Several statistical models have been proposed in the literature to describe the behavior of speckles. Among them, the Nakagami distribution has proven to very accurately characterize the speckle behavior in tissues. However, it fails when describing the heavier tails caused by the impulsive response of a speckle. The Generalized Gamma (GG) distribution (which also generalizes the Nakagami distribution) was proposed to overcome these limitations. Despite the advantages of the distribution in terms of goodness of fitting, its main drawback is the lack of a closed-form maximum likelihood (ML) estimates. Thus, the calculation of its parameters becomes difficult and not attractive. In this work, we propose (1) a simple but robust methodology to estimate the ML parameters of GG distributions and (2) a Generalized Gama Mixture Model (GGMM). These mixture models are of great value in ultrasound imaging when the received signal is characterized by a different nature of tissues. We show that a better speckle characterization is achieved when using GG and GGMM rather than other state-of-the-art distributions and mixture models. Results showed the better performance of the GG distribution in characterizing the speckle of blood and myocardial tissue in ultrasonic images.
Among the noninvasive imaging modalities, probably, the most widespread are the ultrasound imaging. The main reason of its success is that it provides a low-cost way to help diagnosing and can be used for many medical applications. However, ultrasonic (US) images are characterized by the presence of a peculiar granular pattern: the so-called
This term was adopted from the field of laser optics [
The analysis of backscattered echo from tissues needs a proper description of the ultrasonic signals. For this purpose, and due to the random nature of the speckle, several statistical models have been proposed in the literature. This characterization can be used either for segmentation [
The statistical description of US signals provide an important information of the backscattered echo from tissues. The parameters of the statistical models allow identifying the features of tissues and provides important descriptors for classification. Some of the filtering algorithms relay on a Bayesian approach where an accurate statistical model becomes necessary. As a consequence, modeling the amplitude statistics of US signals has been a very active area.
Several statistical models have been proposed in the last decades. Probably the most wellknown is the Rayleigh model, which is a one-parameter distribution which describes the so-called fully formed (or developed) speckle. This probabilistic distribution describes the behavior of a speckle when a high number of effective scatterers are present in the resolution cell. However, real images show a deviation from this model, this non-Rayleigh behavior can be due to a small number of scatterers in the resolution cell or when there are some dominant components in the cell. The most commonly accepted distributions that try to model non-Rayleigh distributions are the Rice (fully resolved speckle),
Although, those models are based on physical assumptions of the backscattering process, some other distributions have proven to provide a good performance on real images. This is the case of Gamma [
The capability of the Nakagami distribution to model the backscattering from tissues for fully resolved and fully formed speckle made it become the most commonly accepted model for tissue characterization. However, the tails of the probabilistic density functions of Nakagami, K, Rayleigh, or Gamma do not show the impulsive response of speckle which originate heavier tails. In order to describe this impulsive response, a generalized Nakagami distribution was proposed by Shankar in [
The different nature of tissues is reflected in a different response of the speckle. Hence, a mixture model has shown to be a natural strategy for statistically describing the features of tissues. This approach has been previously used for segmentation purposes in the case of Nakagami mixture models (NMMs) by Destrempes et al. in [
The EM algorithm cannot be easily applied for the calculation of a Generalized Gamma Mixture Model (GGMM) without an MLE. However, some interesting results have been recently published on the calculation of the MLE of the Generalized Gamma which permit efficient computation of the GGMM.
The aim of this work is to revitalize the use of the Generalized Gamma distribution (also called, Generalized Nakagami Distribution) for tissue characterization. For this purpose, we present two main contributions: first, we propose a simple methodology to calculate the ML estimate which offers robust results comparing to the methods in the literature [
The rest of the paper is structured as follows. In Section
The formation of US images begins with the emission of a pulse packet which travels through the tissue. The backscattering produced by the scatterers in the resolution cell contribute to the change of the pulse shape according to the characteristics of the media, that is, the number of scatterers as well as their size [
The contribution of the backscattered echo,
The fully formed speckle model assumes a high number of scatterers, so the
Then, the envelop of the backscattered signal echo,
Under the assumption of a high number of effective scatterers but with the presence of resolvable structures in the resolution cell (specular component,
When the number of scatterers decreases and the Central Limit Theorem cannot be applied, more complicated distributions are proposed to model the distribution of the envelope. Concretely, the
A generalization of the previous models appears when a specular component is considered and the number of scatterers,
On a completely different approach, Shankar in [
The Nakagami PDF is as follows:
This distribution offers good properties to describe the backscattered echo: the Rayleigh distribution is a particular case of the Nakagami (
In order to describe the impulsive response of scatterers, Shankar proposed in [
In the next section, we describe some methods that have been used in the literature with special attention to methods that provide an ML estimate of the GG parameters. Additionally, we propose a simple method to calculate the ML estimates of the parameters. The results obtained in the derivation of this ML method provide the foundations for the development of the Generalized Gamma Mixture Model, which is the main contribution of this work.
