# A Method to Obtain the Transducers Impulse Response (TIR) in Photoacoustic Imaging
## 4 Tests and Discussion
### The Ability to Solve the TIR  
The digital model of the vascular grid in the K-Wave toolbox will be used to simulate the PAI process and results. Figure 1a describes the photoacoustic data acquisition geometry diagram, and Figure 1b shows the vascular mesh geometry used for the simulation study. In this paper, the simulation is carried out in 2D (to reduce computational complexity), and the initial pressure of the vascular model is loaded from the image and scale using a computational grid of 128 \(\times\) 128 pixels. Here, PML (Set PML = 20) is used to satisfy the boundary conditions, the vascular grid is placed in the center of the computational grid, and the number of transducers is 120. Place the transducers clockwise on a 3/4 circle with a radius of 30 mm to simulate the PAI setup. The No. 1 transducer is placed on the far left of the horizontal line in the center of the grid, as shown in Figure 1a. The transducer is assumed to be a point transducer. The simulation assumes a sound velocity of 1500 m/s. The optical absorption coefficient of the vascular model was set to 2 mm\({}^{-1}\). For simplicity, the medium is considered acoustically homogeneous and has no absorption or diffusion of sound. Of course, any other medium (acoustically inhomogeneous) can also be considered. Since the simulated environment is a near-ideal environment, it is assumed to be unaffected by any noise other than TIR. It is necessary to set the TIR to study and verify whether the algorithm is suitable for PAI techniques. Different types of transducers have different TIR, especially in practical applications. However, the ideal TIR model is generally used in the theoretical or simulation research of PAI methods. The center frequency of the transducer was 3 MHz, the photoacoustic data were collected at a sampling rate of 10 MHz,  
Figure 1: In this paper, the use of the blood vessel grid model, the distribution of the sensor position, and IRT. (**a**) Describe the photoacoustic data acquisition geometry diagram. (**b**) Vascular mesh model in the K-Wave toolbox. (**c**) The hypothesized TIR is in this paper. and the number of temporal samples obtained by the sensor was 300. The bandwidth of the transducer set in this article is 80%. Its image in the time domain is shown in Figure 1c. Using MATLAB 2021b for simulation, the hypothetical TIR is convolved with the PAPS to obtain the OPAS. Here, 120 transducers are used for the calculation, place the transducers clockwise on a 3/4 circle with a radius of 30 mm, and each transducer is 2.25\({}^{\circ}\) apart. The signal length of the OPAS data obtained from each transducer in this paper is 339, so the length of PAPS is assumed to be 339 for the calculation of the algorithm, and the size of the smallest 339 eigenvectors taken is compared. Figure 2a is an image of eigenvalues plotted on a logarithmic scale. It can be seen that it drops significantly after the 300 th eigenvalue and is significantly smaller compared to other eigenvalues. Then, it can be considered that the eigenvalues from the 300 th point to the 339 th point are 0. That is, the number of zero-eigenvalues is 40. It can be seen from the algorithm section, that the accurate PAPS can be obtained when there is only one zero-eigenvalue. And then its length is known. Therefore, Figure 2b shows the relative sizes of the smallest 40 eigenvalues when recalculating the eigenvalues in the above case. It can be seen that the last eigenvalue is significantly smaller than the other eigenvalues. The eigenvector associated with this minimum eigenvalue is an array of 120 PAPS. Thus, the array length of the PAPS can be obtained as K = 300. The calculated PAPS of one of the transducers is compared with the original PAPS as shown in Figure 2c. It can be seen that the shape and trend are the same between them, and the calculated results are correct. The TIR in the PAI system is solved to verify the accuracy of the algorithm. After obtaining the PAPS, the LS algorithm can now be used to solve TIR.