# Rician Beamforming: Despeckle Method via Coarray Projection Stochastic Analysis
## 2 Physical and Mathematical Background  
### Physical Model and SAFT-Based Coarray Projection  
For the stochastic characterization of the underlying physical process, we start from Born model, notwithstanding its reported inaccuracies [15; 19]. Different tissues are modelled as reflective point sources, grouped or not, in high density distributions, within an isotropic minimally dispersive medium. The structure is found in semi-space \(z>d_{min}\), whereas ROI is limited to plane \(y=0\)--see Figure 1a. Regarding instrumentation, transducer has been modelled as a full multiple-input, multiple-output (MIMO) uniform linear array (ULA). It consists of \(N\) reciprocal sources, highly directive on \(\phi\) and cuasi-omnidirectional on \(\theta\). Array elements are interleaved by \(\lambda/2\). Electrical input to this array displays analytical form \(s(t)\in\mathbb{C}\), time separable and band-limited. For simplicity we consider a modulated Gaussian pulse with central frequency \(f_{0}\) and bandwidth BW. Furthermore, the proposed methodology and conclusions are extensible whenever prior restrictions hold. Figure 1: Uniform linear array (ULA) analysis geometry: (**a**) equidistant surface to a transmitter–receiver pair, (**b**) \(y=0\) plane. Hereafter, the bank of received signals is modelled as  
\[s_{\text{i,j}}(t)=\sum_{f=1}^{F}s(t)*g_{\text{i,j}}(t,\mathbf{p}_{f}) \tag{1}\]  
where \(F\) is the particle number, \(g_{i,j}(t,\mathbf{p}_{f})\) is the result of the cascade filters modelling electroacoustic responses of the pairs, as well as the temporary space round trip of source \(f\) placed at \(\mathbf{p}_{f}\). Considering Born model, \((\text{i,j})\)-th signal could be approximated as  
\[s_{\text{i,j}}(t)\approx\sum_{\forall f\in F^{\prime}}\rho_{f}s\left(t-\frac{d _{x_{\text{i,j}}\mathbf{p}_{f},x_{\text{j}}}}{c}\right) \tag{2}\]  
where \(c\) is the medium propagation speed and \(d_{x_{\text{i,j}}\mathbf{p}_{f},x_{\text{j}}}\) is the round trip for transmitter i, reflective point source and receiver j--see Figure 1b. Reflectivity of a point source is represented by \(\rho_{f}\in\mathbb{C}\). \(F^{\prime}\subset F\) is the subset of the sources close to \(y=0\) plane. On weakly dispersive media, it can be argued that for \(\bar{s}(t)\) temporal separability is preserved, as \(g_{\text{i,j}}(t,\mathbf{p}_{f})\) can be approximated by a delayed double derivative process. As SFFT is being used to compose the ultrasonic image, the pixel at \(\mathbf{p}_{a}\) underwent a focusing process at \(t=d_{x_{\text{i,j}}\mathbf{p}_{a},x_{\text{j}}}/c\) on signals \(s_{\text{i,j}}\) in \(\mathcal{S}\) dataset. To simplify Equation (2) UV coordinate system has been used. This introduces the concept of coarray [20]. After these transformations, the following approximation applies, \(d_{x_{\text{i,j}}\mathbf{p}_{f},x_{\text{j}}}\approx 2R_{f}-u_{f}\chi+\xi_{f}\), where \(u_{f}=\sin\theta_{f}\cos\phi_{f}\approx\sin\theta_{f}\), being \(R_{f}\), \(\theta_{f}\) and \(\phi_{f}\) the spherical coordinates where the point source is placed and \(\chi=x_{\text{i}}+x_{\text{j}}\). The additive term \(\xi_{f}\) accounts for errors due to far field approximation at point source \(f\). As a result, the nearer the particle is with respect to the array, the larger the influence of \(\xi_{f}\). In contrast to ULA, the \(\chi\) codomain is unevenly sampled, whereas \(\chi\) values are spatially sampled at \(\lambda/2\). Both are related by a surjective function: where \(N\) is the number of elements of ULA, indexed by set \(\mathcal{N}=\{0,1,\ldots,N-1\}\), considering a pulse-echo working mode, FMC signals are indexed by \(\mathbf{e}=(\text{i,j})\) tuples part of \(\mathcal{N}^{2}=\mathcal{N}\times\mathcal{N}\). The coarray domain is indexed by group \(\mathcal{L}=\mathcal{N}\oplus\mathcal{N}\), where \(\oplus\) denotes Kronecker sum.