Complex Wishart Distribution Research Articles

Underlying Topography Estimation over Forest Using Maximum a Posteriori Inversion with Spaceborne Polarimetric SAR Interferometry

This paper presents a method for extracting the digital elevation model (DEM) of forested areas from polarimetric interferometric synthetic aperture radar (PolInSAR) data. The method models the ground phase as a Von Mises distribution, with a mean of the topographic phase computed from an external DEM. By combining the prior distribution of the ground phase with the complex Wishart distribution of the observation covariance matrix, we derive the maximum a posterior (MAP) inversion method based on the RVoG model and analyze its Cramer–Rao Lower Bound (CRLB). Furthermore, considering the characteristics of the objective function, this paper introduces a Four-Step Optimization (FSO) method based on gradient optimization, which solves the inefficiency problem caused by exhaustive search in solving ground phase using the MAP method. The method is validated using spaceborne L-band repeat-pass SAOCOM data from a test forest area. The test results for FSO indicate that it is approximately 5.6 times faster than traditional methods without compromising accuracy. Simultaneously, the experimental results demonstrate that the method effectively solves the problem of elevation jumps in DEM inversion when modeling the ground phase with the Gaussian distribution. ICESAT-2 data are used to evaluate the accuracy of the inverted DEM, revealing that our method improves the root mean square error (RMSE) by about 23.6% compared to the traditional methods.

Parameter estimation and sensitivities of a Bayesian source localization method in an urban environment

Acoustic signals propagating in urban environments are influenced by rough-surface scattering, multipath reflections, and diffraction. The performance of conventional source localization algorithms often suffers when these effects are present. Bayesian approaches, however, are particularly well suited to incorporating physics-based statistical models for the signal propagation. Previously, we found that the complex Wishart distribution, which describes fully saturated scattered signals across a network of receivers, can be readily employed in a Bayesian framework. In this presentation, a new source localization algorithm based on the Wishart distribution signal model is described and tested on real acoustic data. The experimental data were collected within a mock urban environment as part of a NATO urban acoustics-seismics experiment in Walenstadt, Switzerland. Four acoustic arrays recorded signatures from gunshots and ground vehicles. The present study uses the measured signals to investigate the sensitivity and relative importance of the model parameters, including source and noise amplitudes, frequency band, prior signal sampling, and sensor-sensor correlation behavior. Results are presented in the form of source location probability maps and localization error metrics. Interpretations of the study, including strengths and weaknesses of the new method, are discussed.

Bayesian source localization in an urban environment using scattered signal distributions

Acoustic signals propagating in urban environments are influenced by rough-surface scattering, multipath reflections, and diffraction. Conventional source localization algorithms often perform poorly when these effects are present. Bayesian approaches, however, are particularly well suited to incorporating physics-based statistical models for the signal propagation. Previously, we found that the complex Wishart distribution, which describes fully saturated scattered signals across a network of receivers, can be readily employed in a Bayesian framework. This approach is very general, as it includes source triangulation and trilateration as special cases. Feasibility was initially demonstrated using simulations. In the present work, we describe an experimental demonstration of Bayesian source localization using data recorded in an urban-like environment. The experimental data were collected as part of a NATO urban acoustics-seismics experiment in Walenstadt, Switzerland, in May 2023. A network of four acoustic nodes, each with 12 microphones, was deployed and recorded emissions from a variety of sources. Initial results from the data processing and localization are described.

Knowledge-Aided Wideband Radar Target Detection in the Heterogeneous Clutter

In this paper, the problem of wideband radar target detection in the heterogeneous environment is addressed. The wideband radar return of target with range migration is characterized as a subband model, and the heterogeneous clutter is described with a hierarchical Bayesian model. Both the prior knowledge of the clutter power and the covariance matrix and the dependence of the primary data and the secondary data are characterized with the inverse gamma and the inverse complex Wishart distribution, respectively. Based on the target and the clutter models, knowledge-aided maximum posterior ratio test (KAMAPRT), knowledge-aided Rao (KARao) test and knowledge-aided Wald (KAWald) test for wideband radar target detection in heterogeneous clutter are proposed. Finally, the performance of the proposed detectors is evaluated by simulations with both the simulated clutter generated by the probability model and the synthesized clutter from a real synthetic aperture radar (SAR) complex image. The results show that the proposed knowledge-aided detectors are effective for wideband radar target detection in the heterogeneous clutter.

Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix

The complex Wishart distribution has ample applications in science and engineering. In this paper, we give explicit expressions for E(tr(Wh))g(tr(Wj))i and E(tr(W−h))g(tr(W−j))i, respectively, for particular values of g, h, i, j, g+h+i+j≤5, where W follows a complex Wishart distribution. For specific values of g, h, i, j, we first write (tr(Wh))g(tr(Wj))i and (tr(W−h))g(tr(W−j))i in terms of zonal polynomials and then by using results on integration evaluate resulting expressions. Several expected values of matrix-valued functions of a complex Wishart matrix have also been derived.

Fast Line Segment Detection and Large Scene Airport Detection for PolSAR

In this paper, we propose a fast Line Segment Detection algorithm for Polarimetric synthetic aperture radar (PolSAR) data (PLSD). We introduce the Constant False Alarm Rate (CFAR) edge detector to obtain the gradient map of the PolSAR image, which tests the equality of the covariance matrix using the test statistic in the complex Wishart distribution. A new filter configuration is applied here to save time. Then, the Statistical Region Merging (SRM) framework is utilized for the generation of line-support regions. As one of our main contributions, we propose a new Statistical Region Merging algorithm based on gradient Strength and Direction (SRMSD). It determines the merging predicate with consideration of both gradient strength and gradient direction. For the merging order, we set it by bucket sort based on the gradient strength. Furthermore, the pixels are restricted to belong to a unique region, making the algorithm linear in time cost. Finally, based on Markov chains and a contrario approach, the false alarm control of line segments is implemented. Moreover, a large scene airport detection method is designed based on the proposed line segment detection algorithm and scattering characteristics. The effectiveness and applicability of the two methods are demonstrated with PolSAR data provided by UAVSAR.

Shrinkage estimators of large covariance matrices with Toeplitz targets in array signal processing

The problem of estimating a large covariance matrix arises in various statistical applications. This paper develops new covariance matrix estimators based on shrinkage regularization. Individually, we consider two kinds of Toeplitz-structured target matrices as the data come from the complex Gaussian distribution. We derive the optimal tuning parameter under the mean squared error criterion in closed form by discovering the mathematical properties of the two target matrices. We get some vital moment properties of the complex Wishart distribution, then simplify the optimal tuning parameter. By unbiasedly estimating the unknown scalar quantities involved in the optimal tuning parameter, we propose two shrinkage estimators available in the large-dimensional setting. For verifying the performance of the proposed covariance matrix estimators, we provide some numerical simulations and applications to array signal processing compared to some existing estimators.

PolSAR Models with Multimodal Intensities

Polarimetric synthetic aperture radar (PolSAR) systems are an important remote sensing tool. Such systems can provide high spacial resolution images, but they are contaminated by an interference pattern called multidimensional speckle. This fact requires that PolSAR images receive specialised treatment; particularly, tailored models which are close to PolSAR physical formation are sought. In this paper, we propose two new matrix models which arise from applying the stochastic summation approach to PolSAR, called compound truncated Poisson complex Wishart (CTPCW) and compound geometric complex Wishart (CGCW) distributions. These models offer the unique ability to express multimodal data. Some of their mathematical properties are derived and discussed—characteristic function and Mellin-kind log-cumulants (MLCs). Moreover, maximum likelihood (ML) estimation procedures via expectation maximisation algorithm for CTPCW and CGCW parameters are furnished as well as MLC-based goodness-of-fit graphical tools. Monte Carlo experiment results indicate ML estimates perform at what is asymptotically expected (small bias and mean square error) even for small sample sizes. Finally, our proposals are employed to describe actual PolSAR images, presenting evidence that they can outperform other well-known distributions, such as WmC, Gm0, and Km.

