions. However, these estimators are sensitive to sporadic outliers of large amplitude in the data. As the second contribution of the paper, outlier-robust tri-percentile estimators of the CGNG distributions are proposed. Moreover, experimental results using simulated and measured sea clutter data are reported to show the suitability of the CGNG amplitude distributions and outlier-robustness of the proposed tri-percentile estimators.

An Improved Compound Gaussian Model for Bivariate Surface EMG Signals Related to Strength Training

An Improved Compound Gaussian Model for Bivariate Surface EMG Signals Related to Strength Training

Suzuki Distributed Monostatic and Bistatic S-Band Radar Sea Clutter

Low grazing angle S-band radar sea clutter data, collected with the NetRAD multistatic radar system, are analyzed. The Suzuki distribution, a compound Gaussian model with log normal texture, is found to be a good model for both monostatic and bistatic clutter. Clutter textures from simultaneously recorded monostatic and bistatic data are interpreted using hydrodynamics, providing estimates of the water depth and wave direction.

Dual-Polarimetric Persymmetric Adaptive Subspace Detector for Range-Spread Targets in Heavy-Tailed Non-Gaussian Clutter

To solve the problem of range-spread targets detection in non-Gaussian clutter for dual-polarimetric radar systems, a novel dual-polarimetric persymmetric detector based on the two-step generalized likelihood ratio test (GLRT) is proposed. Firstly, the target signal is modeled as a multi-rank subspace signal, and the clutter is modeled as a compound Gaussian model with deterministic unknown texture. Then, by taking polarimetric and persymmetric characteristics of the clutter into account, the detection performance of the proposed detector is obviously improved. Finally, through the verification via both simulated data and measured data, it is shown that the proposed detector poses better detection performance in comparison with the existing non-polarimetric detectors. Due to the complexity of the detector form, the analytical expression of false alarm probability cannot be derived. Therefore, we verified the constant false alarm ratio (CFAR) characteristic of the proposed detector through simulation experiment.

An Empirical Model of Shape Parameter of Sea Clutter Based on X-Band Island-Based Radar Database

Sea clutter modelling and parameter estimation/prediction are important basis of system design, performance assessment, and target detection of maritime radars. Besides parameter estimation from measured data, parameter prediction from radar parameters, sea state, and the viewing geometry of the radar is an alternative way. In this letter, a dual-polarimetric full-recorded sea clutter database measured by an island-based X-band experimental radar on the Yellow sea of China is introduced. On this database, the model selection reveals that the compound-Gaussian model with inverse Gaussian (CGIG) texture is suited for HH high-resolution sea clutter data, the generalized Pareto (GP) intensity distributions are for HH moderate-resolution data, and the K amplitude distributions are for high and moderate-resolution VV polarized data. Further, three five-parametric empirical formulae are constructed to predict the shape parameter of the CGIG, GP, and K distributions of sea clutter from the area of spatial resolution cell, the grazing angle, the significant wave height, the average wave period, and the wave direction relative to the sight line of radar. The new empirical formulae attain more accurate prediction than existing empirical formula on the database.

Outlier-robust truncated maximum likelihood parameter estimators of generalized Pareto distributions

Compound-Gaussian model with inverse Gamma-distributed texture is widely used to characterize high-resolution sea clutter and is also referred to as the generalized Pareto (GP) distributions. In real oceanic environments, sea clutter data are often corrupted by a small number of outliers of very high intensity, which are probably from radar returns of ships or reefs. Thus, it is necessary to develop outlier-robust parameter estimators of the amplitude/intensity models of sea clutter for maritime radars. This paper proposes outlier-robust truncated maximum likelihood estimators of the GP distributions. To mitigate the influence of outliers on parameter estimation, the first k largest samples are deleted from the data to generate the right truncated data. The truncated likelihood function (TLF) of the two parameters is derived and the truncated maximum likelihood (TML) equations are given. An iterative algorithm is presented to fast attain the solution of the TML equations and outlier-robust TML estimators are developed, which are a generalization of the ML estimator in the case with outlier in data. The TML estimators are compared with outlier-sensitive estimators and existing outlier-robust estimators. Without outliers in data, the TML estimators have comparable performance with the best ML estimator. With outliers in data, the TML estimators attain higher precision than the existing outlier-robust estimators.

