Heavy-tailed Rayleigh Research Articles

Group acceptance sampling plan based on truncated life tests for Type-I heavy-tailed Rayleigh distribution

This study on the Type-I heavy-tailed Rayleigh (TI-HTR) distribution is a special case of Type-I heavy-tailed (TI-HT) family of distributions was studied. The characteristics were derived, including the moment and its measures, quantile function, reliability measures, and other statistical properties as well as parameter estimation using the maximum likelihood method and penalized likelihood estimation. The behavior of its various functions were shown graphically. Analytically, we showed that model linearly grows near the origin and exhibits rapid exponential decay. However, the tail behavior cannot equal the traditional heavy-tail in the power law sense, hence it is called the type-I heavy-tail. Interestingly, we designed a group acceptance plan (GASP) and demonstrated usefulness with both assumed and maximum likelihood estimates. The GASP under the TI-HTR distribution is preferable when the parameter values are small. The distribution was used to model real-life data sets to justify its usefulness. The results of the application of the model to both COVID-19 and Cancer data showed that the model fits the two data better than the competing models and also suggest that inference from the model is better than those of the competitors. In estimating the parameters, the penalized likelihood procedure perform considerably better with minimum standard error of the estimates. From the Cramér-von Mises test results which guides against the heavy-tail sensitivity, the TI-HTR distribution offers a better model for fitting fast decaying exponential data since it has the least statistics in both datasets.

Non-Gaussian amplitude PDF modeling of ultrasound images based on a novel generalized Cauchy-Rayleigh mixture

In this paper, a mixture of generalized Cauchy distribution and Rayleigh distribution that possesses a closed-form expression is proposed for modeling the heavy-tailed Rayleigh (HTR) distribution. This new approach is developed for analytically modeling the amplitude distribution of ultrasound images based on the HTR distribution. HTR as a non-Gaussian distribution is basically the amplitude probability density function (PDF) of the complex isotropic symmetric α-stable (S α S) distribution which appears in the envelope distribution of ultrasonic images. Analytic expression for HTR distribution is a momentous consideration in signal processing with stable random variables. Furthermore, we introduce a mixture ratio estimator based on the energy of amplitude PDF which contains both α and γ parameters. For a quantitative assessment, we compare the accuracy and computational complexity of the proposed mixture with other approximations of HTR distribution through several numeral simulations on synthetic random samples. Experimental results obtained from the Kolmogorov-Smirnov (K-S) distance and Kullback-Leibler (K-L) divergence as the goodness-of-fit tests on real ultrasound images reveal the favor of the new mixture model.

SAR Image Filtering Based on the Cauchy–Rayleigh Mixture Model

In this letter, a novel maximum a posteriori (MAP) filter for synthetic aperture radar (SAR) images is developed. We characterize the return signal of SAR using the Cauchy-Rayleigh mixture model, which is an approximation to the heavy-tailed Rayleigh distribution. The parameters of the Cauchy-Rayleigh mixture model are estimated from the noisy observation by using the expectation-maximization algorithm. Finally, we compare the proposed filter with several classical spatial filtering techniques by applying them on simulated data and various real SAR images. Experimental results show that the Cauchy-Rayleigh-mixture-based MAP filter performs better for speckle removal than the other methods, including Lee, Kuan, and Γ-MAP.

Modeling multilook polarimetric SAR images with heavy-tailed rayleigh distribution and novel estimation based on matrix log-cumulants

With the improvements in modern radar resolution, the Gaussian-fluctuation model based on the central limit theorem does not accurately describe the scattering echo from targets. In contrast, the heavytailed Rayleigh distribution, based on the generalized central limit theorem, performs well in modeling the synthetic aperture radar (SAR) images, whereas its application to multi-look image processing is difficult. We describe successful modeling of multilook polarimetric SAR images with the heavy-tailed Rayleigh distribution and present novel parameter estimators based on matrix log-cumulants for the heavy-tailed Rayleigh distribution including the equivalent number of looks (ENL). First, a compound variable of heavy-tailed Rayleigh distribution is divided into a product of a positive alpha-stable variable and a complex Gaussian variable. The parameter estimations of the characteristic exponent and scale parameter based on log-cumulants in a single polarization channel are then derived. Second, the matrix log-cumulants (MLCs) for full polarization in multilook images are obtained, which can be applied to estimate model parameters. Therefore, a novel ENL estimator based on MLC is presented that describes the model more precisely. Extended to all other multivariable product models, this estimator performs better than existing methods. Finally, calculations on both simulated and real data are performed that give good fits with theoretical results. Multilook processing in one image with a fixed pixel number can improve parameter estimations over single-look processing. Our heavy-tailed Rayleigh model with its parameter estimation provides a new method to analyze the multilook polarimetric SAR images for target detection and classification.

Novel mixture model for synthetic aperture radar imagery

We propose a novel mixture model that combines two special cases of heavy-tailed Rayleigh distribution. These two special families possess the only analytical forms of heavy-tailed Rayleigh distribution. As a consequence, the mixture model has an analytical form. Because heavy-tailed Rayleigh distribution is a member of spherically invariant random process, one can obtain the parameter estimation by the method-of-moments technique. Finally, the mixture model has been tested on various synthetic aperture radar images, and the performance of this model is strong compared with other models such as K distribution, G0 distribution, and heavy-tailed Rayleigh models.

