A Generalized Gamma Mixture Model for Ultrasonic Tissue Characterization

A Generalized Gamma Mixture Model for Ultrasonic Tissue Characterization Gonzalo Vegas-Sanchez-Ferrero, Santiago Aja-Fernandez, Cesar Palencia, and Marcos Martin-Fernandez Laboratorio de Procesado de Imagen, ETSI Telecomunicación Edificio de las Nuevas Tecnologías, Campus Miguel Delibes s/n, Universidad de Valladolid, 47011 Valladolid, Spain Received 1 June 2012; Revised 20 August 2012; Accepted 31 August 2012 Academic Editor: Huafeng Liu Copyright © 2012 Gonzalo Vegas-Sanchez-Ferrero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract 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. 1. Introduction 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 speckle. This term was adopted from the field of laser optics [1] in the early 1960s due to the similarity of the patterns between laser optics and ultrasonics. Although the nature of the speckle in US images stems from a different phenomenon, there still share some similarities. Both patterns come from the random interference of many coherent wave components reflected from different microscopic elements. In the case of US, the volume, the number of effective scatterers, and the acquisition process contribute to the formation of a speckle [2]. 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 [3], classification [4] purposes or for filtering the speckle itself [5–8]. The latter usually considers the speckle as an undesired consequence, since it degrades resolution and adds spatial noise to the image. Thus, filtering is commonly applied as a preprocessing step for further segmentation of regions of interest or to extract relevant measures for physiological analysis. 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), (partially formed speckle), and Homodyned (partially 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 [9–11] and Nakagami [12] distributions. The first is proposed as a two-parameter distribution that describes the result of interpolated/filtered fully formed speckle [9] and also has shown good results in empirical tests among other distributions [10, 11]. The Nakagami distribution proposed by Shankar for the case US characterization [12] is also a two-parameter distribution which generalizes the Rayleigh distribution. This distribution was adopted from the models proposed to describe the statistics of the returned echo radar. 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 [13]. This is a three-parameter model which has shown a better behavior than the Nakagami or Rayleigh, an expected result since it is a generalization of the other models. However, the generalized Nakagami distribution does not have closed-form maximum likelihood estimates (MLE) and, thus, it makes their use difficult. Note that, though Shankar in [13] said that the MLE can be obtained, the equations used were based on the results from Stacy and Mihram [14], which were calculated by the methods of moments and they also expressed the difficulties of obtaining an MLE: “Closed expressions for solutions to the maximum likelihood equations are highly unlikely.” It is important to note that the results of [14] were obtained for the estimation of the Generalized Gamma (GG) distribution which is essentially the same as the generalized Nakagami distribution but with another parametrization. The different nature of tissues is reflected in a different (...truncated)