Persymmetric Adaptive Target Detection With Dual-Polarization in Compound Gaussian Sea Clutter With Inverse Gamma Texture

This paper investigates the dual-polarimetric target detection problem in non-Gaussian sea clutter. The compound Gaussian model with inverse Gamma texture is adopted to describe the real sea clutter of which characteristics are time-varying and heterogeneous. Three novel polarimetric detectors are proposed based on the two-step maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> (MAP) generalized likelihood ratio test (GLRT), MAP Rao, and MAP Wald criteria. Specifically, we assume the polarimetric clutter covariance matrix (PCCM) and the inverse Gamma textures are known in the first step, and the test statistics of the proposed polarimetric detectors are derived. In the second step, we take advantage of the proposed polarimetric persymmetric property and MAP criterion to estimate the PCCM and inverse Gamma textures, respectively. The fully adaptive persymmetric polarimetric detectors are obtained by substituting the estimates into the test statistics of the proposed detectors. Then the proposed detectors are proved to have the constant false alarm rate (CFAR) properties with respect to the real PCCM. The performance assessments are evaluated by contrasting the proposed detectors with their counterparts on the simulated and measured sea clutter data. The numerical results verify the performance of the proposed persymmetric polarimetric detectors.

Adaptive Detection of Radar Targets in Heavy-Tailed Sea Clutter With Lognormal Texture

This article deals with the problem of detecting a marine target with coherent radars in a correlated heavy-tailed sea clutter background. The heavy-tailed sea clutter is modeled by a compound-Gaussian model, and the clutter texture is characterized by the lognormal distribution with a new parameterization form. We develop an adaptive coherent detector on the basis of the two-step generalized likelihood ratio test. The proposed detector can achieve adaptation to sea clutter characteristics by using the maximum a posterior estimate of the clutter texture, the constrained approximate maximum likelihood estimator of the speckle covariance matrix, and the proposed negative- and positive-fractional moment estimate of amplitude parameters of sea clutter. Remarkably, the proposed detector inherently ensures a constant false alarm rate with respect to the clutter power mean and the speckle covariance matrix. Finally, numerical experiments using simulated data and real radar data demonstrate that the proposed estimator and adaptive coherent detector outperform their respective competitors.

Outlier-robust tri-percentile parameter estimation of compound-Gaussian clutter with lognormal distributed texture

Compound-Gaussian model with lognormal distributed texture (CG-LNT) is one of effective amplitude distribution models to characterize high-resolution sea clutter and therefore it is important for high-resolution maritime radars to robustly estimate the parameters of the CG-LNT model from raw radar data, a mixture of sea clutter data and a small number of outliers. Existing estimators, such as moment-based and [zlog(z)]-based ones, are sensitive to outliers and fail to use in practical radars. In this paper, an outlier-robust tri-percentile estimator is proposed to realize the robust parameter estimation of the CG-LNT model of sea clutter. It is shown that the ratio of two percentiles is a monotonic function of the shape parameter and independent of the scale parameter, though without analytical expressions. In this way, the shape parameter is estimated from the ratio of the two sample percentiles by the look-up table method. Later, the scale parameter is estimated by the third sample percentile that is determined by the estimated shape parameter. Moreover, it is found that the positions of the three percentiles affect the performance and thus two empirical formulae are given to improve the estimate precision by numerical computation. Finally, the experimental results using simulated and real sea clutter data show that the proposed tri-percentile estimator implements robust and fast estimation of the parameters of the CG-LNT distribution.

Bi-Phase Compound-Gaussian Mixture Model of Sea Clutter and Scene-Segmentation-Based Target Detection

In high-resolution maritime surveillance radars, sea clutter exhibits highly spatial heterogeneity due to modulation of long waves with wavelengths longer than the width of one range cell. Compound-Gaussian model (CGM) fails to characterize the heterogeneous high-resolution sea clutter in both amplitude distribution and Doppler spectrum. In this article, a bi-phase compound-Gaussian mixture model (BP-CGMM) is proposed to characterize the heterogeneous sea clutter. In the BP-CGMM, spatial resolution cells are grouped into two disjoint sets, and the sea clutter in each set is represented by one CGM with inverse Gamma-distributed texture. The spectral heterogeneity indicates that sea clutter vectors at spatially adjacent resolution cells in one set share the same speckle covariance matrix, while that at two adjacent spatial cells separated in the two sets often have different speckle covariance matrices. The BP-CGMM is validated by a mass of measured high-resolution sea clutter data. Moreover, under the BP-CGMM, a detection method based on batch test is given to detect sea-surface small targets, which is composed of scene segmentation, by the aid of Bayesian threshold and morphological filtering, and adaptive generalized likelihood ratio test linear-threshold detector (GLRT-LTD) separately in each set. The detection method is verified by measured data with small targets under test. The experimental results show that it attains better detection performance than the adaptive GLRT-LTD under the CGM of sea clutter.

