# Adaptive Beamformer Combined with Phase Coherence Weighting Applied to Ultrafast Ultrasound
## 2 Materials and Methods  
### Modified Amplitude and Phase Estimation (mAPES) Beamforming [26]  
The complex ultrasound signals received by individual transducer elements in an ultrasound array probe are defined as follows:  
\[\mathbf{S}=(s_{0},s_{1},\cdots,s_{M-1})^{\mathrm{T}}, \tag{1}\]where \({}^{\rm T}\) and \(M\) denote the transpose and the number of transducer elements, respectively, and \(s_{m}(m=0,1,\cdots,M-1)\) is the complex echo signal received by the \(m\)-th transducer element. The output signal \(u\) is expressed as follows:  
\[u=\mathbf{w}^{\rm H}\mathbf{S}, \tag{2}\]  
where \({}^{\rm H}\) and \(\mathbf{w}\) are Hermitian operator and the weight vector applied to the received echo signals, respectively. Let us define the spatial covariance matrix by \(\mathbf{R}=\mathrm{E}[\mathbf{S}\mathbf{S}^{\rm H}]\), where \(\mathrm{E}[\cdot]\) denotes the expectation. In the APES beamforming, the weight \(\mathbf{w}_{\rm APES}\) is expressed as follows:  
\[\mathbf{w}_{\rm APES}=\frac{\mathbf{Q}^{-1}\mathbf{a}}{\mathbf{a}^{\rm H} \mathbf{Q}^{-1}\mathbf{a}}, \tag{3}\]  
where \(\mathbf{Q}=\mathbf{R}-\mathbf{G}\mathbf{G}^{\rm H}\), \(\mathbf{G}=[g_{0},g_{1},\cdots,g_{M-1}]^{\rm T}\), and \(\mathbf{a}\) is the steering vector expressed as follows:  
\[\mathbf{a}=\left(\begin{array}{c}a_{0}\\ a_{1}\\ \vdots\\ a_{M-1}\end{array}\right)=\left(\begin{array}{c}\exp\left(-j\frac{2\pi f_{0} \sqrt{(x_{0}-x_{f})^{2}-z_{f}^{2}}}{c_{0}}\right)\\ \exp\left(-j\frac{2\pi f_{0}\sqrt{(x_{1}-x_{f})^{2}-z_{f}^{2}}}{c_{0}}\right) \\ \vdots\\ \exp\left(-j\frac{2\pi f_{0}\sqrt{(x_{M-1}-x_{f})^{2}-z_{f}^{2}}}{c_{0}} \right)\end{array}\right), \tag{4}\]  
where \(f_{0}\) and \(c_{0}\) are the ultrasonic center frequency and speed of sound, respectively. The vector \(\mathbf{G}\) corresponds to the desired signal from the receiving focal point \((x_{f},z_{f})\), and \(g_{m}\)\((m=0,1,\cdots,M-1)\) is estimated as follows [26]:  
\[g_{m}=\frac{D(\theta_{m})\mathbf{a}^{\rm H}\mathbf{S}}{\sum_{i=0}^{M-1}D( \theta_{i})}, \tag{5}\]  
where  
\[D(\theta_{m})=\frac{\sin\left(\frac{\pi l}{\lambda}\sin\theta_{m}\right)}{ \frac{\pi l}{\lambda}\sin\theta_{m}}, \tag{6}\]  
\[\theta_{m}=\tan^{-1}\left(\frac{x_{f}-x_{m}}{z_{f}}\right). \tag{7}\]  
In our previous study, the outputs of the sub-aperture beamformers were used instead of \(\mathbf{S}\) in Equation (2) [26]. The output \(y_{k}\) of the \(k\)-th sub-aperture \((k=0,1,\cdots,K-1)\) consisting of \(M_{\rm sub}=M/K\) elements is expressed as follows:  
\[y_{k}=\left(a_{M_{\rm sub},k}^{*},a_{M_{\rm sub},k+1}^{*},\cdots,a_{M_{\rm sub },k+M_{\rm sub}-1}^{*}\right)\left(\begin{array}{c}s_{M_{\rm sub},k}\\ s_{M_{\rm sub},k+1}\\ \vdots\\ s_{M_{\rm sub},k+M_{\rm sub}-1}\end{array}\right)=\mathbf{a}_{k}^{\\[\mathbf{C}=\mathbf{Y}\mathbf{Y}^{\mathrm{H}}-\mathbf{V}\mathbf{V}^{ \mathrm{H}}, \tag{12}\] \[\mathbf{V}=(v_{0},v_{1},\cdots,v_{K-1})^{\mathrm{T}},\] (13) \[v_{k}=\frac{b_{k}\sum_{i=0}^{K-1}y_{i}}{\sum_{i=0}^{K-1}b_{i}},\] (14) \[b_{k}=\sum_{i=0}^{M_{\mathrm{sub}}-1}D(\theta_{M_{\mathrm{sub}} \cdot k+i}), \tag{15}\]  
where \(\mathbf{J}\) is a \(K\) dimensional vector of ones.