Life-Cycle Assessment of Long-Span Bridge’s Wind Resistant Performance Considering Multisource Time-Variant Effects and Uncertainties

This paper examines the life-cycle wind resistant performance of a constructed long-span suspension bridge in the coastal region of China, aiming to quantify the multisource time-variant effects and uncertainties and offering a reference for designs of long-span bridges in the future. Randomness from modal frequencies, damping ratios, and identification uncertainty of flutter derivatives (FDs) was considered; then, their effects on probability of flutter failure and probability of exceeding the predefined buffeting response root-mean square (RMS) are discussed. Firstly, results of full-track tropical cyclone (TC) simulation under various climate warming scenarios are reviewed; then, the time-variant probability density function (PDF) of annual extreme wind speed is discussed. Secondly, 6-year modal frequencies and damping ratios of a long-span suspension bridge with a center-slotted section were extracted by fast Bayesian FFT method with structural health monitoring (SHM) data, which were utilized to explore the deterioration rules of structural properties. Thirdly, FDs were modeled from a probabilistic perspective based on complex Wishart distribution, which were identified in the turbulent flow and the frequency domain by Bayesian inference. The posterior distributions of FDs, namely identification uncertainty, were quantified by Markov chain Monte Carlo (MCMC) sampling. This paper finds that for flutter resistant performance, the time-variant effects (i.e., modal frequencies and PDFs of extreme wind speed) will make the flutter failure probability seven times larger than the initial value; for the probability of exceeding the predefined buffeting response RMS, however, the time-variant effects will make a negligible difference.

Bayesian approach to passive receiver localization in a randomly scattered sound field

A Bayesian approach is described for localizing sources and receivers in a randomly scattered sound field, for which the complex Wishart distribution represents the joint distribution of the signals at multiple, arbitrarily separated receivers. The signals are assumed to be narrowband and fully saturated by scattering from homogeneous turbulence (or similar scattering processes) as they propagate along line-of-sight paths to the receivers. Here, we consider a scenario in which there is an array of receivers at known locations, a source of opportunity with unknown amplitude at an unknown location, and a single receiver at an unknown location. The objective is to passively determine the unknown receiver location even though the source characteristics are not initially known. A slice-sampling strategy is found to be suitable for solving the problem. Simulations show that a small number of independent signal samples at the receivers initially enables the unknown receiver to be localized on a circle of constant amplitude around the source. With more samples, the correlation characteristics of the field enable full determination of the unknown receiver location.

Comparison of infrasound emissions from tornadic and non-tornadic storms

At a previous ASA meeting (Spring 2019), some results corresponding to application of maximum likelihood estimation with the complex Wishart distribution to direction-of-arrival of infrasound from tornadic storms were presented. In this presentation, updated results are presented and compared with estimation of direction-of-arrival of infrasound from two non-tornadic storms that were measured with the same infrasound arrays. One of these storms was similar to the tornadic storm in that it consisted of numerous isolated storm cells with frequent lightning. The other storm was significantly different in that it was a well-organized squall line that produced only limited lightning. The tornadic storm yielded very well-defined direction-of-arrival tracks in contrast to the non-tornadic storms.

Bayesian spectral density approach for identification of bridge section’s flutter derivatives operated in turbulent flow

