Compound Gaussian Distribution Research Articles

Regularized maximum likelihood estimation for radio interferometric imaging in the presence of radiofrequency interferences

We consider a regularized Maximum Likelihood Estimation (MLE) framework to produce images in the context of radio interferometric measurements. Specifically, we consider the class of compound Gaussian distributions to model the additive noise in the presence of radiofrequency interferences. In most cases, direct maximization of the likelihood is not tractable. To overcome this issue, we propose a generic expectation–maximization (EM) algorithm in the presence of a compound Gaussian noise. In addition, we leverage an approximation of the forward radio interferometric operator to derive an original latent data space that allows the use of the FFT in the maximization step, leading to an accelerated extension of the proposed imaging algorithm. The proposed approaches are evaluated on simulated and real data and show a significant improvement in the robustness to the presence of radiofrequency interferences (RFI) in the measurement.

A new OSL dose model to account for post-depositional mixing of sediments

In applications of optically stimulated luminescence (OSL) dating to unconsolidated sediments, the burial age of a sample of grains is estimated using statistical models of the distribution of the experimentally determined equivalent doses of the grains, together with estimates of the environmental dose rate. For grains that have been vertically mixed after deposition (e.g., due to bioturbation), existing dose models may fail to appropriately account for the complexity of the mixing process, thus producing inaccurate age estimates of the original time of deposition of the ‘native’ grains in any particular sample (usually the quantity of most interest). Here we introduce a new dose model, the asymmetric Laplacian mixture model (ALMM), developed for vertically mixed samples with single-grain dose distributions. The approach is based on a continuous statistical mixture that models the displacement of grains in both upward and downward directions. The central dose of the native grains in each sediment sample is estimated by the ALMM, as well as the parameters associated with overdispersion of single-grain dose distributions and the (modelled) mixing process. Using Bayesian methodology, we apply the model to two series of vertically contiguous samples collected at the site of Nawarla Gabarnmang in northern Australia. Independent age estimates obtained from radiocarbon dating of charcoal fragments support the OSL ages for the native grains estimated by the ALMM. Moreover, our study includes sensitivity analyses that show the model is robust to variation in the experimental error of the OSL data, as well as a simulation study that demonstrates the model’s good ability to recover the simulated central dose and its excellent coverage properties. The ALMM is introduced in the context of compound Gaussian distributions, a broadly encompassing statistical framework that includes many of the most commonly used dose models. This unifying and accessible perspective on the statistical modelling of dose distributions will support practitioners in selecting an appropriate model for samples affected by post-depositional mixing, and hopefully stimulate further theoretical developments. A new R package rstanosl is provided that fits the ALMM and other commonly used dose models using Hamiltonian Monte Carlo methods via the Stan programming language.

Statistical Analysis of Sea-Clutter using K-Pareto, K-CGIG, and Pareto-CGIG Combination Models with Noise

In this paper, the combinations of two compound Gaussian distributions plus thermal noise for modeling measured polarimetric clutter data are proposed. The speckle components of the proposed models are formed by the exponential distribution, while the texture components are mainly modeled using three different distributions. For this purpose, the gamma, the inverse gamma, and the inverse Gaussian distributions are considered to describe these modulation components. The study involves the analysis of underlying mixture models at X-band sea clutter data, and the parameters of the combination models are estimated using the non-linear least squares curve fitting method. Compared to existing K, Pareto type II, and KK clutter plus noise distributions, experimental results show that the proposed mixture models are well matched for fitting sea reverberation data across various range resolutions.

Outlier-Robust Truncated Maximum Likelihood Parameter Estimation of Compound-Gaussian Clutter with Inverse Gaussian Texture

Compound-Gaussian distributions with inverse Gaussian textures, referred to as the IGCG distributions, are often used to model moderate/high-resolution sea clutter in amplitude. In moderate/high-resolution maritime radars, parameter estimation of the IGCG distributions from radar returns data plays an important role in adaptive target detection. Due to the inevitable existence of outliers of high amplitude in radar returns data from targets and reefs, parameter estimation must be outlier robust. In this paper, an outlier-robust truncated maximum likelihood (TML) estimation method is proposed to mitigate the effect of outliers of high amplitude in data. The data are first transferred into the truncated data by removing a given percentage of the largest samples in amplitude. From the truncated data, the truncated likelihood function is constructed, and its maximum corresponds to the TML estimates of the scale and inverse shape parameters. Further, an iterative algorithm is presented to obtain the TML estimates from data with outliers, which is an extension of the ML estimation method in the case that data contain outliers. In comparison with outlier-sensitive estimation methods and outlier-robust bipercentile estimation methods, the performance of the TML estimation method is close to that of the best ML estimation method in the case that data are without outlier, and it is better in the case that data are with outliers.