A GLRT-Based Polarimetric Detector for Sea-Surface Weak Target Detection

In this letter, different polarization characteristics of the sea clutter and target are considered to address the sea-surface weak target detection problem. We first construct a logarithmic generalized likelihood ratio test (log-GLRT) problem under the compound Gaussian model by considering multiple texture factors of sea clutter and four polarized channels. Based on the log-GLRT framework, a multivariable optimization problem is formulated to obtain the characteristic parameters of sea clutter and target. Then, under the block majorization-minimization (block-MM) framework, three parameters’ estimation strategies are designed to solve the optimization problem of the proposed parameters. Finally, a log-GLRT-based polarimetric detector is proposed by applying the estimated characteristic parameters. The experimental results on the Intelligent PIxel processing Xband (IPIX) data sets demonstrate that our proposed detector can achieve better detection performance than the several existed detectors.

Two-Dimensional Robust Source Localization Under Non-Gaussian Noise

Various methods have been proposed to estimate the direction of arrival (DOA) of sources under the assumption of Gaussian noise. This assumption, based on the central limit theorem, has been mainly used because it offers an appropriate model in a homogeneous environment. Nevertheless, under certain conditions, the Gaussian hypothesis cannot fully represent the noisy environment. Consequently, these classical estimation methods are no longer suitable, and the use of non-Gaussian noise model is necessary. In this paper, we therefore treat the DOA estimation problem in a non-Gaussian framework based on the maximum likelihood approach using a compound-Gaussian (CG) noise model. We then propose the use of the expectation maximization (EM) algorithm to reduce the computational cost of the proposed algorithm. Several simulations are carried out to illustrate the interest of our approach compared to the state of the art.

A Novel Gamma Distributed Random Variable (RV) Generation Method for Clutter Simulation with Non-Integral Shape Parameters.

Sea clutter simulation is a well-known research endeavour in radar detector analysis and design, and many approaches to it have been proposed in recent years, among which zero memory non-linear (ZMNL) and spherically invariant random process (SIRP) are the most two widely used methods for compound Gaussian distribution. However, the shape parameter of the compound Gaussian clutter model cannot be a non-integer nor non-semi-integer in the ZMNL method, and the computational complexity of the SIRP method is very high because of the complex non-linear operation. Although some improved methods have been proposed to solve the problem, the fitting degree of these methods is not high because of the introduction of Beta distribution. To overcome these disadvantages, a novel Gamma distributed random variable (RV) generation method for clutter simulation is proposed in this paper. In our method, Gamma RV with non-integral or non-semi-integral shape parameters is generated directly by multiplying an integral-shape-parameter Gamma RV with a Beta RV whose parameters are larger than 0.5, thus avoiding the deviation of simulation of Beta RV. A large number of simulation experimental results show that the proposed method not only can be used in the clutter simulation with a non-integer or non-semi-integer shape parameter value, but also has higher fitting degree than the existing methods.

Compound G2 model of coherent radar sea clutter

A two-component compound Gaussian model is proposed for coherent radar sea clutter, with each component having its own spatial correlation. The slow component, centred near zero Doppler, is modulated in both frequency and intensity by gravity waves, while the fast component, offset in Doppler frequency, models scattering from breaking waves. The intensity of each component is represented by an independent gamma variate with its own scale and shape parameter, with the overall texture distributed as the sum of these two gamma variates, denoted G2. The model is compared to data collected at medium grazing angles with the DST Ingara airborne X-band radar. It can produce realistic simulations of coherent radar sea clutter.

Adaptive target detection against spatially correlated compound‐Gaussian clutter with multivariate inverse Gaussian texture

To improve the detection performance in the presence of target-steering vector mismatches, a novel robust adaptive matched filter (AMF) detector for compound Gaussian clutter is proposed. First, the multivariate inverse Gaussian distribution is first introduced to compound Gaussian clutter model. Second, the texture value is estimated with maximum a posteriori (MAP) estimator in spatial domain, which has a closed form and is not affected by the target's steering vector mismatch. A robust AMF detector is derived based on this estimation. Simulation results demonstrate that the proposed detector can achieve better performance in the presence of target-steering vector mismatches.