This study presents a Bayesian spectral density approach for identification of flutter derivatives of bridge sections utilizing buffeting displacement responses, where the wind tunnel test is conducted in turbulent flow. Different from traditional time-domain approaches (e.g., least square method and stochastic subspace identification), the newly-proposed approach is operated in frequency domain. Based on the affine invariant ensemble sampler algorithm, Markov chain Monte-Carlo sampling is employed to accomplish the Bayesian inference. The probability density function of flutter derivatives is modeled based on complex Wishart distribution, where probability serves as the measure. By the Bayesian spectral density approach, the most probable values and corresponding posterior distributions (determined by conditions of measured signals) of each flutter derivative can be obtained at the same time. Firstly, numerical simulations are conducted and the identified results are accurate. Secondly, thin plate model, flutter derivatives of which have theoretical solutions, is chosen to be tested in turbulent flow for the sake of verification. The identified results of thin plate model are consistent with the theoretical solutions. Thirdly, the center-slotted girder model, which is widely-utilized long-span bridge sections in current engineering practice, is employed to investigate the applicability of the proposed approach on a general bridge section. For the center-slotted girder model, the flutter derivatives are also extracted by least square method in uniform flow to cross validate the newly-proposed approach. The identified results by two different approaches are compatible.

Bayesian Detection of Distributed Targets for FDA-MIMO Radar in Gaussian Interference

In this letter, we propose a frequency diverse array multiple-input multiple-output radar detection architecture for distributed targets embedded in Gaussian interference with unknown but stochastic covariance matrix. At the design stage, we model distributed targets within one range cell as a linear combination of several contributions and assume that the interference covariance matrix obeys the inverse complex Wishart distribution. Then, we devise an adaptive decision rule by jointly exploiting the maximum likelihood approach and the Bayesian framework. Unlike existing contributions in this context, the proposed detector does not require the conventional set of training data to estimate the interference covariance matrix. The numerical examples validate the effectiveness of the proposed method also in comparison with suitable counterparts.

Bayesian Detection for Radar Targets in Compound-Gaussian Sea Clutter

We consider the detection problem of maritime radar targets in the training-sample-starved and non-Gaussian sea clutter environment. The performance of conventional detectors for radar targets is seriously degraded due to both the starvation of training samples for estimating the clutter covariance matrix and the non-Gaussianity of sea clutter. In this letter, we adopt the inverse Gaussian distribution and the inverse complex Wishart distribution to model the texture and speckle covariance matrix of sea clutter, respectively. Then an adaptive Bayesian detector is developed based on the two-step generalized likelihood ratio test and the maximum posterior estimates of clutter parameters. Finally, the experimental results on simulated and measured data demonstrate the performance superiority of the proposed detector over its competitors, especially when the training samples are starved.

Discrimination of multi-crop scenarios with polarimetric SAR data using Wishart mixture model

Discrimination of crop varieties spanned over heterogeneous agriculture land is a vital application of polarimetric SAR images for agriculture monitoring and assessment. The covariance matrix of polarimetric SAR images is observed to follow a complex Wishart distribution for major classification tasks. It is true for homogeneous regions, but for heterogeneous regions, the covariance matrix follows a mixture of multiple Wishart distributions. We aim to improve the classification accuracy when the terrain under observation is heterogeneous. For this purpose, Wishart mixture model is employed along with expectation-maximization (EM) algorithm for parameter estimation. Elbow method helps us to devise the number of mixtures. The convergence of the EM algorithm depends on the choice of initial points. So, to improve the robustness of the model, different initialization approaches, such as random, K-means, and global K-means, are embedded in the EM algorithm. Further, the degrees of freedom is one of the crucial parameters of Wishart distribution. Therefore, the impact of different degrees of freedom is analyzed on classification accuracy. The method that is equipped with initialization technique along with optimum degrees of freedom is assessed using three full polarimetric SAR data sets of agriculture lands. The first two are benchmark data sets of Flevoland, Netherlands, region acquired by AIRSAR sensor, and third is our study area of Mysore, India, acquired by RADARSAT-2 sensor.

On characterizing the complex Wishart distribution

In this research paper, we generalized the results of Ignacy in the multivariate case in order to characterize the complex Wishart distribution.