Polarimetric Target Detection in Compound Gaussian Sea Clutter With Inverse Gaussian Texture

This letter addresses the polarimetric target detection problem in compound Gaussian sea clutter. In view of the heavy-tailed characteristic of sea clutter, we model the sea clutter as compound Gaussian distribution with inverse Gaussian texture (IGCG). We propose three polarimetric compound Gaussian detectors based on the two-step Rao test, the Wald test, and the generalized likelihood ratio test (GLRT). Specifically, we assume that the polarimetric clutter covariance matrix (PCCM) and the inverse Gaussian textures are known in the first step, and the test statistics of the proposed polarimetric detectors are derived. In the second step, we use the training data to estimate PCCM and maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> (MAP) to estimate the inverse Gaussian textures, and we obtain the fully adaptive detectors. Then, we give proof of the constant false alarm rate (CFAR) properties of the proposed detectors. We validate the performance of the proposed polarimetric detectors by conducting experiments based on simulated and real sea clutter data. Finally, the numerical results indicate that the proposed detectors exhibit better detection performance than their competitors and are robust when the mismatched signal occurs.

GLRT-Based Polarimetric Detection in Compound-Gaussian Sea Clutter With Inverse-Gaussian Texture

This letter presents the derivation of the generalized likelihood ratio test (GLRT)-based polarimetric detector in the non-Gaussian clutter. We model the non-Gaussian sea clutter as compound-Gaussian distribution with inverse-Gaussian texture (IG-CG), which has better goodness-of-fit for the high-resolution sea clutter. Based on the two-step GLRT criterion, we develop the test statistic of the proposed detector by assuming the texture component is known in the first step. In the second step, with the texture of the secondary data estimated as the power of the clutter, we insert the maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> estimate (MAPE) of the texture of the primary data into the test statistic to achieve the fully adaptive detection. Furthermore, the model-based polarimetric detector, which exploits the independence between the co-polarized and the cross-polarized component, is derived. Finally, we conduct experiments with simulated clutter and real sea clutter to evaluate the performance of the proposed detector. The simulation results verify that the proposed detectors have better detection performance and the MAPE under different hypotheses leads to different levels of robustness to the mismatched signal.

Singular Non-circular Complex Elliptically Symmetric Distributions: New Results and Applications

Absolutely Continuous non-singular complex elliptically symmetric distributions (referred to as the nonsingular CES distributions) have been extensively studied in various applications under the assumption of nonsingularity of the scatter matrix for which the probability density functions (p.d.f's) exist. These p.d.f's, however, can not be used to characterize the CES distributions with a singular scatter matrix (referred to as the singular CES distributions). This paper presents a generalization of the singular real elliptically symmetric (RES) distributions studied by Díaz-García et al. to singular CES distributions. An explicit expression of the p.d.f of a multivariate non-circular complex random vector with singular CES distribution is derived. The stochastic representation of the singular non-circular CES (NC-CES) distributions and the quadratic forms in NC-CES random vector are proved. As special cases, explicit expressions for the p.d.f's of multivariate complex random vectors with singular non-circular complex normal (NC-CN) and singular non-circular complex Compound-Gaussian (NC-CCG) distributions are also derived. Some useful properties of singular NC-CES distributions and their conditional distributions are derived. Based on these results, the p.d.f's of non-circular complex t-distribution, K-distribution, and generalized Gaussian distribution under singularity are presented. These general results degenerate to those of singular circular CES (C-CES) distributions when the pseudo-scatter matrix is equal to the zero matrix. Finally, these results are applied to the problem of estimating the parameters of a complex-valued non-circular multivariate linear model in the presence either of singular NC-CES or C-CES distributed noise terms by proposing widely linear estimators

PERFORMANCE OF QAM-OFDM SYSTEM OVER RAYLEIGH FADING CHANNEL IN WEI BULL NOISE DISTRIBUTION

The phenomenal increase in the usage of mobile devices and wireless networking tools in recent years has resulted in the communication industry needing higher data speeds for connections and bandwidth. As a result, multi-carrier modulation has been suggested as a reliable and effective method of transmitting data over difficult communication networks such as selective fading channels. Orthogonal Frequency Division Multiplexing (OFDM) is a highly effective multi-carrier technique that can meet users' high demands. Many studies have looked into this technique, mostly as a way to counteract fading and Additive White Gaussian Noise (AWGN). As a result, the performance evaluation of the QAM-OFDM system in the presence of multi-path Rayleigh fading in Weibull noise is examined in this article. Furthermore, bit error rate performance (BER) is computed using the optimal derivation of the real system contaminated by compound Gaussian and non-Gaussian (Weibull) noise distributions at the OFDM demodulator output. The derived result is an exact match to the simulated result over various scenarios introduced by the MATLAB software package.

