## 2 The Methods of PAI and LUI
### Spatial Resolution, Sampling, and Limited-View Issues
Spatial undersampling leads to the increased number of artifacts in the image. The particular shape of the artifacts depends on the reconstruction algorithm. For the back-projection algorithm, the artifacts are streak-shaped. In PAI, the streaks have a spherical (or circular) shape, while in LUI with planar probe pulse, the streaks' shape is parabolic. In general, the streaks have a shape of lines of constant times-of-flight of the PA or LU acoustic response. The amplitude of the streak artifacts decreases as \(1/N\), where \(N\) is the number of the sensors, since \(N\) streaks are summed coherently in a single point in space where the source is located. Oversampling in the spatial domain (scanning with step-sizes smaller than the sensor's aperture), as well as oversampling in the time domain, cannot mitigate the blurring if the reconstruction procedure assumes point-like sensors with infinite bandwidth. Hence, sensitivity and spatial resolution maps depend only weakly on the number of sensors [82]. While there are algorithms that may restore the higher frequencies lost during the measurements, they may enhance the high-frequency noise and should be used with care. Limited view scenarios can be analyzed in the far-field approximation. Let the acoustic field be represented as an infinite sum of plane monochromatic waves (by spatiotemporal Fourier transform). Let the sensor at point \(\mathbf{r}(r,\theta_{r},\varphi_{r})=\mathbf{n}r\) (where \(r,\theta_{r},\varphi_{r}\) are the spherical coordinates of the sensor), far away from the source in the origin, detect the plane waves with wave vectors \(\mathbf{k}=\mathbf{\alpha}\omega/c\). If the PA source is an infinitely thin plane normal to vector \(\
u(1,\theta,0)\) (Figure 3), then the PA response will be a plane wave with wave vectors \(\mathbf{k}=\mathbf{r}\omega/c\) that can be detected by the sensors with \(\mathbf{n}=\mathbf{\
u}\). If the LU reflector is the same infinitely thin plane, and the probe pulse is a plane wave with a wave vector \(\mathbf{m}(1,\pi,0)\), then the reflected wave will have the wave vectors \(\mathbf{k}=\mathbf{u}\omega/c\), where \(\mathbf{u}(1,2\theta,0)\). Therefore, the angular aperture necessary to detect the reflected LU pulse should be twice as large as the angular aperture necessary to detect the PA response. In PAI, the angular aperture of \(2\pi\) steradian is sufficient for exact reconstruction, since the PA response in the direction \((-\
u)\) can be expressed using the response in the direction \(\mathbf{\
u}\). In LU imaging, the acoustic inhomogeneities cannot be reconstructed without full tomographic coverage. In general, for PA imaging, the \"visibility\" condition [86] states that the image will be blurred away (\"invisible\") at point \(P\) on the sharp boundary \(L\) of the object, if the normal to \(L\) at \(P\) does not pass through any receiver (there is no sensor that will detect the corresponding plane wave). In LU imaging, the \"visibility\" condition is analogous. Non-linear combination of multiple images for different positions of the sensor array [27], or different positions of the point-like sources inside the object [87], can be used to improve image quality. However, these approaches require longer acquisition times.