## 3 Results and Analyses
### Algorithm Implementation  
In the proposed method, photoacoustic pressure data and photoacoustic image are obtained based on the homogeneous sound speed of 1495 m/s, as shown in Figure 4. This homogeneous sound speed value is selected from empirical values in the published literature. Comparing Figure 4 with the digital breast model in Figure 3a, it is observed that there are many artifacts in the photoacoustic image under a homogeneous sound speed. The boundaries of various tissues are blurred. The significant difference between the homogeneous and the real sound speed field seriously degrades the quality of the photoacoustic image. However, the photoacoustic pressure data obtained at a homogeneous sound speed can be used to provide initial regional position information about the sound speed field in the initial iteration. Figure 5 is a cluster deviation map of the photoacoustic pressure data at a homogeneous sound speed. As shown in Figure 5, the \(K\) value corresponding to the \"elbow\" is the optimal number of clusters, and the elbow method also provides the initial cluster center corresponding to \(K\) = 5. \"K-means + GMM\" is used to divide the photoacoustic pressure data into five types of regions, and the initial clustering result is shown in Figure 6a. Figure 6b is the initial inhomogeneous sound speed field. In practical applications, the substance set in the background area is usually a coupling medium with a known sound speed. The coupling medium in the model is water, so the sound speed in this area can be fixed at 1500 m/s. The sound speeds of the other tissues to be measured are unknown, and the initial sound speed in these regions is set to 1495 m/s based on empirical values. The photoacoustic \begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial Number** & **Coupling Medium and Biological Tissues** & **Real Sound Speed (m/s)** \\ \hline
1 & water & 1500.00 \\
2 & fibroagland & 1515.00 \\
3 & fat & 1440.00 \\
4 & skin & 1540.00 \\
5 & tumor & 1470.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Real sound speed values of various components in the digital breast model [26]. Figure 3: Digital breast model. (**a**) The \(y-z\) slice of the digital breast model at \(x\) = 4.0 \(\times\) 10\({}^{-2}\) m. (**b**) Real sound speed distribution map. image shown in Figure 6c is obtained based on the sound speed field shown in Figure 6b. Since the location information and sound speed of each region are inaccurate in the initial inhomogeneous sound speed field, the boundaries of fat and fibrogland in Figure 6c are confused. The focusing effect of various tissues is poor, and in particular, the shape of tumor 2 is severely distorted. Figure 4: Photoacoustic image obtained by using the homogeneous sound speed field of 1495 m/s. Figure 5: Cluster deviation map for different K-values. Figure 6: Initial iteration results. (**a**) Initial clustering map. (**b**) Initial inhomogeneous sound speed field. (**c**) Photoacoustic image obtained by using the initial inhomogeneous sound speed field. Figure 7 shows the fitting curve of the objective function value and the number of iterations. It can be observed from this figure that there is a transient irregular oscillation phenomenon before the objective function value reaches saturation, which may be caused by the following two reasons:  
1. In the process of solving, the step length of the updated sound speed of some tissues is too large, which affects the trend of gradient direction change and leads to the instability of the iterative process. 2. The objective function may be trapped in a local minimum, and the algorithm tries to escape from the local minimum, resulting in a transient oscillation of the objective function value [27]. The objective function value gradually converges in the middle and late iteration. This shows that the unique advantage of adaptively adjusting step length in the proposed method enhances the convergence of the algorithm.