# Method for Detecting Underwater Microbubbles Using Dual-Mode Fusion of Laser Polarization
## 2 Dual-Mode Fusion Detection Mechanism
### Dual-Mode Fusion Discrimination
Figure 10: Results of different methods to identify bubbles under high-turbidity (attenuation coefficient 2.198 m\({}^{-1}\)) conditions. (**a**) Original image after different ways of processing; (**b**) binarized image after different ways of processing. information. At the same time, the weights based on the target polarization greatly enhance the polarization characteristics of weak targets such as bubbles, increase the gray value of small-sized bubbles in the polarization fusion image, effectively distinguish suspended particles and bubbles, and filter out part of the background scattered light to increase the contrast of the bubble target, making it easier to identify the target in the fused image. As a result, the fused image has a greater advantage in recognition detection rate and multi-frame stability, which can significantly improve the image resolution and help to accurately recognize and detect the features of smaller bubbles. In the dual-mode fusion module, the \(R^{2}\) index is used to verify the degree of fitting between the posterior probability model and the experimental data, so as to evaluate the performance of the prediction module [24]. The \(R^{2}\) index is also called the fit index, as shown in Equation (21). \[R^{2}=1-\frac{SS_{\text{reg}}}{SS_{\text{tot}}} \tag{21}\]  
where \(SS_{\text{reg}}\) is the sum of the squared errors of the statistical distributions, i.e., \(SS_{\text{reg}}=\sum_{i=1}^{M}\left(f_{\text{m,i}}-f_{\text{n,i}}\right)^{2}\), where \(M\) is the number of segments in the histogram of the collected data, \(f_{\text{m,i}}\) and \(f_{\text{n,i}}\) are the measured and predicted probability values corresponding to the i-th segment for a given received intensity level, respectively. \(SS_{\text{tot}}\) is the sum of the squares of the distances between the measured points and their mean values, i.e., \(SS_{\text{tot}}=\sum_{i=1}^{M}\left(f_{\text{m,i}}-\overline{f}\right)^{2}\), where \(\overline{f}=\sum_{i=1}^{M}f_{\text{m,i}}/M\). It is clear that this fitting function fits the experimental data better when the \(R^{2}\) measurements for a given distribution are close to the maximum (i.e., 1). Through simulation and polarization detection, we can know that under a certain degree of confidence, that is, within a certain degree of polarization, the speckle in the image can be determined to be a bubble in water. Therefore, the number density and diameter of a speckle can directly represent the number density and diameter of the bubble under a certain coefficient. Therefore, we can obtain the fitting function and curve of the gray fluctuation of the image and the density and size of bubble values, as shown in Figure 11 and Table 2 below: \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{R^{2}}\) & \(\mathbf{0.9352}\) & \(\mathbf{RMSE}\) & \(\mathbf{0.0503}\) \\ \hline a & 0.4694 & b & 0.020 \\ c & 0.1843 & \(\omega\) & 0.679 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Function fitting parameters. Figure 11: The bubble characteristic fitting results of dual mode fusion. From the analysis in Figure 11 and Table 2, it can be seen that the model is suitable for the feature detection of the mixed bubble layer in the range of 50-1000 \(\upmu\)m and 100-2000 cm\({}^{-3}\), and the fitting degree reaches 0.9352. As illustrated in Figure 11, the image grayscale fluctuation (GF) exhibits a consistent increasing trend when bubble size and bubble number density rise. This is due to the fact that when bubbles are large and dense, there is a greater chance that the image will capture them. The influence of these conditions is demonstrated by the increasing number of scattered bubble spots in the image, the acceleration of the bubbles' floating speed, and the rapid passage of the bubbles through the field of view, which cause significant fluctuations in the image's gray level across multiple frames. The image gray level fluctuation (GF) changes from sporadic weak fluctuation to a vast range of strong fluctuations as the size and density of the bubble rise.