Showing papers on "K-distribution published in 1994"
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01 Jan 1994
TL;DR: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
Abstract: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
7,270 citations
TL;DR: In this paper, a more generalized statistical model, the homodyned K distribution, combines the K and Rice distribution features to better account for the statistics of the echo signal, and two parameters that are useful for media characterization: k, the ratio of coherent to diffuse signals, and, β, which characterizes the clustering of scatterers in the medium.
207 citations
08 Aug 1994
TL;DR: In this article, a K-distribution was developed to characterize the statistical properties of multi-look processed polarimetric SAR data, where the probability density function (PDF) was derived as the product of a gamma distributed random variable and the polarIMetric covariance matrix.
Abstract: A K-distribution has been developed to characterize the statistical properties of multi-look processed polarimetric SAR data. The probability density function (PDF) was derived as the product of a gamma distributed random variable and the polarimetric covariance matrix. The latter characterizes the speckle and the former depicts the inhomogeneity (texture). For multi-look data incoherently averaged from correlated one-look samples, the authors found that, for better modeling, the number of looks has to assume a non-integer value. A procedure was developed to estimate the equivalent number of looks and the parameter of the K-distribution. Experimental results using NASA/JPL 4-look and 16-1ook polarimetric SAR data substantiated this multi-look K-distribution. The authors also found that the multi-look process reduced the inhomogeneity and made the K-distribution less significant. >
145 citations
01 Feb 1994
TL;DR: In this article, the estimation errors of three moment based estimation schemes are compared with the maximum likelihood estimation errors calculated via the Cramer-Rao lower bound, and recommendations are made regarding the number of looks and the parameter estimation scheme that should be used to obtain near optimum estimation performance.
Abstract: Parameter estimation forms an essential part of many signal- and image-processing tasks. In particular, in the analysis of coherent imagery, such as that provided by synthetic aperture radar (SAR), parameter estimation is required to characterise the statistical properties of homogeneous regions for use in segmentation and target detection algorithms. The statistics of SAR imagery can be modelled by the K-distribution, and so it is of interest to study methods for estimating the parameters of this distribution. The estimation errors of three moment based estimation schemes are compared with the maximum likelihood estimation errors calculated via the Cramer-Rao lower bound. On the basis of this comparison, recommendations are made regarding the number of looks and the parameter estimation scheme that should be used to obtain near optimum estimation performance, without resorting to cumbersome numerical evaluations of the maximum likelihood solution. In particular, it is found that an estimator based on the mean and the variance of the data yields large errors, but an estimator based on the mean of the data and the mean of the log of the data is close to optimum.
129 citations
TL;DR: In this article, several conditions are established under which a family of elliptical probability density functions possesses a preferable consistency property, which ensures that any marginal distribution of a random vector whose distribution belongs to a specific elliptical family also belongs to the family.
120 citations
13 Nov 1994
TL;DR: This work uses the Weibull and K distributions to model clutter since they seem to fit observed data better and also include the Rayleigh distribution as a special case in CFAR.
Abstract: Constant False Alarm Rate (CFAR) processing of Synthetic Aperture Radar (SAR) images facilitates target detection in spatially varying background clutter. The traditional Rayleigh distribution does not appear to be a good choice for modeling the natural terrain backscatter in high resolution SAR. We use the Weibull and K distributions to model clutter since they seem to fit observed data better and also include the Rayleigh distribution as a special case. The Cell Averaged CFAR technique works well in situations where a single, small target is present in locally homogeneous clutter. The Order Statistic CFAR is more useful for larger targets and in multiple target situations. Comparisons are made between the various CFAR techniques by applying them to real, high-resolution SAR images, obtained from the MIT Lincoln Laboratory. >
105 citations
TL;DR: In this article, the authors carried out numerical simulations of wave traversing a three-dimensional random medium with Gaussian statistics and a power-law spectrum with inner-scale cutoff and provided the probability density function (PDF) of irradiance.
Abstract: We have carried out numerical simulations of waves traversing a three-dimensional random medium with Gaussian statistics and a power-law spectrum with inner-scale cutoff. The distributions of irradiance on the final observation screen provide the probability-density function (PDF) of irradiance. For both initially plane and initially spherical waves the simulation PDF’s in the strong-fluctuation regime lie between a K distribution and a log-normal-convolved-with-exponential distribution. We introduce a plot of the PDF of scaled log-normal irradiance, on which both the exponential and the lognormal PDF’s are universal curves and on which the PDF at both large and small irradiance is shown in detail. We have simulated a spherical-wave experiment, including aperture averaging, and find agreement between the simulated and the observed PDF’s.
