# Photoacoustic Imaging with a Finite-Size Circular Integrating Detector
## 2 Methods
### Blurring Effects of Circular Integrating Detectors  
According to [31], the finite size of the circular integrating detector causes a spatially variant blur. The PSF exhibits spatially variant characteristics. The blurred pressure distribution can be expressed as the convolution of the initial pressure distribution \(p_{0}(\mathbf{r^{\prime}})\) and PSF \(h(\mathbf{r})\)[27]:  
\[p(\mathbf{r})=\int\limits_{\Re^{2}}h(\mathbf{r}\text{-}\mathbf{r^{\prime}})p_ {0}(\mathbf{r^{\prime}})dA(\mathbf{r^{\prime}}) \tag{6}\]  
Figure 3 provides a simple example to illustrate this phenomenon visually. Figure 3a shows the initial pressure distribution consisting of four points. Figure 3b shows the image reconstructed using a detector with an aperture of \(2\theta\). The relationship between the arc aperture size and its radius is proportional, and the blurring effect increases linearly with the distance from the center of the reconstruction domain. The blur of the point sources near the detector approaches the width of the detector. A similar phenomenon occurs when a camera or an object being photographed rotates rapidly around a central point; during the exposure, the resulting image exhibits blur in the arc direction. This angular blur, which is proportional to the distance from the rotation center, is referred to as \"spin blur\".  
Figure 3: (a) Initial pressure distribution; (b) reconstruction in Cartesian coordinates; (c) reconstruction in polar coordinates.  
Figure 2: Schematic diagram of the circular array PAI system.  
The \"spin blur\" image can be described using the following formula:  
\[p^{\theta}(r,\phi)=\int\limits_{-\theta}^{\theta}p_{0}(r,\phi+\theta^{\prime})d \theta^{\prime}=\int\limits_{0}^{2\pi}h_{\theta}(\phi-\phi^{\prime})p_{0}(r,\phi ^{\prime})d\phi^{\prime} \tag{7}\]  
where \(p^{\theta}(r,\phi)\) is the blurred pressure distribution in the radial \(r\) and angular \(\phi\) directions, and \(p_{0}(r,\phi^{\prime})\) represents the initial pressure distribution. The blurring kernel is represented as: \(h_{\theta}(\phi)=\begin{cases}1,&|\phi-\phi^{\prime}|\leq\theta\\ 0,&\text{otherwise}\end{cases}\), where \(\theta\) represents half of the angle of the detector aperture.  
In Cartesian coordinates, the spatially varying nature of the PSF leads to complex coupling between pixels. However, in polar coordinates, the blurring effect remains constant in the \(\phi\) direction and is independent of \(r\) (as shown in Figure 3c), which simplifies the problem. On this basis, we can convert the spatially varying blurring problem into a spatially invariant blurring problem through coordinate transformation. Specifically, the image is transformed from Cartesian coordinates to polar coordinates, and pixel information for the polar coordinate image \(P_{pol}\) is obtained using bilinear interpolation. In \(P_{pol}\), each column corresponds to a fixed radius, so the blurring effect in that column is independent of the pixels in other columns. As a result, after transforming the \(n\times n\) Cartesian image into a \(\frac{n}{2}\times 2n\) polar image, the problem is decomposed into \(\frac{n}{2}\) independent blurring problems, each with a length of 2 \(n\), and all of these problems share the same blurring kernel.  
We represent the blurring problem at each specific radius as:  
\[p^{\theta}(\phi)=p^{0}(\phi)*h_{\theta}(\phi) \tag{8}\]  
At this point, the initial pressure distribution of interest is denoted by \(p^{0}(\phi)\), and the blurred pressure distribution is denoted by \(p^{\theta}(\phi)\). We focus on the initial pressure distribution. Therefore, in photoacoustic image processing, the general procedure involves estimating the blurring kernel and then removing its effect on the image through deconvolution.