## 2 The Methods of PAI and LUI
### Spatial Resolution, Sampling, and Limited-View Issues  
While the closed-form solution exists for the inverse problem of PA imaging, in practice the measured data is not perfect. The experimental signals contain noise. The sensors have finite bandwidth and aperture, while the reconstruction formulas assume they are point-like. The sensor array contains a finite number of sensors, and the experimental signals contain a finite number of samples, so the problem is discrete, rather than continuous. Finally, the investigated object may not be accessible from all angles, so the sensor array has limited angular aperture (limited view). In addiction, the analytical solutions usually assume lossless acoustically homogeneous medium, which may not be the case in practice. All these imperfections affect the image quality. The effects of imperfect imaging setup can be quantified using the point-spread function (PSF). PSF is the image of a point source reconstructed by the system. For the perfect system, the PSF is a Dirac delta function. Generally, for a given imaging setup, \(PSF(\mathbf{r}|\mathbf{r}_{0})\) functionally depends on the image spatial coordinates \(\mathbf{r}\), and parametrically depends on the coordinates of the point source \(\mathbf{r}_{0}\). The imaging system is usually assumed to be linear. In this case, the image of a non-point-like source is a spatial convolution of the source with the PSF. The sizes (e.g., full width at half-maximum, FWHM) of the PSF along three spatial directions--axial, lateral, and elevation--are the spatial resolutions of the system. PSF depends on both the geometrical characteristics of the sensor array and the reconstruction algorithm [82]. Manufacturing of sensor arrays remains a complicated procedure. Therefore, characterization and comparison of sensor array geometries is an important task. In [82; 83], the authors propose the sensitivity maps and spatial resolution maps for characterization of the toroidal sensor arrays with limited view and back-projection reconstruction algorithm. Toroidal arrays are promising for real-time 2D imaging with high lateral resolution [84]. Such arrays are curved, both in the image plane (to achieve high lateral resolution) and perpendicular to it (to achieve high elevation resolution). Construction of the sensitivity and spatial resolution maps involves placing a point PA source, calculating the response of all the sensors, accounting for their finite size and bandwidth, reconstructing the image of the source (PSF), and calculating its FWHM in axial and lateral directions (spatial resolutions--dimensions of \(PSF(\mathbf{r}|\mathbf{r}_{0})\)) and maximum amplitude (sensitivity is \(\underset{\mathbf{r}}{\max}PSF(\mathbf{r}|\mathbf{r}_{0})\) and depends on \(\mathbf{r}_{0}\)). Figure 2 shows the examples of such maps for cylindrical and spherical sensor arrays. Figure 2: Sensitivity and spatial resolution maps 32 \(\times\) 32 mm for a cylindrical (**top** row) and spherical (**bottom** row) arrays with 64 elements, 0.5 mm wide with 0.5 mm pitch and a curvature radius of 40 mm [83]. The sensor array is located below the maps symmetrically, with respect to the \(z=0\) mm line. The cylindrical sensor array is located at \(x=-40\) mm, and the center of curvature of the spherical array is \(x=0\) mm, \(z=0\) mm. Adapted with permission from [83], Springer Nature, 2018.