Simulation and comparison of distributed target detectors in compound Gaussian background

In the non Gaussian background modelled by compound Gaussian distribution, a variety of distributed target detectors are addressed. Firstly, the detection statistics of different detectors are given. Then, the detection performance of five detectors is compared when the non Gaussian clutter covariance matrix is known. Secondly, when the non Gaussian clutter covariance matrix is unknown, a two-step architecture is adopted. Based on the same estimation method for non-Gaussian clutter covariance matrix structure, seven adaptive distributed target detectors are introduced and compared. Finally, through comparative analyses, the applicable conditions and application range of different detectors are summarized, which provide a theoretical reference for the practical application of radar target detection.

Adaptive Waveform Design with Multipath Exploitation Radar in Heterogeneous Environments

The problem of detecting point like targets over a glistening surface is investigated in this manuscript, and the design of an optimal waveform through a two-step process for a multipath exploitation radar is proposed. In the first step, a non-adaptive waveform is transmitted and a constrained Generalized Likelihood Ratio Test (GLRT) detector is deduced at reception which exploits multipath returns in the range cell under test by modelling the target echo as a superposition of the direct plus the multipath returns. Under the hypothesis of heterogeneous environments, thus by assuming a compound-Gaussian distribution for the clutter return, this latter is estimated in the range cell under test through the secondary data, which are collected from the out-of-bin cells. The Fixed Point Estimate (FPE) algorithm is applied in the clutter estimation, then used to design the adaptive waveform for transmission in the second step of the algorithm, in order to suppress the clutter coming from the adjacent cells. The proposed GLRT is also used at the end of the second transmission for the final decision. Extensive performance evaluation of the proposed detector and adaptive waveform for various multipath scenarios is presented. The performance analysis prove that the proposed method improves the Signal-to-Clutter Ratio (SCR) of the received signal, and the detection performance with multipath exploitation.

Waveform Optimization for Multistatic Radar Imaging Using Mutual Information

This article addresses the problem of radar waveform design for imaging targets in a cluttered environment. A multistatic radar scenario is considered for sparse and random sensor positions. The target impulse response is modeled using the simulated frequency response of a buried metallic mine, and the clutter is modeled using the compound Gaussian (CG) distribution. We explore novel waveform design techniques with respect to the mutual information (MI) criterion based on the CG clutter. The first method presents a waveform that exploits the CG distribution of the scene reflectivity function when projected onto a sparse basis. This is compared to the second method, called the target-specific approach that uses knowledge of the target and clutter frequency responses to optimize a matched illumination waveform. In both cases, the Taguchi and particle swarm optimization solvers are employed for MI-based waveform design optimization. To validate and compare the effectiveness of the optimized waveforms, the resulting scene reflectivity function is estimated using the sparsity-driven regularization radar imaging method. Our experimental results demonstrate that both waveform optimization techniques result in significantly better image reconstruction performance than the traditional linear-frequency-modulated waveform, and that the target-specific approach additionally suppresses clutter information in the scene.

Riemannian geometry for compound Gaussian distributions: Application to recursive change detection

A new Riemannian geometry for the zero-mean Compound Gaussian distribution with deterministic textures is proposed. In particular, the Fisher information metric (up to a factor) is obtained, along with corresponding geodesics and distance function. This new geometry is applied on a change detection problem on Multivariate Image Times Series: a recursive approach based on Riemannian optimization is developed. As shown on simulated data, it allows to reach optimal performance while being computationally more efficient.

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.

Performance analysis of LMS filters with non-Gaussian cyclostationary signals

Abstract The least mean-square (LMS) filter is one of the most common adaptive linear estimation algorithms. In many practical scenarios, e.g., digital communications systems, the signal of interest (SOI) and the input signal are jointly wide-sense cyclostationary. Previous works analyzing the performance of LMS filters for this important case assume Gaussian probability distributions of the considered signals. In this work, we provide a transient and steady-state performance analysis that applies to non-Gaussian cyclostationary signals. In the considered analysis, the SOI is modeled as a perturbed response of a linear periodically time-varying system to the input signal. We obtain conditions for convergence and derive analytical expressions for the non-asymptotic and steady-state mean-squared error. We then show how the theoretical analysis can be effectively applied for two common non-Gaussian classes of input distributions, the class of elliptical compound Gaussian distributions, and the broad class of Gaussian mixture distributions. The accuracy of our analysis is illustrated for system identification and signal recovery scenarios that involve linear periodically time variant systems and non-Gaussian inputs.

