Parameter estimation of the homodyned K distribution based on neural networks and trainable fractional-order moments

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Parameter estimation of the homodyned K

distribution based on neural networks and trainable

fractional-order moments

Michal Byra∗†, Ziemowit Klimonda†, Piotr Jarosik†

†Institute of Fundamental Technological Research,

Polish Academy of Sciences, Warsaw, Poland

∗Corresponding author, e-mail: mbyra@ippt.pan.pl

Abstract—Homodyned K (HK) distribution has been widely

used to describe the scattering phenomena arising in various re-

search ﬁelds, such as ultrasound imaging or optics. In this work,

we propose a machine learning based approach to the estimation

of the HK distribution parameters. We develop neural networks

that can estimate the HK distribution parameters based on

the signal-to-noise ratio, skewness and kurtosis calculated using

fractional-order moments. Compared to the previous approaches,

we consider the orders of the moments as trainable variables

that can be optimized along with the network weights using

the back-propagation algorithm. Networks are trained based on

samples generated from the HK distribution. Obtained results

demonstrate that the proposed method can be used to accurately

estimate the HK distribution parameters.

Index Terms—homodyned K distribution, neural networks,

parameter estimation, quantitative ultrasound.

I. INTRODUCTION

Homodyned K (HK) distribution has been widely used to

describe the scattering phenomena arising in various research

ﬁelds. In ultrasound (US) imaging, the HK distribution has

been utilized to model the backscattered echo amplitude and

quantitatively assess tissue structure [1]. For example, the

HK distribution was applied for ultrasound based temperature

monitoring and tissue characterization [2], [3], [4], [5].

Various methods have been developed for the estimation

of the HK distribution parameters. Hruska and Oelze pro-

posed a level-set estimation technique based on the signal-to-

noise ratio, skewness and kurtosis parameters calculated using

fractional-order moments [6]. Destrempes et al. proposed an

iterative estimation technique based on the ﬁrst moment of the

intensity and two log-moments, namely the X- and U-statistics

[7]. Building on the previous works, Zhou et al. utilized an

artiﬁcial neural network (ANN) to estimate the parameters of

the HK distribution [8]. Authors utilized the signal-to-noise

ratio, skewness, kurtosis, X- and U- statistics as the input to

the feed-forward neural network.

In this work, we propose a machine learning based tech-

nique for the estimation of the HK distribution parameters.

Similar to Zhou et al., we train our neural network based on

the SNR, skewness and kurtosis statistics [8]. However, in

our case the orders of the moments used for the calculations

are not ﬁxed. Hruska and Oelze presented that the choice of

the moments is important for the accurate estimation of the

HK distribution parameters [6]. To improve the estimation,

we treat the orders of the moments as trainable variables that

can be optimized along with the network weights using the

back-propagation algorithm.

II. METHODS

A. Homodyned K distribution

The probability density function of the HK distribution can

be expressed in the following way:

p(A) = A

∞

hJ0(sh)J0(Ah)1 + h2σ2

2u−udh, (1)

where Astands for the amplitude, J0is the zero-th order

Bessel function of the ﬁrst kind and variable his used for

the integration. Parameters s2and σ2stand for the coherent

and diffusive signal power. HK distribution has two param-

eters used for the quantitative assessment of the scattering

phenomena in US. The ﬁrst parameter, u, is the scatterer

clustering parameter reﬂecting the number of the scatterers

in the resolution cell. The second quantitative parameter of

the HK distribution is expressed as the ratio k=s

σand is

related to the spatial periodicity of the scatterer distribution.

B. The RSK estimator

Hruska and Oelze proposed the level-set method for the

estimation of the HK distribution parameters based on the

signal-to-noise ratio (R), skewness (S) and kurtosis (K) of the

amplitude, denoted as the RSK estimator [6]. These three can

be calculated with the following equations:

R(v) = E[Av]

(E[A2v]−E2[Av])1/2,(2)

S(v) = E[A3v]−3E[Av]E[A2v]+2E3[Av]

(E[A2v]−E2[Av])3/2,(3)

arXiv:2210.05833v2 [cs.LG] 16 Dec 2022

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摘要：

ParameterestimationofthehomodynedKdistributionbasedonneuralnetworksandtrainablefractional-ordermomentsMichalByray,ZiemowitKlimonday,PiotrJarosikyyInstituteofFundamentalTechnologicalResearch,PolishAcademyofSciences,Warsaw,PolandCorrespondingauthor,e-mail:mbyra@ippt.pan.plAbstractHomodynedK(HK)dist...

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Parameter estimation of the homodyned K distribution based on neural networks and trainable fractional-order moments

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