# Rician Beamforming: Despeckle Method via Coarray Projection Stochastic Analysis
## 2 Physical and Mathematical Background
### SAFT Stochastic Behaviour  
In recent years, statistical studies on radio frequency (RF) signals and ultrasonic image themselves have attracted increasing interest as they relate stochastic process models [21] with tissue physical parameters. One of the magnitudes attracting special interest is the envelope, which has been related to particle spatial organisation and reflectivity [22; 23; 24]. Initial approximations were carried directly on B-Scan signals envelope \(E\)[22], but nowadays, work is around specific image regions or global appearance [25; 26]. Starting from (2) and its analytic form, knowing that \(s(t)\) is a modulated pulse:  
\[ \begin{array}{ll}s_{\mathrm{ij}}(t)=X\cos(2\pi f_{0}t)-i\,Y\sin(2\pi f_{0}t) &X=\sum\limits_{f=1}^{F^{\prime}}\alpha_{f}\cos(\phi_{f})\\ &Y=\sum\limits_{f=1}^{F^{\prime}}\beta_{f}\sin(\phi_{f})\end{array} \qquad E= \sqrt{X^{2}+Y^{2}} \tag{6}\]  
where \(F^{\prime}\) denotes all insonsificated scatterers near \(y=0\). In the general case, \(E\) matches a homodyned K-distribution [27; 28]. This is a variant of the Gamma distribution that yields a complex fitting process [29; 30], but describes a large number of possible scenarios, i.e., flexible modelling of \(X\) and \(Y\). However, these fitting processes lower parameter estimation accuracy as scatterers number increases [27]. Table 1 summaries distribution functions usually employed in ultrasound imaging to characterise envelop behaviour. Parameters could be interpreted as coherent signal power \(\varepsilon^{2}\), variance of scatterer strength \(\sigma^{2}\) and \(\alpha\) the scatterer clustering degree. These models can be extrapolated for DAS conformed ultrasonic imaging, as for SAFT the image results from the absolute value of focused samples mean at \(\mathbf{p}_{a}\). In analytic form  
\[\left.\mathcal{S}\right|_{\mathbf{p}_{a}}=X-iY\quad\longrightarrow\quad\mu \right|_{\mathbf{p}_{a}}=\mathbb{E}\left(\left.\mathcal{S}\right|_{\mathbf{p}_ {a}}\right)=\sqrt{\mathbb{E}^{2}(X)+\mathbb{E}^{2}(Y)} \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Probability Density Function** & **Asymptotic Values** \\ \hline Homodyned K-distribution & \(P_{\mathrm{HK}}(E|\varepsilon,\sigma^{2},\alpha)=E\int\limits_{0}^{\infty}\mu |_{0}(ue)I_{0}(ue)\left(1+\frac{u^{2}\sigma^{2}}{2}\right)^{-\alpha}\mathrm{d}u\) & \\ K-distribution & \(P_{\mathrm{K}}(E|\sigma^{2},\alpha)=\frac{4\varepsilon^{2}}{(2\sigma^{2})^{4 \varepsilon}(1)^{1/2}\Gamma(a)}K_{a}-\left(\sqrt{\frac{2}{\sigma}}E\right)\) & \(\varepsilon\to 0\) \\ Rice distribution & \(P_{\mathrm{Ri}}\left(E|\varepsilon,\sigma^{2}\right)=\frac{E}{\sigma^{2}}I_{0} \left(\frac{\varepsilon}{\sigma}E\right)\exp\left(\frac{-\left(\varepsilon^{2} +E^{2}\right)}{2\sigma^{2}}\right)\) & \(\varepsilon>0\quad\alpha\rightarrow\infty\) \\ Rayleigh distribution & \(P_{\mathrm{Ra}}\left(E|\sigma^{2}\right)=\frac{E}{\sigma^{2}}\exp\left(\frac{-E ^{2}}{2\sigma^{2}}\right)\) & \(\varepsilon\to 0\quad\alpha\rightarrow\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Distribution functions used to describe soft tissues behaviour. Overview of models in [29]. The approximated expected value is estimated by the standard average out of \(N^{2}\) samples. Once focused, the casuistic related to the \(X\) and \(Y\) variables modelling is dramatically reduced. This is because the same focusing process centres the zone under study, in spite of the integration inherent to the wave propagation process into the medium. Basically, \(X\) and \(Y\) reflect geometrical characteristics of the problem, displaying three different trends:  
* High particle count regions with low reflectivity discrepancies. In this case, both variables can be modelled as Gaussian noise with variance higher than the variables mean. The coarray projection in this zone shows random values of \(\alpha_{f}\) and \(\beta_{f}\) in Equation (5). As a result, \(E\) can be modelled as a traditional random walk, following a Rayleigh distribution.