This method was proposed by Stacy in [
This is the definition of the GG distribution hereafter. For a given
Now, let
For this RV, the central moments,
Additionally, it is easy to show that, given a RV,
So, a new RV
The moment generating function of
Finally, the three first central moments are defined as:
These equations can be used to estimate the parameters of the
The estimates are derived by means of calculating the value
This method, though provides a quite straight-forward calculation of the parameters, can provide estimates which are outside the parameter space. Yet, it is highly sensitive to the number of samples.
In order to avoid the problems associated to the moments method, some heuristic methods have been proposed in the literature. As examples, Gomes et al. [
This method presents some shortcomings. First, the parameters of the Gamma distributed data were calculated by the moments method, so the problems associated to the moments method are not circumvent. However, even if a good estimate is calculated, the
Other heuristic method is the one presented by Wingo in [
Now, calculating the derivatives with respect to the parameters and setting it equal to zero, one can obtain the ML equations:
This system of equations can be reduced to a single nonlinear equation with
In [
Though this method can provide an ML estimate of the parameter by solving (
A very interesting analysis was recently published by Noufaily and Jones in [
In that work, the log-likelihood equations were calculated following the re-parametrization proposed in [
So, in the end, the following equations have to be solved:
The important result of [
This method provides a fundamental result about the behavior of the log-likelihood equations, and guarantees their solution. However, the method does not provide any proof concerning its convergence or the uniqueness of the ML. Yet, this method needs to solve two nonlinear equations by numerical techniques, whereas the method proposed by Wingo in [
We propose a simple but efficient method to calculate the ML estimates of the GG distribution. The main advantage of the method is that it can be easily implemented and has the same properties of the method of [
The method consists in transforming the RV,
In order to see if this method provides a proper solution, we first demonstrate that the ML estimate of the parameters of the new random variable
First, we calculate the ML estimates of the parameters of (
The maximum with respect to the parameter
Now, by introducing
This result guarantees that there exists always a solution for the ML estimate of the GG distribution (
Additionally, since the MLE of a Gamma distribution always exist for whatever positive
The search method for
This method does not demonstrate the uniqueness of
The main advantage of the method here proposed is that it is easy to implement and only one non linear equation has to be solved, whereas the method of [
An additional advantage of the proposed method for the calculation of the MLE parameters for the GG distribution is that it can be easily adapted for the calculation of the parameters of GG Mixture Models (GGMM).
There were some attempts in the literature to obtain the parameters of a GGMM. Concretely, in [
In this section, we derive the GGMM by applying the Expectation-Maximization methodology [
Let
The joint distribution of IID samples is given by
The EM is applied here to maximize the log-likelihood function when some hidden discrete random variables
Now, defining
In the maximization step, the new estimate
The application of the EM algorithm for estimating the parameters of mixture models has been applied for several distributions, see, for example, [
In order to derive the estimates of the parameters in each iteration, we first define the joint distribution of IID samples
Now, the log-likelihood function can be defined in the following way:
The expectation of the log-likelihood function with respect to the hidden RVs when data
The probability
Note that (
Now, calculating the derivative with respect to each
By summing both terms of the equation over
For the calculation of the maximum of (
The result is
Now, plugging (
Note that (
The interval where the Brent’s algorithm is performed can be derived by means of the following property:
So, the desired value of
This property can be found in [
Now, the problem can be stated in the same way as was done for the ML estimate proposed in Section
It is important to note that the parameter estimates can be also solved by extending the ML method of [
The log-likelihood equations to be solved are completely equivalent to (
So, in the case of the parameter
This equation is well behaved and all the theoretical demonstrations obtained in [
As a conclusion, (
The solution is in the interval
So, finally, from an initial guess of
This methodology generalizes the proposed method of [
In this section, we detail the implementation of both of the proposed methods for the GGMM.
