# Photoacoustic Imaging with a Finite-Size Circular Integrating Detector
## 3 Algorithms
### Improved Algorithm Under Noise Conditions  
In the presence of noise, recovering the initial pressure distribution from the blurred pressure distribution becomes more challenging because noise can cause deviations in the zero eigenvalues of the matrix \(\mathbf{\hat{E}}\), or even introduce additional near-zero eigenvalues. These deviations can significantly affect the stability of the solution. To achieve a more stable recovery of the initial pressure distribution \(\mathbf{\hat{P}^{0}}\), we propose an improved algorithm that leverages multiple near-zero eigenvalues and their corresponding eigenvectors.  
The matrix \(\mathbf{\hat{E}}\) has the following two characteristics related to the initial pressure distribution [32]. In the absence of noise, the correct length \(K\) ensures that the null space of the matrix \(\mathbf{\hat{E}}\) is directly associated with the initial pressure distribution. Therefore, the initial pressure distribution can be uniquely determined by this zero eigenvector. In the presence of noise, the null space of the matrix \(\mathbf{\hat{E}}\) will expand, leading to multiple eigenvectors corresponding to zero eigenvalues. As a result, the initial pressure distribution becomes non-unique.  
We determine that \(K=K^{\prime}\) (\(K^{\prime}\) is not the correct length of the initial pressure distribution); then, matrix \(\mathbf{\hat{E}}\) will find zero eigenvalues of \(L=K^{\prime}-K+1\).  
To account for these additional eigenvalues, we structure the zero eigenvectors as:  
\[\mathbf{y}=\left(\begin{array}{c}\mathbf{r_{1}}\\ \mathbf{0}\\ \vdots\\ \mathbf{0}\\ \mathbf{r_{2}}\\ \mathbf{0}\\ \vdots\\ \mathbf{r_{j}}\end{array}\right) \tag{24}\]  
The first eigenvector consists of \(J\) initial pressure distribution vectors \(\mathbf{r_{j}}\) arranged in sequence, with each vector being appended by L additional zeros. The second eigenvector is generated as a cyclically shifted copy of the first eigenvector, and so on. The length of the blurred pressure data in each column of the image is \(M\), meaning that \(\mathbf{y}\) will be a \(MJ-L+1\) dimensional vector.  
When solving for the eigenvectors of matrix \(\mathbf{\hat{E}}(0)\) to determine the correct value of K, there exists an L-dimensional vector \(\mathbf{s_{j}}\) such that the eigenvector corresponding to the zero eigenvalue satisfies the following relationship with vector \(\mathbf{y}\):  
\[\mathbf{w_{j}}=\mathbf{y}*\mathbf{s_{j}}=\mathbf{Y}\mathbf{s_{j}}=\mathbf{S_{ j}}\mathbf{y} \tag{25}\]  
where \(\mathbf{w_{j}}\) represents the L eigenvectors corresponding to the zero eigenvalue of \(\mathbf{\hat{E}}(0)\). \(\mathbf{Y}\) denotes a C-matrix constructed from the vector \(\mathbf{y}\) with dimensions \(MJ-L+1\). \(\mathbf{S_{j}}\) represents the C-matrix constructed from the vector \(\mathbf{s_{j}}\). Among these, \(J_{\rho}\) eigenvectors \(\mathbf{w_{j}}\) are selected. Using these vectors, a matrix \(\mathbf{\hat{W}}\) is constructed, which is the same as the matrix in Equation (19).  
Similarly, \(\mathbf{\hat{W}}(MJ-L)\) also has a singular vector, for example, the singular value of \(\mathbf{u}\) is zero. It is equal to:  
\[\mathbf{s}=\left(\begin{array}{c}\mathbf{s_{1}}\\ \mathbf{s_{2}}\\ \vdots\\ \mathbf{s_{l}}\mathbf{v}\end{array}\right) \tag{26}\]  
Once we have the vector \(\mathbf{s_{j}}\), we can use Equation (25) to solve the initial pressure distribution \(\mathbf{y}\).