104 citations
TL;DR: In this article, the probability density function (pdf) of optical signal intensity in an optical communication channel impaired by motion-induced beam jitter and turbulence is derived and the conditions for which these approximations seem to be valid are also discussed.
Abstract: Expressions for the probability density function (pdf) of optical signal intensity in an optical communication channel impaired by motion-induced beam jitter and turbulence are derived. It is assumed that the optical beam possesses a Gaussian profile, generated by a pulsed laser, and that the beam scintillation is governed by either log-normal distribution for weak turbulence or K distribution for moderate to strong turbulence in the saturation region. For extreme propagation distances or very strong turbulence, a negative exponential pdf is used to model turbulence. For the aforementioned beam scintillation statistics, approximate pdf's for the signal intensity are also obtained and the conditions for which these approximations seem to be valid are also discussed.
74 citations
TL;DR: In this paper, various transformations commonly employed in order to obtain near normal distributions of precipitation data are discussed, in particular the properties of the square-root-normal distribution and its relationship to the other commonly used two-parameter distributions.
Abstract: This paper deals with various transformations commonly employed in order to obtain near normal distributions of precipitation data. In particular, we discuss the properties of the square-root-normal distribution and its relationship to the other commonly used two-parameter distributions. Similarities of different distributions are discussed with the aid of the normal probability graph and the moment-ratio diagram. The latter has the coefficient of variation on the abscissa and the coefficient of skewness on the ordinate. The examination of some historical data sets with the help of these diagrams demonstrates a large variety of forms, but also points to some of their common characteristics.
28 citations
TL;DR: The S-distribution as discussed by the authors is defined by the ordinary differential equation dF/dX = α(Fg − FhFo = F(Xo), where F is the cumulative distribution of the random variable X, and α, g, h, and Fo are parameters.
Abstract: The S-distribution is defined by the ordinary differential equation dF/dX = α(Fg — FhFo = F(Xo), where F is the cumulative distribution of the random variable X, and α, g, h, and Fo are parameters. The S-distribution was recently described in this journal as a tool for the approximation and classification of univariate, unimodal continuous probability distributions. This article shows that the S-distribution rather accurately models the commonly used univariate discrete distributions.
24 citations
TL;DR: In this paper, a comprehensive study of the shape of discrete random walks, considering the general case of arbitrary length and any space dimension, is presented, and the probability distributions for several magnitudes such as the principal inertia moments, the asphericity and the angle between the principal axis of inertia and the end-to-end vector are evaluated numerically.
Abstract: A comprehensive study of the shape of discrete random walks, considering the general case of arbitrary length and any space dimension, is presented. The probability distributions for several magnitudes such as the principal inertia moments, the asphericity and the angle between the principal axis of inertia and the end-to-end vector are evaluated numerically. In most cases, and especially in low-dimensional spaces, these probability distributions spread widely and possess a non-zero skewness. This implies that any description of the shapes of random walks which is based only on mean values of related magnitudes is incomplete. This situation does nor vary when random walks of large lengths are considered since the relative fluctuations (ratio between the standard deviation and the mean value) do not go to zero in that limit. On the other hand, when the space dimension grows, the distributions become more peaked, reaching the expected asymptotic limit for infinite dimension. The present study also indicates that the probability distributions for the principal inertia moments have approximately the form of chi-squared distributions. Analysing the probability distribution for the asphericity, it is shown that the distributions for the different moments are not completely independent.
08 Aug 1994
TL;DR: Methods are described for the generation of realisations from a correlated K distribution with specified correlation properties, including higher order correlations, for target detection algorithms for radar remote sensing.
Abstract: Discusses clutter simulation which is an important element in the development of target detection algorithms for radar remote sensing. SAR images are well represented by the K distribution and an important feature of SAR clutter is the autocorrelation function. Methods are described for the generation of realisations from a correlated K distribution with specified correlation properties. Higher order correlations are also considered. >
23 May 1994
TL;DR: A construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of [13] and can be efficiently combined with the method of conditional probabilities to yield improved NC algorithms for many problems.
Abstract: We present two techniques for approximating probability distributions. The first is a simple method for constructing the small-bias probability spaces introduced in [21]. This construction can be efficiently combined with the method of conditional probabilities to yield improved NC algorithms for many problems such as set cover, set discrepancy, finding large cuts in graphs etc. The second is a construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of [13].
08 Aug 1994
TL;DR: In this paper, a nearly complete analysis of the key distributions encountered in single and multi-look polarimetric and interferometric synthetic aperture radar (SAR) data, under a Gaussian or multi-variate K distribution model, is presented.
