## 2 Methods
### Proposed Methods
#### 2.5.2 Adaptive Diagonal Reducing for CMSB  
The user-defined constant diagonal reducing factor \(\delta\) in CMSB has an influence on the imaging performance. The contrast of CMSB improves with the decreasing \(\delta\) while the speckle quality degrades [39]. The speckle quality of CMSB is insufficient in phantom experiments. In other words, a tradeoff exists between contrast and lesion detectability in CMSB due to the constant \(\delta\). Therefore, an accurate selection of \(\delta\) for different imaging targets has the potential to overcome this problem. Here, we introduce adaptively choosing \(\delta\) for diagonal reducing in CMSB by combining CF and the normalized reciprocal of ASD to maintain a balanced imaging performance with an improved speckle quality.  
As an adaptive weighting, CF combined with synthetic aperture focusing has been applied in intravascular ultrasound imaging, showing an improved image quality [44; 45]. The CF is calculated as the ratio of coherent to incoherent energy of echo signals,\[\text{CF}(p)=\frac{|\sum_{n=1}^{N}y_{n}(p)|^{2}}{N\sum_{n=1}^{N}|y_{n}(p)|^{2}}, \tag{15}\]  
When signals are perfectly coherent, the CF value equals 1, while, in the case of incoherent noise, it falls to 0. CF has the ability to reduce clutter and noise effectively; however, it often degrades the speckle statistics [25].  
Nevertheless, the speckle statistics of CF can be improved by adjustment using the normalized reciprocal of ASD (i.e., \(\sigma^{\prime}\)). The adaptive diagonal reducing factor selected using the aperture coherence (i.e., CF(\(p\)) and \(\sigma^{\prime}(p)\)) for imaging point \(p\) is defined as  
\[\delta_{ac}(p)=\text{CF}(p)^{\sigma^{\prime}(p)}\times\delta_{max}, \tag{16}\]  
where \(\delta_{max}\) is the maximum diagonal reducing factor. According to (16), the value of \(\delta_{ac}(p)\) is in the range from 0 to \(\delta_{max}\). The diagonal reducing factor selected for coherent signals is slightly larger than that for speckle signals, which is much larger than that for incoherent noise.  
According to (10), the covariance matrix through the pixel-based adaptive diagonal reducing is obtained as  
\[\widetilde{\mathbf{R}}(p)=\tilde{\mathbf{R}}(p)-\delta_{ac}(p)\tilde{\mathbf{R}}(p)\cdot \mathbf{I}, \tag{17}\]  
Then, the weight \(\widetilde{w}_{i}(p)\) of CMSB with adaptive diagonal reducing is calculated according to (11) as  
\[\widetilde{w}_{i}(p)=\frac{\frac{1}{L(p)}\sum_{n=1}^{L(p)}\widetilde{\mathbf{R}}_ {i,n}(p)}{\sqrt{\frac{1}{L(p)}\sum_{n=1}^{L(p)}(\widetilde{\mathbf{R}}_{i,n}(p)- \frac{1}{L(p)}\sum_{n=1}^{L(p)}\widetilde{\mathbf{R}}_{i,n}(p))^{2}}}, \tag{18}\]  
where \(i\in[1,L(p)]\) is the row index.  
The weights estimated from incoherent noise are always very low under different \(\delta\)[39]. In addition, the weights estimated from speckle signals and coherent signals are more close due to the reason that the \(\delta\) for speckle signals is slightly smaller than that for coherent signals. As a result, this will lead to an enhanced speckle intensity and smoothness in the CMSB image.  
The output of CMSB with adaptive diagonal reducing factor \(\delta_{ac}(p)\), referred to as CMSB (\(\delta=\delta_{ac}\)), is obtained as  
\[V_{\text{CMSB}(\delta=\delta_{ac})}(p)=\frac{1}{N-L(p)+1}\sum_{l=1}^{N-L(p)+1} \frac{\widetilde{w}^{H}(p)\hat{y}_{l}(p)}{L(p)}, \tag{19}\]  
where \(\widetilde{w}(p)=[\widetilde{w}_{1}(p),\widetilde{w}_{2}(p),...,\widetilde{w} _{L(p)}(p)]^{T}\), and \(\hat{y}_{l}(p)=[y_{l}(p),y_{l+1}(p),...,y_{l+L(p)-1}(p)]^{T}\).