## 2 The Methods of PAI and LUI
### Spatial Resolution, Sampling, and Limited-View Issues
Having analyzed more than a hundred sensitivity and spatial resolution maps, the authors provided approximate formulas for the FWHM of the sensitivity map along \(z\)-axis \(D\) and the lateral resolution \(Lat\)[82]:  
\[D^{2}(R)=D_{F}^{2}+4(R-F)^{2}\sin^{2}\biggl{[}\left(D_{\infty}^{2}-D_{F}^{2} \right)^{1/2}/(2R)\biggr{]},\]  
\[Lat^{2}(R)=Lat_{F}^{2}+4(R-F)^{2}\sin^{2}\biggl{[}\left(Lat_{\infty}^{2}-Lat_ {F}^{2}\right)^{1/2}/(2R)\biggr{]},\]  
where \(R\) and \(F\) are the radii of curvature of the sensor array in the image plane and perpendicular to it, and \(D_{F}\), \(Lat_{F}\) are FWHM of the sensitivity region and the lateral resolution for a spherical array (\(R=F\)), \(D_{\infty}\), \(Lat_{\infty}\)--for a cylindrical array (\(R\rightarrow\infty\)). As the radius of curvature increases (sensor array is getting \"more cylindrical\" and less focused in the image plane) the size of the region with high sensitivity increases, and the lateral resolution worsens. Spherical arrays have the best lateral resolution and the smallest region with high sensitivity. Toroidal geometry of the array can be optimized for a specific task using formulas above. Earlier the practical limitations for PA imaging were addressed in [85]. Here, we will briefly review the results and try to draw parallels between PA and LU imaging. In the case of a finite bandwidth of the sensors and full-view reconstruction in ideal medium, the PA PSF is isotropic, spatially independent, and depends on the sensor's temporal response \(h(t)\): \(PSF(\mathbf{r}|\mathbf{r}_{0})=PSF(|\mathbf{r}-\mathbf{r}_{0}|)=-(h^{\prime}(|\mathbf{r}-\mathbf{r}_{0}| /c)+h^{\prime}(-|\mathbf{r}-\mathbf{r}_{0}|/c))/(4\pi|\mathbf{r}-\mathbf{r}_{0}|)\). In the case of a finite sensor aperture, the registered signal is a spatial average of the actual ultrasonic field. The PSF is typically spatially dependent and anisotropic. If the sensor's bandwidth is infinite, the angular size of the PSF (as seen from the center of curvature of the detection surface) is generally equal to the angular aperture of the sensor. The spatial resolution maps show that increasing the radius of curvature in the image plane decreases the lateral resolution. If the detection surface is an infinite plane, the PSF will be spatially independent. The axial resolution in PAI is determined by the bandwidth of the sensors \(f_{sens}\), the lateral resolution by the angular size of the sensors in the image plane, and the elevation resolution by the width of the sensors in the elevation direction and focusing. The axial spatial resolution in LUI seems to be twice as good as the axial resolution in PAI. Let two point-like PA sources (or LU scatterers) be separated by the distance \(d\) in the medium with speed-of-sound \(c\). The duration \(\delta t\) of the temporal response from each source/scatterers is limited by the bandwidth of the sensors (and the duration of the probe pulse in LUI): \(\delta t\gtrsim 1/f_{sens}\). The temporal responses from the two point-like PA sources will not overlap when \(d/c\gtrsim\delta t\gtrsim 1/f_{sens}\), since the PA pulses travel from the sources to the sensors only once. The temporal responses from the two point-like LU scatterers will not overlap when \(2d/c\gtrsim\delta t\gtrsim 1/f_{sens}\) since, first, the LU probe pulse should travel the distance \(d\) between the scatterers, and then the reflected pulse should travel the distance \(d\) back to the sensor. For the same reason, the LU PSF (the LU image of a point-like acoustic scatterer) is half as thick in the longitudinal direction, in comparison with PA PSF. If, however, the acoustic boundary is highly reflective, the travel-time can be estimated using dedicated time-delay estimation algorithms. In this case, the distance to the boundary can be found with much better accuracy than the size of LU PSF [52; 53]. Temporal sampling usually does not pose a problem, provided the sampling frequency of the analog-to-digital converter (ADC) fulfills the Nyquist criterion.