# Rician Beamforming: Despeckle Method via Coarray Projection Stochastic Analysis
## 3 Rician Beamformer  
The variability of the stochastic behaviour of the random variables involved in the focusing and projection processes of the coarray demands the integration of all three scenarios, prior to further analysis. For this, we exploit the space diversity principle. The latter is commonly used in radar environments, not new to ultrasonic ULA systems, but up to now was mainly used for side lobes' suppression [32]. Sparse coarray sampling has proved valuable on modifying ULA directivity properties [33; 34; 35; 36]. This is an usual strategy for resource minimisation [37; 38]--i.e., coarray elements decimation--or to formulate effective acquisition strategies, at the expense of a minimum performance loss. All these approaches struggle between resolution improvement and lobes artefacts minimisation. Space diversity is here exploited for sparse coarray random selection and applied on projected data, see Equation (5):  
\[\hat{\mathcal{S}}_{p}:=\left\{\left.s(\chi)\right|_{\mathbf{p}_{\mathbf{z}}} \right\},\qquad A_{\mathbf{t}}\subset\hat{\mathcal{S}}_{p}\mid\mathrm{card} \left(A_{\mathbf{t}}\right)=L=\left\lfloor\frac{2}{3}\ (2N-1)\right\rfloor \tag{8}\]  
First side lobe suppression criterion has been considered to determine required sparse level [39]. In this particular case, the projection \(\left.s(\chi)\right|_{\mathbf{p}_{\mathbf{z}}}\) is a 1.5 periods wavelet. A new random variable \(Z_{p}\) is defined per each analysis point:  
\[Z_{p}=\left|\frac{1}{L}\sum_{s_{\mathbf{t}}\in A_{\mathbf{t}}}s_{\mathbf{t}}\right| \tag{9}\]  
By virtue of the central limit theorem, once a large enough number \(N_{z}\) of virtual arrays groups \(A_{\mathbf{t}}\) have been formed, averages for the real and imaginary parts are equivalent to a pair of Gaussian random variables, identically distributed, with potentially non-zero mean. This implies that in the three proposed scenarios in Section 2.2, random variable \(Z_{p}\) can be modelled as a Rician,  
\[f_{Z_{p}}(z\mid\varepsilon,\sigma)=\frac{z}{\sigma^{2}}\exp\left(\frac{-(z^{2} +\varepsilon^{2})}{2\sigma^{2}}\right)I_{0}\left(\frac{z\varepsilon}{\sigma^{ 2}}\right) \tag{10}\]  
and fitting of \(Z_{p}\) by means of the maximum likelihood method [40] provides an unbiased estimate of the Rician distribution. Figure 2 shows a naive example of several image points for a one spot source case. For all evaluated cases measured contributors are due to the electrical and acoustic noises, except for the first case. * Case 1: Real target, hence signal to noise ratio (\(\varepsilon/\sigma\)) is high. In this scenario \(Z_{p}\) acts as a Gaussian with \(\sqrt{\varepsilon^{2}+\sigma^{2}}\) mean and \(\sigma^{2}\) variance. In this specific case, \(\left.\mu\right|_{\mathbf{p}_{a}}\rightarrow\varepsilon\). * Cases 2 and 3: Most part of the noise is of acoustic nature and (\(\varepsilon/\sigma\)) ratio is low. Depending on the nature of \(\alpha\) and \(\beta\), see (5) (\(\varepsilon/\sigma\)) may be biased due to acoustic noise. Attending to (9), \(\left.\mu\right|_{\mathbf{p}_{a}}\rightarrow\varepsilon\) and \(\sigma^{2}\) represents noise power and can be used as a penalty. * Case 4: Signal to noise ratio is low and most part of the noise contribution is due to thermal-electrical causes. These points can be used to determine the inherent noise power, using \(\sigma^{2}\) estimator. From these considerations, a new pixel-wise estimator is proposed for ultrasonic imaging, based on (10) fitting:  
\[\mathrm{RiB}(\kappa)=F_{Z_{p}}^{-1}\left(\kappa\cdot F_{Z_{p}}\left(\left.\mu \right|_{\mathbf{p}_{a}}\right)\right) \tag{11}\]  
where \(F_{Z_{p}}\) is the cumulative distribution function for \(Z_{p}\) variable. The \(\kappa\) parameter controls penalty imposed by the proposed local expander--see Appendix A. Figure 2 depicts results from the proposed and DAS beamformers with \(\kappa=0.01\) for each of the last mentioned cases. For simplicity, Figure 2b depicts \(Z_{p}\) fitting applied to the four considered cases, normalized by their maximum values.