Abstract: Provides a nearly complete analysis of the key distributions encountered in single and multi-look polarimetric and interferometric synthetic aperture radar (SAR) data, under a Gaussian or multi-variate K distribution model. It contains new analytic results on the moments of phase difference in single look data, and on multi-look distributions of amplitude and phase. As yet no analytic results for the moments of multi-look phase difference have been found, except in limiting cases. The maximum likelihood estimators of the covariance matrix of two jointly Gaussian channels is derived, along with their asymptotic variances. A more complete discussion of these ideas is presented by Tough et al. (1994). >
08 Aug 1994
TL;DR: In this article, the order parameter of the K distribution of agricultural targets in low-resolution AirSAR polarimetric data has been investigated and the results indicate that five parameters are sufficient to describe distributed agricultural targets at higher frequencies, but an extra parameter is available at lower frequencies.
Abstract: Agricultural areas in low resolution AirSAR polarimetric data exhibit K distributed statistics. The order parameter of the K distribution shows considerable variation with frequency and vegetation type. At C-band, all vegetation types give large order parameters, so that they are essentially gaussian, but order parameters decrease with frequency, markedly so for cereal crops. Coniferous woodland is gaussian at all frequencies. Root crops show intermediate behaviour, being better fitted by K distributions whose order parameters are not as low as those of cereals. These results indicate that five parameters are sufficient to describe distributed agricultural targets at higher frequencies, but an extra parameter is available at lower frequencies. >
TL;DR: In particular, the exact order of uniform convergence is s −p, where p = min(1, α) as mentioned in this paper, where α is the negative binomial distribution function with parameters α and a/s, s > 0.
Abstract: Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p , where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.
08 Aug 1994
TL;DR: It is shown that, mainly for forest data, the fit with the K multilook distribution is superior to some of other distributions that frequently appear in the literature.
Abstract: The K distribution has been used as a flexible tool for the modelling of SAR data over non-homogeneous areas. It is characterized by three real-valued parameters; one of these parameters, the number of looks, is related to the kind of processing the raw data suffer in order to become an image. This distribution has been mostly used for one look data. In this paper the multilook case is considered for both quadratic and linear detections. A closed (recursive) computational form is provided for the K cumulative distribution function, as well as the estimators derived from the substitution method. The sensitivity of the cumulative distribution function, with respect to possible discretizations of the parameters due to limitations imposed by the recursive form is discussed. The recursive form of the cumulative distribution function of K multilook random variables is used to perform the Kolmogorov-Smirnov (KS) test of goodness of fit over SAREX data. It is shown that, mainly for forest data, the fit with the K multilook distribution is superior to some of other distributions that frequently appear in the literature. Specifically, the use of the normal distribution for this kind of data is discarded systematically. >
TL;DR: In this paper, several probability and statistical methods are discussed for detecting spatial randomness in two dimensions, and one method is derived and proposed for its ease of application. Monte Carlo simulation is used for the detection.
Abstract: Several probability and statistical methods are discussed for detecting spatial randomness in two dimensions. One method is derived and proposed for its ease of application. Monte Carlo simulation ...
21 Dec 1994
TL;DR: In this paper, a technique for refining a previous segmentation by using different properties of K-distributed SAR clutter simultaneously is proposed, which is based on a maximum entropy approach which assumes that only two moments of the data can be estimated with sufficient accuracy over a small sample region.
Abstract: This paper proposes a technique for refining a previous segmentation by using different properties of K-distributed SAR clutter simultaneously. We first consider approximate forms for the K distribution based on a Maximum Entropy approach which assumes that only two moments of the data can be estimated with sufficient accuracy over a small sample region. The choice of moments defines the form of the approximate probability density functions (PDF). After the initial segmentation we then propose a post-processing stage in which the values of the moments of the complete previously-identified segments are assumed exact and an optimum fit to the edge position is defined. We demonstrate that joint estimation of the edge position, based on estimates of the mean of the data and of its logarithm, provides a close approximation to the full K- distribution treatment, while being significantly simpler to implement.
TL;DR: In this paper, various probability distributions which have been proposed to explain the charged particle multiplicity distributions in high energy collisions are shown to arise from the evolution equation of a pure birth process subject to appropriate initial conditions.
Abstract: Various probability distributions which have been proposed to explain the charged particle multiplicity distributions in high energy collisions are shown to arise from the evolution equation of a pure birth process subject to appropriate initial conditions For example, both the negative binomial distribution (NBD) as well as the partially coherent laser distribution (PCLD) can be obtained in this way New interrelations between some of these probability distributions are also brought out
TL;DR: A procedure for obtaining the mean density of clusters and their associated structures in a very general type of distribution is presented and how this quantity may be evaluated for different types of distributions is shown.