Change Detection in Multilook Polarimetric SAR Imagery With Determinant Ratio Test Statistic

In this article, we propose a determinant ratio test (DRT) statistic to measure the similarity of two covariance matrices for unsupervised change detection in polarimetric radar images. The multilook complex covariance matrix is assumed to follow a scaled complex Wishart distribution. In doing so, we provide the distribution of the DRT statistic that is exactly Wilks’s lambda of the second kind distribution, with density expressed in terms of Meijer G-functions. Due to this distribution, the constant false alarm rate (CFAR) algorithm is derived in order to achieve the required performance. More specifically, a threshold is provided by the CFAR to apply to the DRT statistic producing a binary change map. Finally, simulated and real multilook polarimetric SAR (PolSAR) data are employed to assess the performance of the method and is compared with the Hotelling–Lawley trace (HLT) statistic and the likelihood ratio test (LRT) statistic.

CP-IRGS: A Region-Based Segmentation of Multilook Complex Compact Polarimetric SAR Data

The Canadian RADARSAT constellation mission (RCM) is represented by three synthetic aperture radar (SAR) satellites, each of which includes a compact polarimetry (CP) mode. CP is advantageous because it provides increased backscatter information relative to single and conventional dual-polarized modes and has larger swath widths relative to a quad polarization mode. CP captures single-look complex data which can be used to derive the multilook complex (MLC) coherence matrix, or, equivalently, the Stokes vector data of the backscattered field. The challenge is to develop computer vision algorithms that can be used to effectively segment the scene using this new data source. An unsupervised region-based segmentation approach has been designed and implemented that utilizes the complex Wishart distribution characteristic of the MLC CP data. The segmentation method is based on the iterative region growing with semantics algorithm originally designed for single and dual pol intensity SAR data. The algorithm has been tested using both simulated CP SAR images and a pair of available quad polarization SAR images. The results demonstrate that the CP-IRGS algorithm provides more accurate segmentation images than those using only the RH and RV channel intensity images.

A family of divergence-based classifiers for Polarimetric Synthetic Aperture Radar (PolSAR) imagery vector and matrix features

ABSTRACT Polarimetric Synthetic Aperture Radar (PolSAR) is one of the most important remote sensing tools. However, PolSAR images are strongly contaminated by a multidimensional interference (called speckle noise), making their processing (e.g. in the classification context) difficult. In terms of structure, multilook PolSAR data follow a definite positive hermitian behaviour and, therefore, require tailored classifiers for their features. Some classic classifiers – such as Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), K-Nearest Neighbours (KNN), and Support Vector Machine (SVM) – have been yielding unacceptable performance to these data, when applied directly. One justification is because they do not often take into account neither the speckle presence nor properties which are inherent to under-study relieves. This paper addresses a collection of PolSAR divergence-based classifiers, deduced from the normal, skew-normal, t-Student, and skew-t vector models as well as the scaled complex Wishart (SCW) distribution. The last model is a standard supposition to describe multilook PolSAR data, having two parameters: covariance matrix (which is directed to data nature) and number of looks (which controls the speckle noise effect). The considered remainder laws aim to model the main diagonal of these data, known as multivariate intensities. The performance of proposed methods is quantified and compared with those due to the Kullback-Leibler (KL) distance for multivariate normal distribution and to LDA, QDA, KNN, and SVM methods. Experiments with both artificial and real PolSAR data are considered. Results favour optimal Rényi classifiers for an Airborne Synthetic Aperture Radar (AIRSAR) image of San Francisco and the t-Student KL classifier for an SAR image system of the Electromagnetics Institute (EMISAR) image of Foulum.

General formulation of source localization for multiple receivers with fully saturated scattered signals

A general formulation is provided for the problem of localizing a source from signals at multiple, arbitrarily separated receivers. The signals are assumed to be narrowband and fully saturated by scattering from homogeneous turbulence as they propagate along line-of-sight paths to the receivers. In this case, the complex Wishart distribution can be used to describe the joint distribution of the received signals. Previously derived results from the theory of wave propagation in random media can furthermore be used to model the covariance matrix between the received signals. Then, given a prior distribution on the source location, Bayes's theorem enables the maximum a posteriori (MAP) estimate for the source location to be formulated. This general approach incorporates source localization by beamforming and by signal amplitude as special cases. The behavior of the solution is illustrated with numerical examples.