Fractional moments and their usefulness in atmospheric laser scintillation
01 Sep 1998-Pure and Applied Optics: Journal of The European Optical Society Part A (IOP Publishing)-Vol. 7, Iss: 5, pp 1013-1032
TL;DR: In this article, a comparison procedure based on two low-order measured fractional moments is described and tested by utilizing Monte Carlo samples from three distributions commonly used in atmospheric propagation (Ln, LnME and K).
Abstract: Use of moments of non-integer order (called fractional moments) in statistical optics is proposed for discriminating between candidates for probability density functions (PDFs) of the scintillation intensity to overcome the experimental problems encountered by using integer moments. Low-order (< 2) fractional moments allow comparison of experimental data with theoretical PDFs of the intensity in different scintillation regimes. A comparison procedure, based on two low-order measured fractional moments, is described and tested by utilizing Monte Carlo samples from three distributions commonly used in atmospheric propagation (Ln, LnME and K). The bin width required to discriminate between different distributions and the effect of noise are investigated. Examples of application to experimental data are presented.
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TL;DR: A modified scheme which exploits the relationship between the Gamma-Gamma shape parameters in free-space optical communication and can achieve satisfactory performance for a wide range of turbulence conditions is proposed.
Abstract: We study the parameter estimation problem for the Gamma-Gamma turbulence model for free-space optical communication. An estimation scheme for the shape parameters of the Gamma-Gamma distribution is proposed based on the concept of fractional moments and convex optimization. To improve the estimation performance, we further propose a modified scheme which exploits the relationship between the Gamma-Gamma shape parameters in free-space optical communication. Simulation results reveal that the modified estimation scheme can achieve satisfactory performance for a wide range of turbulence conditions.
91 citations
TL;DR: In this paper, the authors discuss several aspects of the MIG distribution useful for modelling positive data, specifically transformations, the derivation of moments, fitting of models, and a shape analysis of the transformations.
Abstract: Skewed models are important and necessary when parametric analyses are carried out on data. Mixture distributions produce widely flexible models with good statistical and probabilistic properties, and the mixture inverse Gaussian (MIG) model is one of those. Transformations of the MIG model also create new parametric distributions, which are useful in diverse situations. The aim of this paper is to discuss several aspects of the MIG distribution useful for modelling positive data. We specifically discuss transformations, the derivation of moments, fitting of models, and a shape analysis of the transformations. Finally, real examples from engineering, environment, insurance, and toxicology are presented for illustrating some of the results developed here. Three of the four data sets, which have arisen from the consulting work of the authors, are new and have not been previously analysed. All these examples display that the empirical fit of the MIG distribution to the data is very good.
82 citations
Cites background from "Fractional moments and their useful..."
...Fractional moments are hardly used in statistics, but are widely applied in some other fields such as environmental sciences; see [12], image processing; see [13], and queueing theory; see [14]....
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TL;DR: The statistics of the radio frequency signal in the case of partially developed speckle are studied using the K distribution framework and the ability of the proposed distribution to model RF echographic signals from cardiac tissues is evaluated from data acquired in vivo.
Abstract: We study in this paper the statistics of the radio frequency (RF) signal in the case of partially developed speckle. Using the K distribution framework, we give the probability density function of the associated distribution, the corresponding moments, and estimators for the parameters of the distribution. The consistency of the proposed estimators is evaluated in terms of their bias and variance through numerical simulations. The ability of the proposed distribution to model RF echographic signals from cardiac tissues is evaluated from data acquired in vivo
37 citations
TL;DR: In this article, the authors presented the joint probability density function (PDF) between the bucket signals and reference signals in thermal light ghost imaging, by regarding these signals as stochastic variables.
Abstract: We present the joint probability density function (PDF) between the bucket signals and reference signals in thermal light ghost imaging, by regarding these signals as stochastic variables. The joint PDF allows us to examine the fractional-order moments of the bucket and the reference signals, in which the correlation orders are fractional numbers, other than positive integers in previous studies. The experimental results show that various images can be reconstructed from fractional-order moments. Negative (positive) ghost images are obtained with negative (positive) orders of the bucket signals. The visibility and peak signal-to-noise ratios of the diverse ghost images depend greatly on the fractional orders.
