# A Method to Obtain the Transducers Impulse Response (TIR) in Photoacoustic Imaging
## 3 Algorithms
The element along the diagonal of \(\mathbf{S}\) is called the singular value of \(\hat{\mathbf{Y}}\) and is equal to the positive square root of the eigenvalues of \(\hat{\mathbf{Y}}^{T}\hat{\mathbf{Y}}\). In this step, only the number of singular values which are equal to zero needs to be required. In the next step, the singular vector corresponding to the smallest singular value is used. It is the array of the PAPS. (2) Recalculate the eigenvector that \(\hat{\mathbf{E}}(K)\) has only one minimum eigenvalue (the numerical infinitely approaches zero). This eigenvector is the array of the PAPS. Break down the eigenvectors into \(J\)-group PAPS, and the length of the PAPS obtained for each transducer is \(K\). (3) TIR is found by the LS algorithm. It maps each of these \(J\) PAPS arrays to the corresponding transducer. When the matrix is solved, the result is only one of the general solutions, which may lead to the opposite result of the experiment. In this case, the PAPS needs to be determined to be correct. As in experiments and simulations, the starting point of the power amplifier signal always tends to rise. If the result is contrary to this, then it needs to be corrected by adding a minus sign in front of the PAPS data and then continuing to solve for the TIR. Next, the LS algorithm for solving the TIR in step (3) is introduced. In some optimization problems, such as curve-fitting problems, the objective function consists of the sum of squares of several functions [37; 38]. It can generally be written as  
\[F(\mathbf{h})=\sum_{i=1}^{M}f_{i}^{2}(\mathbf{h}), \tag{21}\]  
in (21), we assume  
\[f_{i}(\mathbf{h})=x_{i}*\mathbf{h}-\mathbf{y}_{i}.i=1,2,\cdots,M, \tag{22}\]where \(\mathbf{h}=(\mathbf{h}_{1},\mathbf{h}_{2},\ldots,\mathbf{h}_{L})^{T}\), which is the TIR vector of the transducer. The general assumption is that \(M\geq L\). Where \(x_{i}\) is the PAPS vector and \(y_{i}\) is the OPAS vector, Equation (22) is expressed in matrix form as  
\[f_{i}(\mathbf{h})=\mathbf{A}\mathbf{h}-\mathbf{B}=0, \tag{23}\]  
where \(\mathbf{A}\) is the C-matrix composed of the PAPS, and it is a matrix of \(M\) by \(L\)-dimensional. \(\mathbf{B}\) is a matrix of OPAS, which is an \(M\)-dimensional column vector. Let \(\mathbf{A}\) be full rank, and \(\mathbf{A}^{T}\mathbf{A}\) is a symmetric positive-definite matrix of order \(n\). The TIR formula derived from the LS algorithm derived from the literature [38] is  
\[\mathbf{h}=(\mathbf{A}^{T}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{B}. \tag{24}\]  
For linear LS problems, as long as \(\mathbf{A}^{T}\mathbf{A}\) is not singular, it can be solved with the Formula (24). Then, the resulting \(\mathbf{h}\) is the TIR signal. It should be noted that the algorithm in this paper is not suitable for regular-shaped circular and spherical biological tissue models. Under normal circumstances, we will arrange all transducers in a ring. If the shape of the measured biological tissue is round or spherical, and it is placed in the center of the calculation grid, then the OPAS received by each transducer is the same data. Using these data to solve TIR, the linear homogeneous system will be unsolvable. In this case, the method proposed in this paper is not applicable. If there is noise in OPAS data, then the algorithm must be improved. The matrix \(\hat{E}\) has two properties:  
(1) When the correct value of \(K\) is determined, the matrix \(\hat{E}\) has only one zero-eigenvalue and its corresponding eigenvector is PAPS. (2) If the value of \(K\) cannot be determined, the matrix \(\hat{E}\) has more than one zero-eigenvalue. If we determine that \(K=K^{\prime}\) (\(K^{\prime}\) is not the correct length of the PAPS signal), then the matrix \(\hat{E}\) will find \(L=K^{\prime}-K+1\) zero-eigenvalues. Accordingly, these zero eigenvectors will have the following structure:  
\[\mathbf{r}= \begin{pmatrix}\mathbf{r}_{1}\\ 0\\ \vdots\\ 0\\ \mathbf{r}_{2}\\ 0\\ \vdots\\ \mathbf{r}_{l}\end{pmatrix} .