A Numerical Integral Algorithm Based on the CAPSO to Improve the Estimation for the Parameters of the Homodyned-K Distribution

Wang, Yuqian; Zhang, Yufeng; Zhao, Wei; Zhu, Hongxuan

doi:10.1109/icct.2018.8599949

Cited by 1 publication

(8 citation statements)

References 9 publications

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“…In Figure 5, we use the adaptive numerical method based on CAPSO. 40,42,43 The step size of the numerical integration can adaptively change by minimizing the fitness function f t = ∑ j = 1 n 1 2 ( x j + 1 − x j ) ( f ( x j + 1 ) − f ( x j ) ) to obtain a more accurate result. The maximum number of iterations MaxIterations was set to 20.…”

Section: Methodsmentioning

confidence: 99%

“…Second, both the article Hu et al 39 and Wang et al 40 use the Newton-Raphson method to find the critical point that maximizes the log-likelihood equation. And the authors believe that sole solution can be obtained based on the Newton-Raphson algorithm to solve the log-likelihood equation numerically.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…And the authors believe that sole solution can be obtained based on the Newton-Raphson algorithm to solve the log-likelihood equation numerically. 40 However, this statement is true unless the log-likelihood equation has only one critical point in the parameter domain because Newton-Raphson method can only find one root near the initial value. Recall that the Maximum likelihood estimation can be defined as a point θ ¯ * in the parameter domain at which (i) the gradient of the log-likelihood function log ( L ( θ ¯ )) vanishes (i.e., θ ¯ * is a critical point of the log-likelihood function); (ii) log ( L ( θ ¯ * )) is maximal among all critical points of the log-likelihood function (at least in some bounded domain).…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…Therefore, we need to further explore a method that can find a point that maximizes the log-likelihood equation (at least in some bounded domain) and always converges. In addition, relevant contents such as the range of the various functions and the setting of the initial values in an implementation have not been sufficiently discussed in the article Hu et al 39 and Wang et al 40…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…They used a high-order global adaptive Gauss-Kronrod quadrature method to calculate the probability density function (PDF) of the HK distribution, and then used the Newton-Raphson method to numerically solve the log-likelihood equation. Moreover, Wang et al 40 proposed a similar MLE for parameter estimation of HK distribution but adapted a cloud adaptive particle swarm optimization (CAPSO) to calculate the PDF of the HK distribution. However, these studies have significant limitations, and some of the statements are misleading.…”

Section: Introductionmentioning

confidence: 99%

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An Improved Parameter Estimator of the Homodyned K Distribution Based on the Maximum Likelihood Method for Ultrasound Tissue Characterization

Liu

Zhang

et al. 2022

Ultrason Imaging

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The homodyned K distribution (HK) can generally describe the ultrasound backscatter envelope statistics distribution with parameters that have specific physical meaning. However, creating robust and reliable HK parameter estimates remains a crucial concern. The maximum likelihood estimator (MLE) usually yields a small variance and bias in parameter estimation. Thus, two recent studies have attempted to use MLE for parameter estimation of HK distribution. However, some of the statements in these studies are not fully justified and they may hinder the application of parameter estimation of HK distribution based on MLE. In this study, we propose a new parameter estimator for the HK distribution based on the MLE (i.e., MLE1), which overcomes the disadvantages of conventional MLE of HK distribution. The MLE1 was compared with other estimators, such as XU estimator (an estimation method based on the first moment of the intensity and tow log-moments) and ANN estimator (an estimation method based on artificial neural networks). We showed that the estimations of parameters α and k are the best overall (in terms of the relative bias, normalized standard deviation, and relative root mean squared errors) using the proposed MLE1 compared with the others based on the simulated data when the sample size was N = 1000. Moreover, we assessed the usefulness of the proposed MLE1 when the number of scatterers per resolution cell was high (i.e., α up to 80) and when the sample size was small (i.e., N = 100), and we found a satisfactory result. Tests on simulated ultrasound images based on Field II were performed and the results confirmed that the proposed MLE1 is feasible and reliable for the parameter estimation from the ultrasonic envelope signal. Therefore, the proposed MLE1 can accurately estimate the HK parameters with lower uncertainty, which presents a potential practical value for further ultrasonic applications.

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