# Photoacoustic Imaging with a Finite-Size Circular Integrating Detector
## 4 Test and Discussion
### Algorithm Performance Testing in a Noisy Environment  
Gaussian noise with an SNR of 40 dB was added to the simulation data, and the improved algorithm's feasibility and robustness were tested. Columns 57, 58, and 67 of the pressure distribution were chosen as test data due to their rich information and their tail values, which were close to zero, reducing the impact of noise on performance. In the first step of the optimization algorithm, eigenvalue decomposition was performed on the constructed matrix, as shown in Figure 10a. Apart from the smallest eigenvalue, the remaining eigenvalues exhibit a stable decay trend. To analyze the variations in \begin{table}
\begin{tabular}{c c c} \hline \hline
**Image Condition** & **PSNR (dB)** & **SSIM** \\ \hline Blurred image & 10.41 & 0.6245 \\ Wiener deconvolution image & 23.26 & 0.7866 \\ Processed image (proposed) & 32.09 & 0.9823 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantitative evaluation of image quality for different processing methods (under noise-free conditions). Figure 9: Comparison of pixel values between two algorithms and unprocessed blurred images (**a**). Unprocessed blurred pressure distribution contour; (**b**) Wiener deconvolution algorithm (blue); (**c**) proposed algorithm (red). the tail region, after removing the last abrupt drop in the eigenvalue, the variations between the preceding eigenvalues can be observed more clearly, resulting in a more accurate estimation of the blurring kernel, as shown in Figure 10b. Through the analysis of Figure 10b, it can be observed that the eigenvalues gradually decrease without the noticeable \"jump\" seen in the noise-free case, which typically indicates the determination of the blurring kernel length. However, some \"inflection points\" can still be observed, where the difference between certain eigenvalues is more significant than the difference between others. An inflection point appears at the 26 th eigenvalue, which suggests that the blurring kernel length may be too short; another inflection point occurs at the 15 th eigenvalue, which corresponds to a more reasonable blurring kernel length. Therefore, we determine that L = 20. Although the blur length was set to L = 25 during the simulation, the actual blurring kernel of the noisy image is significantly affected by the noise in the data. During the optimization process, the algorithm interprets part of the original blurred distribution (especially the weaker signal components) as part of the initial pressure distribution rather than an extension of the blurring kernel. In this case, the blurring kernel length calculated according to the optimization algorithm will be shorter. After determining that L = 20, the algorithm proceeded to the second step, where the eigenvector corresponding to the smallest eigenvalue was used to reconstruct the initial pressure distribution. This eigenvector was decomposed into three groups of pressure sequences, and the LS method was applied to calculate the blurring kernel and deblur the entire image, yielding the processed pressure distribution image. Figure 11 shows the comparison results of pressure distribution curves before and after denoising processing: (a) the pressure distribution curve with noise exhibits significant blurring kernel effect and noise interference, and (b) after being processed using the optimization method proposed in this article, the pressure distribution curve exhibits more accurate feature extraction. In the absence of noise and when the blurring kernel is error free, the LS can serve as an effective approach for recovering the acoustic pressure distribution. However, in practical applications, both the observed data and the blurring kernel are typically subject to errors, and the least squares method cannot handle the errors in the blurring kernel effectively, resulting in the inability to completely eliminate the influence of the blurring kernel. This issue relates to the ill-posed nature of inverse problems.