Abstract: We present a procedure for obtaining the mean density of clusters and their associated structures in a very general type of distribution. We first consider the one-dimensional case, and then use it to develop the procedure according to the d-dimensional case. These procedures require the probability that a randomly placed body should contain no points. We show how this quantity may be evaluated for different types of distributions. The particular case of spherical clusters in three dimensions is treated in detail.
TL;DR: In this paper, an improved and simplified version of the unifying probability density function of [1] was presented, which is shown to be the parent of the Rayleigh distribution in addition to the Weibull-, gamma-, Erlang-, χ 2 -and exponential distributions.
01 Jan 1994
TL;DR: In this article, a variant of the p-adic theory of probability, where probabilities belong to spaces of distributions (generalized functions), is proposed. But it is not a generalization of quantum probability.
Abstract: Unboundedness of the p-adic Gaussian distribution is the strong reason to create a variant of the p-adic theory of probability, where probabilities belong to spaces of distributions (=generalized functions). This chapter is devoted to this problem. This theory is very similar to the ordinary quantum probability (over the field of real numbers). Both these formalisms develop without measure-theoretical constructions.
01 Jan 1994
TL;DR: In this article, the class of bivariate distributions having the almost-lack-of-memory property (ALM-distribution) is introduced and the exact form of these distributions in a sub-class with almost independent components is derived.
Abstract: The class of bivariate distribution having the almost-lack-of-memory property (ALM-distribution) is introduced and the exact form of these distributions in a sub-class with almost independent components is derived. Some of the corresponding probability properties are discussed.
08 Aug 1994
TL;DR: Experimental data from a high resolution airborne SAR system are proved to K statistics model and compared with Weibull, lognormal and gamma density functions.
Abstract: In the last decade many authors prefer to describe the statistical distribution of backscattering amplitudes in single-look synthetic aperture radar (SAR) images by K distribution. In this paper experimental data from a high resolution airborne SAR system are proved to K statistics model and compared with Weibull, lognormal and gamma density functions. >
01 Jan 1994
TL;DR: In this article, an insurance policy where the claim amount has a mixed lognormal distribution was considered and the individual distribution of claim amount, given the value of the parameter for the median of that distribution, was assumed to follow a normal, Laplace, uniform, power function and gamma distribution.
Abstract: Consider an insurance policy where the claim amount has a mixed lognormal distribution. In this paper, we assume that the individual distribution of claim amount, given the value of the parameter for the median of that distribution, follows a lognormal distribution. To model the heterogeneity in the population, we assume that the median is disbursed according to normal, Laplace, uniform, power function and gamma distributions. The exact ultimate forms for the probability density function of the portfolio claim amount are given for the former three distributions. Whereas, a difference equation representation is given for the latter two distributions.
TL;DR: In this paper, the amplitude intensity distribution of the sea clutter measured by an X-band radar is studied quantitatively by means of the Akaike Information Criterion (AIC) under an assumption that the intensity distribution is Weibull, log-normal and K-distributions.
Abstract: In radar signal processing, it is important to eliminate unwanted noise called clutter and to detect the desired target so that the degree of discrimination is improved. To this end, the statistical characteristics of the clutter have been investigated. In such instances, it is necessary to have a process which obtains a constant false alarm rate (CFAR).
In this paper, the amplitude intensity distribution of the sea clutter measured by an X-band radar is studied quantitatively by means of the Akaike Information Criterion (AIC) under an assumption that the intensity distribution is Weibull, log-normal and K-distributions. It is confirmed that, for a high sea state with wave height of 6 to 9 m and wind velocity of 25 m/s, the sea clutter follows a log-normal distribution as a whole and a K-distribution within the antenna beamwidth.
With regard to the CFAR circuit by normalization of the standard deviation which is considered effective for suppression of the clutter following a Weibull and a log-normal distribution, a CFAR processing was carried out with both Monte Carlo simulation and raw data. From the results of computer simulation, it was confirmed that the CFAR circuit is effective for suppression of clutters following a K-distribution. Also, by processing the raw data by this circuit, it was successful in correctly detecting targeted ships while the clutter was suppressed.
05 Aug 1994
TL;DR: The results of experimental study of probability density function in case when a number of scatterers in the resolution element of optical system is small are presented in this article, where measurements have been carried out with ground glass and roughly polished plexiglass specimens at (lambda equals 0.63 micrometers.
Abstract: The results of experimental study of probability density function in case when a number of scatterers in the resolution element of optical system is small are presented. The measurements have been carried out with ground glass and roughly polished plexiglass specimens at (lambda) equals 0.63 micrometers . Experimentally derived histograms of probability density function were in good agreement with K-distribution model.