In Algorithm
each component
maxIter Tol Err
err
In the case of Algorithm
each component
maxIter Tol err
maxIterML TolML
errML err
The computational complexity of the previous GGMM methods when compared to the calculation of a simple GG depends on the number of components,
When other mixture models such as RMM, NMM, and GMM are considered, the complexity of the EM method is similar to the GGMM. Note that both the GMM and the NMM need to solve a non-linear equation similar to (
The estimated times in a Matlab (R2011a) implementation running in an ASUS G53SW laptop (Intel Core i7 2630QM Processor, 2.2 GHz, 8 GB RAM) were
In this section, we show the performance of the proposed methods for calculating the parameters of a GG distribution. For this purpose, we performed
The synthetic data was calculated in the same way as was done in [
We choose this interval since lower values than
Some examples of the GG distribution for the parameters of the synthetic dataset.
The number of iterations for the proposed method and for the method of [
The Kolmogorov-Smirnov statistic is the uniform norm of the cumulative distribution function (CDF), defined as
In Figure
Results for
In order to see the effect of this, we also show in Figure
Results for the relative error of the estimates for
An example of the fitting performance of the methods is shown in Figure
Example of the fitting performance for
Following, we analyze the dependence of the estimates with the number of samples. The same experiment is repeated considering 500 samples. The results of both goodness-to-fit measures are shown in Figure
Results for
Results for the relative error of the estimates for 500 samples. Methods: Stacy and Mihram [
The better performance of Noufaily and the proposed methods are seen in Figure
In this section, we test the performance of the GG distribution for characterizing tissues of real images. For this purpose, we used a set of 518 real US images (
In Figure
Example of an image of the data set. The red contour is the segmented areas of blood which are considered in the study, while the green contour is the segmented areas of tissue. The intersection of both regions was rejected in the study.
Additionally, the histogram of the image was depicted for the blood region as well as the fitted distributions most commonly used to characterize tissue. From the whole data set, a total number of
In the case of Figure
The performance of the GG distributions was tested by estimating the PDFs for both tissue classes (myocardial tissue and blood) for the following distributions: Exponential, Rayleigh, Weibull, Normal, Nakagami, Gamma, and GG. The PDFs were compared by means of both the
Results for the relative error of the estimates for 500 samples. Methods: Stacy Stacy and Mihram [
The results are shown in Tables
Results of the
Blood |
|
Hypothesis |
---|---|---|
Nakagami versus Gamma | <10−15 |
|
Gamma versus GG |
|
|
Nakagami versus GG | <10−15 |
|
Results of the
Myocardial tissue |
|
Hypothesis |
---|---|---|
Nakagami versus Gamma |
|
|
Gamma versus GG |
|
|
Nakagami versus GG |
|
|
Mean values for
Nakagami | Gamma | Generalized Gamma | |
---|---|---|---|
Blood |
|
|
|
Myocardial |
|
|
|
In this section, we test the performance of the proposed GGMM methods in three different scenarios. First, we test the necessity of using more than a simple GG for describing tissues with an increasing echolucent response of the effective scatterers. The case of a variation of the number of effective scatterers is also considered. This behavior can be found in structures with an increasing deterministic response that changes the speckle nature from fully formed speckle to fully resolved speckle. The variation of the number of effective scatterers can be found in structures which change their scattering cross-section.
In order to simulate B-mode US images, we followed the same methodology proposed in [
As a first example, we simulate an increasing echolucent tissue which varies its intensity from 0 to 255 from left to right. The sampling process and the resulting B-mode image are shown in Figures
Simulation of spatial variant echolucent response of tissue. (a) Sampling of an increasing echolucent tissue. (b) Resulting B-mode image obtained by corrupting the samples by a random walk process of
Sampling
Resulting B-mode image
GG and GGMM fitted to data
The fitted GG and GGMM with
As an additional experiment, in Figure
Simulation of spatial variant density of scatterers. The number of scatterers per resolution cell decreases from left to right in order to simulate fully formed speckle in regions with low density and partially resolved speckle in regions with high density.
Simulated B-mode image
GG and GGMM fitted to data
The speckle PDF in this case becomes more impulsive in areas with more effective scatterers (left part of Figure
In the last synthetic experiment for testing the necessity of GGMM, we simulate an anatomic phantom of a kidney scan. For this purpose, we used the artificial kidney scan proposed by Jensen [
Simulation of an anatomic phantom of a kidney scan.
Anatomic phantom of a kidney
Simulated B-mode image
GG and GGMM fitted to data
Probability of belonging to each component of the GGMM fitted to the image in Figure
For testing the performance of the proposed GGMM methods with real data, we use the same data set used in the previous section. The number of components is set to two: blood and myocardial tissue. In order to compare the performance of the GGMM methods, we also fit a Gamma Mixture Model and a Nakagami Mixture model to the data [
The lower values of
Results for
A proper characterization of the speckle by means of suitable distributions can be used to guide segmentation algorithms as the one in [
This last filter includes the probability of belonging each tissue class and adapts the diffusion tensor. Concretely, it calculates the structure tensor of the posterior probability and detects the most probable edges of the image. This information is used to define the diffusion tensor which provides a better behavior in the boundaries of the image.