Model for Non-Gaussian Sea Clutter Amplitudes Using Generalized Inverse Gaussian Texture

In this letter, we focus on the statistical modeling of sea clutter amplitudes. Due to its non-Gaussian nature, the existing statistical models are sometimes difficult to represent well the heavy-tailed portion of amplitude distribution. To address this problem, we propose a compound Gaussian (CG) model with a generalized inverse Gaussian (GIG) texture to describe sea clutter amplitudes. In this regard, the probability density function and the cumulative distribution function of the clutter amplitudes for the proposed model are derived. Moreover, we provide an approach to estimate the unknown parameters of the proposed CG-GIG distribution. The experimental results indicate that the CG-GIG distribution is more suitable to describe the amplitudes of non-Gaussian sea clutter than its competitors.

Outlier-robust parameters estimation for compound-Gaussian Clutter using inverse gamma texture based on truncated moments

ABSTRACTIn this letter, a family of estimators based on truncated moments are proposed to estimate the parameters of compound-Gaussian model with inverse gamma texture. The truncated-moments estimation method generalizes the previous bi-percentile estimators, and the proposed estimator is robust to outliers. The experiments based on simulated and real sea clutter data show that the proposed estimator outperforms the available estimators when the samples contain a small number of outliers.

Explicit bipercentile parameter estimation of compound‐Gaussian clutter with inverse gamma distributed texture

Compound-Gaussian model with inverse Gamma distributed texture is a powerful tool to characterise heavy-tailed sea clutter. The parameters of the model need to be estimated in advance or online to assist the optimum detection. Moment-based estimators are suitable for a limited range of shape parameter, the maximum likelihood (ML) estimator lacks an explicit expression and fast iterative solution, and the method of log-cumulants (MoLC) can work in all cases but is not good for large shape parameters. Moreover, their performance is sensitive to outliers in samples. Explicit bipercentile estimators are proposed to estimate the scale and shape parameters, which are computationally efficient and outliers robust. Without outliers in samples, the bipercentile estimators are comparable with the moment-based estimators and MoLC estimator in performance except when the number of samples is very small. With outliers in samples, the bipercentile estimators are much better than the moment-based and ML estimators. Moreover, the ability of the bipercentile estimators is verified by measured sea clutter data.

Closed‐form estimators of CGIG distributed parameters

The interpolation method has been employed in non-integer-order moments estimators obtained in a previous work for the parameters of compound-Gaussian clutter model with inverse Gaussian texture (CGIG). Closed-form parameter estimators for the CGIG distributed clutter are derived with no special functions. These are achieved after a simple combination of three basic expressions: non-integer positive moments, non-integer negative moments and Bessel recurrence relation. Simulation results show that the derived estimators are well suited for multilook data and have a comparable accuracy as well as the numerically solved maximum-likelihood approach, but they are much simpler and faster to compute

On the Amplitude Distributions of Bistatic Scattered Fields From Rough Surfaces

Non-Rayleigh distributed radar clutter is widely reported in studies of radar scattering from sea and land surfaces. Existing models of scattered field amplitude distributions have been developed primarily through empirical fits to the statistics of radar backscatter measurements. In contrast, this paper investigates a physics-based approach to determine the amplitude distributions of fields scattered from rough surfaces using Monte Carlo simulations and analytical methods, for both backscattering and bistatic configurations. The rough surface is represented using a “two-scale” model. An individual surface facet contains “small-scale” roughness, for which scattered fields are evaluated using the second-order small slope approximation. Individual surface facets are tilted by the slopes of the “large-scale” roughness in a given observation. The results show that non-Rayleigh amplitude distributions are obtained when tilting is performed, and that the departure from the Rayleigh distribution becomes more significant as the variance of the tilting slope increases. Further analysis shows that this departure results from variations in the mean scattering amplitude from a facet (the texture) as tilting occurs. The distribution of the texture is studied and compared with existing models. Finally, the distribution of the scattered field amplitude is modeled through the compound Gaussian model, first using the distribution of the texture, and then in terms of the probability density function of tilting slopes (which avoids the requirement of the knowledge of the texture distribution). The results from the above two methods are in good agreement and both agree well with the Monte Carlo simulation.