# Photoacoustic Imaging with a Finite-Size Circular Integrating Detector
## 4 Test and Discussion
### The Influence of the Number of Bloured Pressure Distributions on the Results  
In the algorithm section, we mentioned using two columns of data for the blurring kernel estimations (\(j=57,58\)). To validate the effectiveness of this approach, we selected two columns of data with abundant and effective information from the polar coordinate image. The selection of these two columns was based on their significant structural features within the image, which provided sufficient information to support the calculation of the blurring kernel while avoiding the inclusion of excessive irrelevant information that could introduce computational interference. This selection method reflects the need for data representativeness. However, in some cases, the initial part of the calculated initial pressure distribution \(p_{j}^{0}(\phi)\) exhibited a downward trend that was inconsistent with the actual situation, as shown in Figure 6a,b. To address this issue, we hypothesized that a sign error might have occurred during the calculation.  
Therefore, a negative sign was added to \(p_{j}^{0}(\phi)\) for correction, and the corrected results are shown in Figure 6c,d. By convolving the corrected \(p_{j}^{0}(\phi)\) with the blurring kernel \(h_{\theta}(\phi)\), the final calculation results matched the blurred pressure distribution data exactly, validating the reliability of the method.  
To further evaluate the computational efficiency and accuracy of using two columns of data, we tested the effects of using more data columns. Figure 7 illustrates the feature value calculation results for different numbers of data columns. Figure 7a-d shows the relative sizes of the 536 feature values calculated using three columns of data, while Figure 7e-h presents the results for four columns of data. Regardless of whether three or four columns of data were used, the 25 recalculated feature values showed no significant improvement in accuracy compared to the results obtained using two columns of data.  
Figure 5: The results obtained by the algorithm: (**a**) distribution of the relative sizes of 536 eigenvalues; (**b**) relative sizes of the 20 eigenvalues after recalculation; (**c**) comparison between the calculated pressure distribution (CPD, represented by a blue dashed curve) and the theoretical initial pressure distribution (IPD, represented by a red solid curve).  
We conducted quantitative analyses using the computation time, peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM). As shown in Table 1, processing with two columns of data required 3.53 s, while using three and four columns increased the computation time to 21.04 s and 73.53 s, respectively. However, the PSNR only improved from 32.09 dB to 33.56 dB and 34.44 dB, and the SSIM increased from 0.9823 to 0.9860 and 0.9872. Although there was a marginal improvement in accuracy, the additional computational cost outweighed the benefits.  
Based on the above analysis, it can be concluded that selecting two columns of data with abundant effective information not only ensures sufficient accuracy in the calculation but also significantly improves computational efficiency.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Columns** & **Computation Time (s)** & **PSNR (dB)** & **SSIM** \\ \hline
2 & 3.53 & 32.09 & 0.9823 \\
3 & 21.04 & 33.56 & 0.9860 \\
4 & 73.53 & 34.44 & 0.9872 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computational efficiency and image quality metrics versus varying data column numbers.  
Figure 6: (a) IPD calculated from two sets of data; (b) \(h_{\theta}(\phi)\) calculated based on negative results; (c) IPD with negative sign added; (d) \(h_{\theta}(\phi)\) calculated after correction results.