## 2 The Methods of PAI and LUI
### Forward and Inverse Problems of PAT and LUT--Reconstruction Algorithms
#### 2.2.2 Algorithms for LUT  
In the majority of applications reviewed in the present article, shear waves can be neglected. Therefore, here, we consider a forward problem of ultrasonic tomography in approximation of propagation of longitudinal waves in an infinite medium with no boundaries [77]:  
\[\
abla^{2}p-\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}=\gamma_{ \kappa}(\mathbf{r})\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}+\
abla \cdot\big{[}\gamma_{\rho}(\mathbf{r})\
abla p\big{]}, \tag{12}\]where \(\kappa,\rho\) are constant compressibility and density of the infinite medium, \(\widetilde{\kappa},\widetilde{\rho}\) are actual (inhomogeneous) compressibility and density, \(\gamma_{\kappa}(\mathbf{r})=(\widetilde{\kappa}(\mathbf{r})-\kappa)/\kappa\), \(\gamma_{\rho}(\mathbf{r})=(\widetilde{\rho}(\mathbf{r})-\rho)/\widetilde{\rho}(\mathbf{r})\)their deviations in a region with an object, \(c^{-2}=\rho\kappa\), and \(c\) is a speed of sound in the infinite medium. Even in this simplified formulation of the inverse problem, finding reflectivity of the medium is very complicated. Generally, all ultrasonic algorithms for solution of the inverse problem can be divided into two groups. Algorithms of the first group are based on a ray approach, and the second wave approach to solution of the inverse problem. Wave algorithms provide better resolution; however, they have strong requirements for memory and computational capacity. Therefore, it is more reasonable to use a ray-based approach for real-time imaging applications, which will be shortly reviewed here. _Back-projection._ There are few exact inversion formulas derived for ultrasonic tomography, and all of them are derived for soft biological tissues. Most of the inverse problems are solved under Born or Rytov approximations, or iteratively under these approximations. Single scattering covers the majority of regions of interest in nondestructive testing and evaluation (NDT&E) and biological imaging. [PERSON] and [PERSON] [77] derived one of such solutions for calculation of unknown parameters from backscattering measurements, for the general case of broadband insonification by a spherically diverging probe pulse under the assumption of the first Born approximation. They have shown that, in this case, simple back-projection gives a solution close to exact, under general practical conditions. The back-projection process is an easily interpreted reconstruction procedure, which assumes summing up all the signals recorded by all the sensors, which are known to have been scattered from the specified point. _Synthetic aperture focusing._ Many systems for ultrasonic tomography of solids are based on the synthetic aperture focusing technique (SAFT) [78; 79; 80]. In this technique, an acoustic array of piezoelectric transducers is used in a pulse-echo mode. A single ultrasonic image is obtained by a series of electric pulses. In each \"shot\", different time delays are applied to each element of the array in different sequences, to effectively focus the imaging system onto different points of the inspected object. Scattered waves are detected, and electrical signals are used to calculate \"one-shot\" images by, e.g., back-propagation. The resulting image is obtained by summing up of all \"one-shot\" images. The multiple \"shot\" reconstruction leads to a sufficiently long reconstruction process which, however, can be optimized for real-time applications. A filtered back-projection algorithm can be used in LUT [81]. The PA generator is a plane-parallel plate; therefore, the probe plane wave travels along \(z\)-axis towards the acoustic inhomogeneity, located in immersion liquid at the point \(\mathbf{r}_{ij}=\left(x_{i},z_{j}\right)\) of image. The scattered and reflected waves are detected by the \(m\)-th sensor at the point \(\mathbf{r}_{m}=\left(x_{m},z_{m}\right)\), \(m=1,\ldots,N\). The total travel time is calculated as \(\mathcal{T}_{ijm}=\left(z+\left|\mathbf{r}_{m}-\mathbf{r}_{ij}\right|\right)/c_{0}\).