 Research
 Open Access
NonGaussian amplitude PDF modeling of ultrasound images based on a novel generalized CauchyRayleigh mixture
 Mohammadreza Hassannejad Bibalan^{1} and
 Hamidreza Amindavar^{1}Email author
https://doi.org/10.1186/s136400160148z
© The Author(s) 2016
 Received: 24 May 2016
 Accepted: 20 November 2016
 Published: 19 December 2016
Abstract
In this paper, a mixture of generalized Cauchy distribution and Rayleigh distribution that possesses a closedform expression is proposed for modeling the heavytailed 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 nonGaussian distribution is basically the amplitude probability density function (PDF) of the complex isotropic symmetric αstable (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 KolmogorovSmirnov (KS) distance and KullbackLeibler (KL) divergence as the goodnessoffit tests on real ultrasound images reveal the favor of the new mixture model.
Keywords
 Generalized CauchyRayleigh model
 Heavytailed Rayleigh distribution
 αstable distribution
 NonGaussian amplitude PDF
 Ultrasound images
1 Introduction
In the medical context, ultrasound provides a noninvasive technique of imaging human anatomy with good visualization characteristics and relatively easy management [1]. As a widely used medical imaging modality, ultrasound applications include cardiology, urology, obstetrics and gynecology, general abdominal imaging, vascular imaging, ophthalmology, orthopedics, and surgical procedures [2, 3]. Bmode (brightness mode) has been the widely accepted method for ultrasound imagery. In this mode of ultrasound, a linear array of transducers simultaneously scans a plane through the body that can be viewed as a 2D image on screen [4]. Modeling the distribution statistics of ultrasound images has its own importance and strongly depends on a comprehensive knowledge of tissue scattering mechanism [5]. In recent years, there has been a growing interest in modeling the amplitude probability density function (PDF) of ultrasound images specifically in speckle denoising [6].
Several models have been utilized to statistically characterize the envelope distribution of ultrasound returns [7]. Three major categories used in the amplitude PDF estimation of ultrasound images are summarized into parametric, nonparametric, and mixture models [8]. The most used models for this purpose are the Rayleigh model [9], Kdistribution [10], Nakagami distribution [11], and generalized Nakagami distribution [12]. In addition to these methods, one can directly use the return data to construct the amplitude distribution of ultrasound images, for example, the heavytailed Rayleigh (HTR) distribution. The HTR distribution arises naturally in scenarios involving scattering effects with scatterers with crosssection distributions that are heavy tailed, and it has found applications in many domains including ultrasound imaging and SAR. Specifically, the HTR model is considered as the most theoretically wellfounded statistical model at present. The HTR distribution can model many classes of ultrasound images [13]. The main consideration in signal processing with HTR distribution is having no closedform expression for its statistics. In other words, there is no analytic formula for the PDF of the HTR distribution. One solution to overcome this limitation is utilizing the mixture approximation. This drawback also makes it difficult to estimate its parameters (it is difficult to derive the maximum likelihood (ML) estimators). We propose a new approximation with a tractable density, with a novel estimator for its mixture ratio.
For the HTR distribution to be practically used for any ultrasound imagery application, one must be competent to estimate the parameters characteristic exponent, α, and dispersion, γ, from the observed data. The ML estimate can be obtained by letting the derivatives of the loglikelihood function to zero and solving the resultant transcendental equations. Although, it is time consuming and is not an effective procedure. Moreover, the most important disadvantage of this method is having no explicit expressions for parameter solutions. In the method of moments (MOM), only the negativeorder moments of the distribution were exploited to estimate the parameters [14]. The method of logcumulants (MOLC) is an extension of MOM, by utilizing the Mellin transform (MT) in place of the usual Fourier and Laplace transforms in statistical computations [15]. Still by analogy with classical statistic for scalar real random variables defined in \({\mathbb {R}^{+}}\), the second characteristic function (CF) of the second kind is defined as the natural logarithm of the first CF of the second kind. Parameter estimation based on the MT is a high accuracy and precision method to extracting the statistical features of ultrasound images.
A typical problem in ultrasound signal processing for coherent image formation is that the tissue scatterer is influenced by multiplicative speckle noise [16]. The multiplicative model was proposed for describing the statistical properties of the ultrasound returns. According to the multiplicative model, several PDFs are developed such as lognormal distribution, Kdistribution, and Gdistribution. Kdistribution is a particular form of the G model, which assumes both the tissue and the speckle component as gamma distribution [17]. Rician inverse Gaussian (RiIG) distribution is derived under the assumption that the scattering process acts as a Wiener Brownian motion with drift, superimposed on an inverse Gaussian (IG) distributed stopping time [18]. In many respects, this model is similar to the K model, but it has a flexible parameterization, which makes it more versatile.
Through this paper, we statistically model the envelope distribution of ultrasound images as a new generalized CauchyRayleigh mixture approximation based on HTR distribution. Moreover, analytical derivation for mixture ratio estimation based on the characteristic exponent parameter and the dispersion parameter of HTR distribution which has closedform expression is derived.
The rest of this paper is organized as follows. Section 2 provides the problem statement for describing the statistics behavior of ultrasound images and related parameter estimation methods. Specifically, three major mixture approximations of HTR distribution are given. In Section 3, our novel mixture approximation for HTR distribution is proposed and derivation of the mixture ratio estimator is provided. We also evaluate the performance of our proposed model through numerical simulations in Section 4. Furthermore, Section 5 provides experiments on real ultrasound images of different kinds. Finally, the paper is concluded in Section 6.
2 Amplitude statistics of ultrasound images
In this section, the prominent models used for modeling the amplitude PDF of ultrasound images are examined. We consider the RiIG model, the K model, and the HTR model with their parameter estimation methods.
2.1 RiIG distribution
The exact solution to this optimization must be numerically accomplished. The aforementioned steps should continue till a maximum number of iterations has been reached or till the desired convergence is achieved. From another point of view, the RiIG distribution has similarity to the K model; whereas its parameter estimation procedure is more computationally expensive, RiIG posterior distribution can be expressed in analytic form, which makes it facile to establish a maximum a posteriori (MAP) speckle filter.
2.2 KDistribution
where \(\tilde {k}_{n}\) denotes the logcumulant of degree n.
2.3 HTR distribution
which corresponds to the generalized Cauchy distribution. Furthermore, for other values of α, the integral in (9) has no closedform solutions; however, asymptotic series expansion exists.
Now, we consider the problem of parameter estimation of the proposed mixture model. The underlying parameter estimation based on the secondorder cumulants is derived using MOLC as an approach for statistical parameter estimation different from the MOM [15].
Finally, by estimating \(\hat \alpha \) and \(\hat \gamma \), the corresponding mixture ratio estimate is procured.
2.3.1 Existing approximations