Segmentation of Skin Lesions in 2-D and 3-D Ultrasound Images Using a Spatially Coherent Generalized Rayleigh Mixture Model

This paper addresses the problem of jointly estimating the statistical distribution and segmenting lesions in multiple-tissue high-frequency skin ultrasound images. The distribution of multiple-tissue images is modeled as a spatially coherent finite mixture of heavy-tailed Rayleigh distributions. Spatial coherence inherent to biological tissues is modeled by enforcing local dependence between the mixture components. An original Bayesian algorithm combined with a Markov chain Monte Carlo method is then proposed to jointly estimate the mixture parameters and a label-vector associating each voxel to a tissue. More precisely, a hybrid Metropolis-within-Gibbs sampler is used to draw samples that are asymptotically distributed according to the posterior distribution of the Bayesian model. The Bayesian estimators of the model parameters are then computed from the generated samples. Simulation results are conducted on synthetic data to illustrate the performance of the proposed estimation strategy. The method is then successfully applied to the segmentation of in vivo skin tumors in high-frequency 2-D and 3-D ultrasound images.

Modeling high-resolution synthetic aperture radar images with heavy-tailed distributions

Statistical distributions of synthetic aperture radar （SAR） images based on central limit theorem cannot reflect the statistical characteristics of sharp peak and heavy tail of high-resolution SAR images. By using the generalized central limit theorem, the heavy-tailed distributions （heavy-tailed Rayleigh distribution for amplitude image and heavy-tailed exponential distribution for intensity image） are obtained from the symmetric stable distributions of real and imaginary parts of echoes. Taking the heavy-tailed Rayleigh distribution as an example, the algebraic tails of heavy-tailed distributions are explained as well as the statistical properties of sharp peak and heavy tail. In order to model the high-resolution SAR images with the heavy-tailed distributions, based on second-kind statistical, Characteristics the log-cumulant estimator is proposed to efficiently estimate the parameters of the heavy-tailed distributions. Modeling experiments on real SAR images demonstrate that the heavy-tailed distributions based on the generalized central limit theorem can accurately describe the sharp-peaked and heavy-tailed statistical characteristics of high-resolution SAR images.

Heavy-tailed Rayleigh Distribution: Basic Properties and Their Applications

摘要: 针对使用拖尾Rayleigh分布对合成孔径雷达(Synthetic aperture radar, SAR)幅值图像建模时遇到的问题, 本文讨论了拖尾Rayleigh分布的相关性质及其应用. 首先, 基于负数阶矩理论, 本文提出了拖尾Rayleigh分布的比值估计、对数矩估计和迭代对数矩估计三种参数估计方法, 并通过Monte Carlo仿真实验比较了它们的估计性能. 其次, 本文使用渐近级数计算拖尾Rayleigh分布的概率密度函数, 基于插值多项式拟合, 提出了高效计算密度函数的三步方法. 最后, 本文给出了SAR幅值图像基于拖尾Rayleigh分布的建模实例. 结果表明, 和一般的Rayleigh分布相比, 拖尾Rayleigh分布可以精确反映SAR幅值图像尖峰厚尾的统计特征, 因此它是SAR幅值图像建模的有效工具. 关键词: SAR幅值图像建模 / 拖尾Rayleigh分布 / 负数阶矩 / Monte Carlo仿真 / 渐近级数 / 插值多项式拟合

Maximum a posteriori filtering for synthetic aperture radar images based on heavy-tailed Rayleigh distribution of speckle

In order to reflect the statistics of high peak and heavy tail, speckle in synthetic aperture radar images is modeled as heavy-tailed Rayleigh distribution. First, based on Gamma prior distribution and heavy-tailed Rayleigh distribution of speckle, the maximum a posteriori filtering equation is proposed and its analytical form is provided in given characteristic parameter. Second, parameters of heavy-tailed Rayleigh distribution are estimated from the observed image using Mellin transformation. Last, maximum a posteriori de-speckling experiments and their quantitative measures are given. In order to eliminate the influence of window size and noise intensity on de-speckling results, dynamic relations of the de-speckling capability to noise variance and window size are suggested respectively. Results demonstrate that the heavy-tailed Rayleigh distribution accords with the real statistics of speckle, so the maximum a posteriori filter in heavy-tailed Rayleigh distribution of speckle has higher capability of noise reduction compared to the one in Rayleigh distribution of speckle and the Kuan filter.

SAR image filtering based on the heavy-tailed Rayleigh model

Synthetic aperture radar (SAR) images are inherently affected by a signal dependent noise known as speckle, which is due to the radar wave coherence. In this paper, we propose a novel adaptive despeckling filter and derive a maximum a posteriori (MAP) estimator for the radar cross section (RCS). We first employ a logarithmic transformation to change the multiplicative speckle into additive noise. We model the RCS using the recently introduced heavy-tailed Rayleigh density function, which was derived based on the assumption that the real and imaginary parts of the received complex signal are best described using the alpha-stable family of distribution. We estimate model parameters from noisy observations by means of second-kind statistics theory, which relies on the Mellin transform. Finally, we compare the proposed algorithm with several classical speckle filters applied on actual SAR images. Experimental results show that the homomorphic MAP filter based on the heavy-tailed Rayleigh prior for the RCS is among the best for speckle removal.