# Photoacoustic Imaging with a Finite-Size Circular Integrating Detector
## 4 Test and Discussion
### Ability to Calculate \(h(\phi)\) Without Noise  
To reduce the computational load, this simulation is conducted in a two-dimensional space. The initial pressure of the vascular model is loaded from an image and scaled accordingly, using a computational grid discretized into 256 \(\times\) 256 pixels (128 \(\times\) 128 mm), with a grid spacing of 0.5 mm. To satisfy boundary conditions, a perfectly matched layer (PML) with a thickness of 20 grid cells is applied. The vascular model is positioned at the center of the computational grid. A total of 120 detectors are arranged uniformly along a circular trajectory with a radius of 45 mm to simulate the signal acquisition configuration of a PAI system. Each detector is assumed to be a point detector, with its specific arrangement shown in Figure 4a. In the simulation, the speed of sound is set to 1500 m/s, and the optical absorption coefficient of the vascular model is defined as 1 mm\({}^{-1}\). To simplify the computation, the medium is assumed to be acoustically homogeneous, with no sound absorption or scattering. The simulation environment is considered nearly ideal, accounting only for the SIR while ignoring other noise sources. In practical applications, variations in detector size affect the SIR, which in turn influences the PSF of the reconstructed image. To simulate the circle detector PAI system, we set the angular coverage of the detector to \(2\theta=10^{\lx@math@degree}\), corresponding to a box function with \(2\theta=10^{\lx@math@degree}\). We then convolved the image with the blurring kernel. Figure 4b shows the image after blurring, Figure 4c shows the result after converting the blurred image into polar coordinates, and Figure 4d displays the blurring kernel used for the convolution. The method we propose for extracting blurring kernels from polar-coordinate pressure distributions involves coordinate transformation using bicubic interpolation. With the O (d\({}^{4}\)) error convergence characteristic of bicubic interpolation, the interpolation error under a grid spacing of d = 0.5 mm remains below \(10^{-3}\) pixel intensity levels. This error is within an acceptable range for the blurring kernel extraction computation. In the tests conducted in this section, we successfully extracted the blurring kernel using the proposed algorithm and reconstructed a clear image. First, we calculated the eigenvalues to determine the length of each column of data in the real discrete initial pressure distribution image. The length of the blurred pressure data in the polar coordinate image used in this study is 536. Therefore, we assumed the length of the initial pressure data to be 536 and compared the sizes of the smallest 536 eigenvectors. Figure 5a shows the distribution of eigenvalues in logarithmic coordinates. It can be observed that, after the 511 th eigenvalue, there is a significant drop, and the eigenvalues after this point are much smaller than the earlier ones. This indicates that the eigenvalues from the 512 th to the 536 th are close to zero and can be considered zero, totaling 25 zero eigenvalues. This step confirmed that the length of the blurring kernel is 25. Considering the relationship between the blurred pressure distribution, the initial pressure distribution, and the blurring kernel, \(M=K+L-1\). Figure 4: (**a**) An ideal initial pressure distribution image of the vascular model (red lines: PML; dots: detectors); (**b**) the blurred image of the vascular model; (**c**) the polar coordinate representation of the blurred image of the vascular model; (**d**) the blurring kernel set in this study. We considered the relationship between the discrete lengths of the blurred pressure distribution, the initial pressure distribution, and the blurring kernel: \(M=K+L-1\). We determined the length of the initial pressure distribution to be \(K=512\). When only one zero eigenvalue exists, the initial pressure distribution can be accurately determined.