LM1 method:$$ \varepsilon_{\alpha} = 2\left(\frac{\alpha1}{\alpha}\right), $$(16)

LM2 method:$$ \varepsilon_{\alpha} = \frac{4}{3}\left(\frac{\alpha^{2}1}{\alpha^{2}}\right), $$(17)

FLOM method:$$ \varepsilon_{\alpha} = \frac{\Gamma(1\frac{p}{\alpha})\Gamma(1\frac{p}{2})}{\Gamma(1p)\Gamma(1\frac{p}{2})}, 2<p<\alpha, $$(18)
where ε _{ α } is the mixture ratio and p denotes the pthorder moment which is not necessarily an integer and positive. The main disadvantage of these methods is the utilization of S α S mixture ratio for HTR distribution, whereas the HTR distribution is the amplitude PDF of a complex S α S random variable and is not symmetric.
In the next section, we propose our model for amplitude PDF of ultrasound images based on a mixture approximation of HTR distribution.
3 Proposed model
To provide a closedform expression for the amplitude PDF of ultrasound images which have been modeled as a HTR distribution, the generalized CauchyRayleigh mixture approximation is proposed. Our proposed method constructs a new mixture for HTR distribution in the context of ultrasound image applications in which its mixture ratio contains both a characteristic exponent and a dispersion of the HTR distribution, α and γ. Since the S α S random variable representation as a scale mixture of the Gaussian random variable is traditional and therefore PDF of S α S can be approximated by a finite Gaussian mixture model (GMM) [29], it suggests that a similar nonGaussian mixture model may be useful for HTR random variables. In the following, the proposed mixture approximation based on α and γ with mixture ratio estimation is introduced.
3.1 Proposed mixture approximation
It is noted that (20) is analytically characterized, and hence, mixture ratio estimation with a closedform expression is achievable. In the following, we introduce an estimator for the mixture ratio which is basically based on the energy stored in the CF of the distribution.
3.2 Mixture ratio estimation based on α and γ
The proposed mixture approximation is a model for the amplitude PDF of complex S α S or HTR distribution. So, only the ultrasound image in which its related RF signal is symmetric (with no skewness) can be modeled with this approach. However, in many applications, the symmetric distribution is considered.
4 Simulation results
In this section, for quantitative evaluation of the proposed mixture, simulations in terms of error, KS distance, KL divergence as the goodnessoffit tests, and computation time are provided. The appropriate choice of moment order plays an important role on FLOM technique for estimating the parameters, −2<p<−0.5 [25]. According to the simulation results, we use p=−0.5 as the best choice while estimating the parameters of the HTR distribution.
4.1 Proposed mixture approximation assessment
Even though the behavior of the samples is very impulsive due to the small values of α, the performance of the proposed method is significantly better than the other models. Figure 3 b also examines the effect of sample size on the error between the empirical PDF and approximations through 1000 simulations. It is simply seen that the value of error decreases by increasing the sample size. In this figure, only different sample sizes form 10,000 to 100,000 are plotted. Since LM1, LM2, FLOM, and the proposed mixture ratios have the same value (0) for α=1 (generalized Cauchy distribution) and the same value (1) for α=2 (Rayleigh distribution), their performance and resultant error are the same at the beginning and at the end of Fig. 3 a.
4.2 KS distance
4.3 Computation time
Now, the performance of the HTR mixture ratio estimators is verified in the context of time complexity and in terms of execution time on a machine.
Mean computation time (in milliseconds) obtained from the 1000 simulation runs for the HTR mixture approximations based on α=1.5 and γ=3
LM1  LM2  FLOM  Proposed  