14 citations
06 Apr 2006
TL;DR: The ability of the proposed distribution to model RD echocardiographic signals from cardiac tissue and blood region is demonstrated on data acquired in vivo.
Abstract: We study in this work the statistics of the radio frequency (RF) signal for both fully and partially developed speckle in echocardiographic images in the context of image segmentation and classification. From physical image formation model, we first derive the probability density function (PDF) of the RF signal using the K distribution framework. We then show that this pdf may be reliably approximated through a generalized Gaussian distribution. The ability of the proposed distribution to model RD echocardiographic signals from cardiac tissue and blood region is demonstrated on data acquired in vivo.
13 citations
References
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TL;DR: In this article, it was shown that a class of modified Bessel-function distributions may have special significance in describing the statistics of radiation scattered by media characterized by a wide range of length scales, and that these distributions may be obtained mathematically by applying a limit procedure to the random walk problem with a variable number of steps.
Abstract: Evidence is presented which suggests that a class of modified Bessel-function distributions may have special significance in describing the statistics of radiation scattered by media characterized by a wide range of length scales. It is shown that these distributions may be obtained mathematically by applying a limit procedure to the random-walk problem with a variable number of steps. The choice of distribution for the step-number fluctuations is briefly discussed.
286 citations
TL;DR: In this article, Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a = 1 and an optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints.
279 citations
TL;DR: The Radon-Wigner transform as mentioned in this paper is the squared modulus of the fractional Fourier transform, and it can be used to translate signal and image processing results between different signal representations.
Abstract: Two recently described transforms are shown to be related. The Radon–Wigner transform is the squared modulus of the fractional Fourier transform. This new theorem may serve to translate signal and image processing results between different signal representations. Some consequences regarding moments are presented, including a new fractional-Fourier-transform uncertainty relation. Implications for processing are suggested.
211 citations
TL;DR: In this article, a new probability density function (PDF) was proposed for irradiance scintillations in the case of strong scintillation (i.e., irradiance variance decreases with further increases in path-averaged refractive-index turbulence).
Abstract: A new probability-density function (PDF) is proposed for irradiance scintillations in the case of strong scintillation (i.e., irradiance variance decreases with further increases in path-averaged refractive-index turbulence). This new PDF is named the log-normally modulated exponential PDF. This PDF is compared with experimental PDF’s of both irradiance and photon counts obtained from atmospheric laser propagation; the agreement is excellent (superior to that of the K PDF). Receiver–aperture-averaged irradiance is shown to be log normal for sufficiently large apertures. Comparison of moments of measured irradiance with theoretical moments shows that the moment comparison method of testing irradiance statistics can be very misleading because of the limitations of receivers and the sensitivity of high-order moments to very large irradiances.
129 citations
TL;DR: In this paper, the probability distribution function (PDF) from simulations for an initially spherical wave propagated through homogeneous atmospheric turbulence was calculated and the onset of strong scintillation was calculated.
Abstract: We calculated the probability distribution function (PDF) from simulations for an initially spherical wave propagated through homogeneous atmospheric turbulence. The onset of strong scintillation was calculated. By onset of strong scintillation, we mean conditions of weak scintillation changing to the condition of strong focusing. In addition, one case in the saturation regime was calculated. The simulations’ PDF’s are compared with several heuristic models of the PDF and are seen to progress from close to log normal for the cases of weakest scintillation to close to the log normally modulated exponential PDF (LNME PDF) for the cases of strong scintillation. The simulations’ PDF’s are not in agreement with the K PDF for any of the calculated cases. The best agreement was obtained in comparison with Beckmann’s PDF [P. Beckmann, Probability in Communication Engineering (Harcourt, Brace, & World, 1967)]. Beckmann’s PDF varies from being the log-normal PDF for weak scintillation to being the LNME PDF for strong scintillation and progresses further to the theoretically expected exponential PDF in the limit of saturated scintillation. We recommend that simulation be used to predict the irradiance PDF for plane and diverged waves in homogeneous turbulence in preference to using heuristic models. More complicated propagation cases remain in the domain of heuristic argumentation.
112 citations