## 2 Methods
### Computation of MICF Using Adaptive OTSU Algorithm  
The clutter in images is reduced by utilizing the CF for weighting the focal point intensities [30]. The CF is calculated as follows:  
\[CF_{\varepsilon}=\frac{\left(\sum_{n}\sum_{i,j}s_{i,j,e}(n)\right)^{2}}{\sum_{n }\sum_{i,j}\left(s_{i,j,e}(n)\right)^{2}}, \tag{3}\]  
where \(\left(\sum_{n}\sum_{i,j}s_{i,j,e}(n)\right)^{2}\) denotes the square of the coherent sum between the signals of different channels, and \(\sum_{n}\sum_{i,j}\left(s_{i,j,e}(n)\right)^{2}\) denotes the incoherent sum between the signals of different channels. As shown in Figure 3a, the CF values between the signals are highest at the focal point where the tumor appears, and the time-delayed signals are aligned peak-to-peak and valley-to-valley. Multiplying with a high CF value preserves the intensity value of the tumor focal point. At some focal points without tumors, the time-delayed signal peaks and valleys are not aligned, as shown in Figure 3b, and the peaks and valleys cancel out after the signals are summed, so the coherent sum is lower and the CF is lower. Multiplying with a low CF reduces the intensity value of the tumor focal point. However, in some focal points without tumors, such as S16 and S23 in Figure 3c, there is no temporal overlap between the tumor scattering signals to offset the signal intensities by summation, and the higher values of the coherent sum result in higher values of CF. There is another  
Figure 2: Block diagram of the MICF-DMAS algorithm for reconstructing the image of position \(\ddot{m}_{e}\) in the breast phantom. case, in which the focal point has no tumor and the peaks and valleys of some of the signals are aligned after a time delay, such as for S16, S23, and S37 in Figure 3d, where higher CF values are also produced. In both cases, the higher CF value multiplied by the focus intensity results in a higher intensity at the focal point where there is no tumor, creating clutter on the image. The first step of the MICF is to find pairs of tumor signals for which there exists a temporal non-overlap. Following pre-artifact removal, signal \(s_{i,j,e}(n)\) contains high-intensity tumor scattering signals and other low-intensity residual signals. In this paper, the absolute values of the signal \(s_{i,j,e}(n)\) were represented as a grayscale histogram, the OTSU algorithm [23] was used to adaptively select the appropriate threshold value, and the signal was divided into the tumor signal region and other regions according to the intensity strength. In order to maintain the integrity of the tumor signal, the first local minimum before the first point in the signal that exceeds the threshold (point n1 in Figure 4) is used as the starting moment of the tumor signal. The first local minimum after the last point in the signal that exceeds the threshold (point n2 in Figure 4) is taken as the moment of termination of the tumor signal. By comparing the tumor signal regions across different channels, we determine whether there is a temporal overlap of tumor signals. We assume that the start and end times of the tumor signal region in signal \(s_{i,j,e}(n)\) are \(n_{i,j,e,start}\) and \(n_{i,j,e,stop}\). We assume that the intensity values of signal \(|s_{i,j,e}(n)|\) range from \([A_{\min},A_{\max}]\). \(A_{f}\) is one of the intensity values in signal \(|s_{i,j,e}(n)|\). The number of points with intensity value \(A_{f}\) in signal \(|s_{i,j,e}(n)|\) is \(e_{f}\). The total number of points of the discrete signal \(|s_{i,j,e}(n)|\) is \(U\). The normalized probability of occurrence of \(A_{f}\) is \(b_{f}\), which can be expressed as  
Figure 3: Schematic diagrams of S16, S23, S37, and S62 waveforms with different focal points. The signal strength is normalized for ease of observation: (**a**) Focal point of tumor presence. High coherence of different channel waveforms. (**b**) No tumor focal point. Low coherence of different channel waveforms. (**c**) No tumor focal point. Low coherence of different channel waveforms. S16 and S23 do not overlap in time.