Low Grazing Angle에서의 코히어런트 해상 클러터 시뮬레이션

The probability density function(PDF) for the amplitude of the reflectivity of low-grazing-angle sea clutter has generally been modeled by a compound-Gaussian distribution, rather than by the Rayleigh distribution, owing to the intensity variation of each clutter patch over time. The texture component forming the reflectivity has been simulated by combining Gamma distribution and memory-less nonlinear transformation(MNLT). On the other hand, there is no typical method available that can be used to simulate the speckle component. We first review Watt’s method, wherein the speckle is simulated starting from the Doppler spectrum of the received echoes that is modeled as having a Gaussian shape. Then, we introduce a newly proposed method. The proposed method simulates the speckle by manipulating a clutter covariance matrix through the Cholesky decomposition after minimizing the effect of adjacent clutter patches using an equalizer. The feasibility of the proposed method is validated through simulation, wherein the results from two methods are compared in terms of the Doppler spectrum and the correlation function.

Radar sea‐clutter modelling using fractional generalised Pareto distribution

A model for radar sea-clutter derived from the fractional differentiation of compound Gaussian distribution with inverse gamma texture is presented. Different stages of fractional calculus are conducted to obtain the new probability density function (PDF) and cumulative distribution function with fractional order. To investigate the fitting of the proposed statistical model to real data, the curve fitting estimation method based on the Nelder–Mead algorithm is used to optimise the shape parameter, the scale parameter and the fractional derivative order. Compared to the generalised Pareto model with the maximum likelihood estimator, it is shown that the proposed fractional PDF models well the tail of IPIX real data especially when medium range resolutions are at hand.

A Nonlinear ARMA-GARCH Model With Johnson $S_u$ Innovations and Its Application to Sea Clutter Modeling

In this paper, a novel time series heteroskedastic model is proposed for sea clutter modeling application. In the light of characteristics of the practical clutter at low grazing angle, the original generalized autoregressive conditional heteroskedasticity process, which has been widely used in various fields of economics, is extended from three aspects. First, the autoregressive moving-average terms are introduced for modeling the temporal correlation of both clutter returns and innovations. Second, the exponential of the conditional variance is generalized from one to arbitrary positive value, to capture the nonlinearity existing in the practical clutter. Third, the traditional Gaussian innovation is replaced by the Johnson $S_{u}$ random variable, which is a monotonic transformation of the Gaussian random variable and is capable of modeling the skewness and kurtosis. By systematically analyzing a large number of practical sea clutter data sets, we show that the proposed time series model fits the data better than some commonly used statistic-based distributions, such as the Weibull and compound Gaussian distributions.

Noncentral limit theorem for the generalized Hermite process

We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Hermite processes $Z_\gamma $ with kernels defined by parameters $\gamma $ taking values in a tetrahedral region $\Delta $ of $\mathbb{R} ^q$. We prove that, as $\gamma $ converges to a face of $\Delta $, the process $Z_\gamma $ converges to a compound Gaussian distribution with random variance given by the square of a Hermite process of one lower rank. The convergence in law is shown to be stable. This work generalizes a previous result of Bai and Taqqu, who proved the result in the case $q=2$ and without stability.

A hierarchical Bayesian-MAP approach to inverse problems in imaging

We present a novel approach to inverse problems in imaging based on a hierarchical Bayesian-MAP (HB-MAP) formulation. In this paper we specifically focus on the difficult and basic inverse problem of multi-sensor (tomographic) imaging wherein the source object of interest is viewed from multiple directions by independent sensors. Given the measurements recorded by these sensors, the problem is to reconstruct the image (of the object) with a high degree of fidelity. We employ a probabilistic graphical modeling extension of the compound Gaussian distribution as a global image prior into a hierarchical Bayesian inference procedure. Since the prior employed by our HB-MAP algorithm is general enough to subsume a wide class of priors including those typically employed in compressive sensing (CS) algorithms, HB-MAP algorithm offers a vehicle to extend the capabilities of current CS algorithms to include truly global priors. After rigorously deriving the regression algorithm for solving our inverse problem from first principles, we demonstrate the performance of the HB-MAP algorithm on Monte Carlo trials and on real empirical data (natural scenes). In all cases we find that our algorithm outperforms previous approaches in the literature including filtered back-projection and a variety of state-of-the-art CS algorithms. We conclude with directions of future research emanating from this work.

Tyler's Covariance Matrix Estimator in Elliptical Models With Convex Structure

We address structured covariance estimation in elliptical distributions by assuming that the covariance is a priori known to belong to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization applied to robust Tyler's scatter M-estimator subject to these convex constraints. Unfortunately, GMM turns out to be non-convex due to the objective. Instead, we propose a new COCA estimator-a convex relaxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the advantages of COCA in synthetic simulations with structured compound Gaussian distributions. In these examples, COCA outperforms competing methods such as Tyler's estimator and its projection onto the structure set.