N=10^{3}  0.58  0.62  0.65  0.78 
N=10^{4}  0.90  0.94  0.95  0.99 
N=10^{5}  5.15  5.29  5.61  5.95 
N=10^{6}  52.1  53.0  54.6  59.3 
5 Experiments on real ultrasound images
Comparison of the six models in the context of error (dB)
K  RiIG  HTRLM1  HTRLM2  HTRFLOM  HTRProposed  

CCA  −14.7025  −12.7357  −14.6616  −12.9651  −13.6374  −19.0420 
Thyroid cyst  −16.6479  −13.8617  −16.7142  −16.7285  −16.6356  −16.8583 
Pancreas  −16.1737  −11.5753  −16.2033  −16.1832  −16.2709  −16.2894 
Breast mass  −18.3238  −10.2990  −18.0846  −18.0010  −18.3902  −18.6290 
KS distance and parameter estimation for K, RiIG, HTRLM1, HTRLM2, HTRFLOM, and HTRproposed models
Model  Parameters  CCA  Thyroid cyst  Pancreas  Breast mass  

KS dis  KS dis  KS dis  KS dis  
K  \(\hat \nu \)  −0.3089  0.0732  0.4786  0.0227  1.6197  0.0343  1.2159  0.0172 
\(\hat \sigma \)  25.8518  55.9408  38.7482  28.3202  
RiIG  \(\hat a\)  3.0394  0.1349  3.2811  0.1893  4.4463  0.1538  3.5398  0.1709 
\(\hat \lambda \)  0.1725  0.4372  0.4095  0.4302  
\(\hat \delta \)  1.9891  1.2166  1.3952  1.2472  
HTRLM1  \(\hat \alpha \)  1.2042  0.0982  1.9505  0.0231  1.8160  0.0369  1.6939  0.0213 
\(\hat \gamma \)  18.8584  36.5655  98.9915  40.7420  
\(\hat \varepsilon _{\alpha }\)  0.3392  0.9548  0.8327  0.6568  
HTRLM2  \(\hat \alpha \)  1.2042  0.1517  1.9505  0.0232  1.8160  0.0391  1.6939  0.0241 
\(\hat \gamma \)  18.8584  36.5655  98.9915  40.7420  
\(\hat \varepsilon _{\alpha }\)  0.4139  0.9532  0.8167  0.6333  
HTRFLOM  \(\hat \alpha \)  1.2042  0.1446  1.9505  0.0225  1.8160  0.0272  1.6939  0.0200 
\(\hat \gamma \)  18.8584  36.5655  98.9915  40.7420  
\(\hat \varepsilon _{\alpha }\)  0.0155  0.9623  0.8420  0.6649  
HTRProposed  \(\hat \alpha \)  1.2042  0.0671  1.9505  0.0217  1.8160  0.0272  1.6939  0.0169 
\(\hat \gamma \)  18.8584  36.5655  98.9915  40.7420  
\(\hat \varepsilon _{\alpha,\gamma }\)  0.2169  0.9603  0.8053  0.6288 
5.1 KL divergence
Numerical results obtained from KL divergence test for four ultrasound images
K  RiIG  HTRLM1  HTRLM2  HTRFLOM  HTRProposed  

CCA  0.1041  0.1328  0.0994  0.1296  0.1315  0.0767 
Thyroid cyst  0.0068  0.2974  0.0033  0.0044  −0.0015  −0.0002 
Pancreas  0.0497  0.2243  0.0431  0.0419  0.0499  0.0407 
Breast mass  0.0215  0.2671  0.0249  0.0230  0.0246  0.0211 
5.2 L _{ m }Norm
6 Conclusions
In this paper, an effective scheme for statistically nonGaussian modeling of the amplitude PDF of ultrasound images is proposed. The basic idea of the model is to represent a mixture approximation for HTR distribution with a closedform formula whose mixture ratio is a function of both α and γ. In particular, we not only consider the characteristic exponent on estimating the mixture ratio but also the influence of dispersion has been investigated. Performance comparison between the proposed method and the existing ones is adopted through several simulations, and the experimental results obtained from real ultrasound images in terms of error, KS distance, KL divergence, and L _{ m }norm confirm the plausibility of the new approach in terms of accuracy and fitness.
Declarations
Authors’ contributions
MHB elaborated the novel generalized CauchyRayleigh mixture, carried out the experiments and statistical analysis of the images, and participated in drafting the manuscript. HA participated in the conception and design of the study, offered useful suggestions, and helped to draft the manuscript and revise it critically for important intellectual content. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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