The structure tensor of the probability density function for each tissue class is calculated as:
Let us consider the following diffusion equation:
Given a diffusion tensor,
As an example, when one eigenvalue is equal to
The POSRAD philosophy makes use of the structure tensors determined out of the probability maps to obtain the most probable structures. In that case, the diffusion filter should be anisotropic. When no probable structures are detected, the diffusion should be isotropic.
Since we have
The interpretation of this choice is that we choose as boundary the one with the maximal gradient of the probability density function over all tissue classes. This way, the most probable boundary is preserved in the filtering process. In the basis of
In Figure
Probability of belonging to each tissue class, where the class
Anisotropic behavior of the filter. The most probably edges of the image are described by the lower values of
Finally, the resulting image after
Results of the POSRAD filter. The anisotropic behavior of the filter is appreciated in the preserved details of the myocardial tissues.
Original image
Filtered image
As a final application of the GGMM, one can make use of the pixel-wise probability of belonging to each tissue class to obtain a spatially coherent probability by introducing an undirected graph where the nodes (each pixel of the image) represent a random variable and the edges of the graph represent the relationships between nodes as it is represented in Figure
Undirected graph. The nodes represent a random variable
In the end, the problem is faced as a discrete MRF where the labels,
The output of the LBP is a belief of node
Probability of belonging to each tissue class after the LBP, where the class
These coherent probability maps can be of help for classifying purposes or as prior information for segmentation algorithms. The valuable information that they provide can be seen in a simple experiment in which we consider the classification of two tissues (blood and myocardial tissue) and we compare the results with the
Simple example of the valuable information of the posterior probability obtained from the GGMM with spatial coherence.
Classification with
Classification with the GGMM probability maps after LBP
Throughout this work we have analyzed the advantages of using a GG distribution for characterizing the speckle in ultrasound images. This distribution offers a suitable way to deal with the impulsive behavior of speckle which causes heavier tails in the distributions. Additionally, the GG is a natural generalization for many distributions commonly used to characterize the speckle: Rayleigh, Gamma, Nakagami, Weibull, Exponential, and Rician [
Although some approaches have used this distribution in the literature, the inconveniences of estimating its parameters make this option thorny and not attractive. The problem stems from the inaccurate estimate of the moments method proposed in [
In this work, we have proposed a simple methodology to calculate the ML estimate which offers robust results comparing to the methods in the literature [
Results with
The formulation of the proposed method allows to generalize this methodology to a GGMM. These mixture models are of great value due to the different nature of the echogenic response of tissues in the received signal. Two different methods were proposed for the calculation of the GGMM parameters GGMM1 and GGMM2. Both were developed by applying the EM method in the derivation of the proposed ML method, the optimization technique for GGMM1 follows the same approach used for the proposed ML method. The GGMM2 method makes use of the optimization technique proposed by [
Through this paper we showed the better behavior of the GGMM methods when compared to the RMM, NMM, and GMM for the case of cardiac imaging. The potentials of mixture models have proven a good classification performance in intravascular ultrasonic images for RMM [
We think the GGMM methods here proposed can be used with good results in the aforementioned modalities since they generalize the RMM, NMM, and GMM in a natural way and allow to describe heavier tails of the PDFs that the RMM, NMM, and GMM fail to fit. Many other US modalities such as breast, liver, and kidney should be considered. We hope the proposed GGMM methods can encourage future research for tissue characterization in those different US modalities.
Finally, we want to recall that the potential applications of GGMM do not confine to those proposed in this paper. We hope the results of this work can revive the use of the GG distribution and its extension, the GGMM, in many other areas.
The authors would like to thank Marta Sitges, Etelvino Silva (Hospital Clinic, IDIBAPS, Universitat de Barcelona, Spain), Bart Bijnens (Instituco Catalana de Recerca i Estudis Avan cats (ICREA) Spain), and Nicolas Duchateau (CISTIB-Universitat Pompeu Fabra, Ciber-BBN, Barcelona, Spain) for providing the images. The authors acknowledge Junta de Castilla y León for Grants VA37A611-2 and VA0339A10-2, Ministerio de Ciencia e Innovación for Grants CEN-20091044, TEC2010-17982, and MTM2007-63257, and Instituto de Salud Carlos III for Grant PI11/01492.