## 2 The Methods of PAI and LUI
### Forward and Inverse Problems of PAT and LUT--Reconstruction Algorithms
#### 2.2.1 Algorithms for PAT
\tag{4}\]  
To find the reconstructed initial pressure distribution \(p_{0}^{r}(\mathbf{r})=p_{tr}(\mathbf{r},2t_{0})\), the solution can be carried out using, for example, Fourier transform in combination with [PERSON]'s function. One of the key advantages of these types of inversion algorithms is that they do not assume the medium to be acoustically homogeneous, and can even be generalized to account for acoustic absorption and dispersion. However, such algorithms are quite computationally expensive, and still cannot be used for real-time imaging. Fourier transform-based algorithmFourier-domain algorithms for PAT [73] are based on the decomposition of the acoustic field. Such algorithms are usually applied when planar detector geometry is used, and the medium is assumed to be acoustically homogeneous. In this case, the acoustic field can be decomposed into the sum of planar monochromatic waves using efficient Fourier transform algorithms. In the cases of spherical and cylindrical detector geometries, the acoustic field has to be decomposed into the sum of spherical or cylindrical waves, respectively, which cannot be done with such high efficiency. They operate using spatial and temporal frequency components \(\mathbf{k}=\big{(}k_{x},k_{y},k_{z}\big{)}\) (wave-vector) and \(\omega\) (temporal circular frequency) of solution:  
\[p(\mathbf{r},t)=(2\pi)^{-3}\int P_{0}(\mathbf{k})\cos(\omega t)\exp(i\mathbf{k}\cdot\mathbf{r} )d^{3}\mathbf{k},\omega=c|\mathbf{k}|=c\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}, \tag{5}\]  
where \(P_{0}(\mathbf{k})=\int p_{0}(\mathbf{r})\exp(-i\mathbf{k}\cdot\mathbf{r})d^{3}\mathbf{r}\). The spatial spectrum of the pressure field acquired by the planar detector at the plane \(z=0\) is  
\[A\big{(}k_{x},k_{y},t\big{)}=\int\int p(x,y,z=0,t)\exp\big{[}-t\big{(}k_{x}x+k _{y}y\big{)}\big{]}dxdy. \tag{6}\]  
The spatial spectrum of the reconstructed initial pressure distribution \(P_{0}^{r}(\mathbf{k})\) can be obtained by the inverse spatial Fourier transform of the expression:  
\[P_{0}^{r}\Big{[}k_{x},k_{y},k_{z}=\pm\sqrt{\left(\omega/c\right)^{2}\ -k_{x}^{2}-k_{y}^{2}}\Big{]}=\pm 2 \frac{c^{2}}{\omega}\sqrt{\left(\omega/c\right)^{2}\ -k_{x}^{2}-k_{y}^{2}}\int_{0}^{ \infty}A\big{(}k_{x},k_{y},\ t\big{)}\cos(\omega t)dt. \tag{7}\]  
Back-projection algorithmXu and [PERSON] [74] derived the exact inversion back-projection type formula for the cases of planar, cylindrical, and spherical detector geometries under the assumption of uniform speed of sound in the medium:  
\[p_{0}^{r}(\mathbf{r})=\int_{S_{0}}\frac{d\Omega_{0}}{\Omega_{0}}\Big{[}2p(r_{S},t) -2t\frac{\partial p(r_{S},t)}{\partial t}\Big{]}_{t=|\mathbf{r}-r_{S}|/c}, \tag{8}\]  
where integration is preformed over the detection surface \(S_{0}\), \(d\Omega_{0}\) is the solid angle of the detection surface element \(dS_{0}\) at point \(r_{S}\), and \(\Omega_{0}\) is the solid angle of the whole surface \(S_{0}\) as seen from point \(\mathbf{r}\)Expression (8) can be easily interpreted. If there is a point-like PA source at the point, \(\mathbf{r}\), the travel time \(t(\tau_{S})\) of the acoustic wave, from this source to the detector at the point \(\mathbf{r}_{S}\), will be \(|\mathbf{r}-\tau_{S}|/c\). Therefore, for each detector, a \"probability sphere\" of radius \(ct(\mathbf{r}_{S})\) can be constructed, and the PA source will be located at the intersection of all the \"probability spheres\" for all the detectors. Reconstruction of the amplitude of the PA source requires the quantity in square brackets in (8) to be projected onto the \"probability sphere\" and all the results for all the detectors to be summed. _Model-based algorithms._ Model-based algorithms generally use a discrete representation of (2), with decomposition of initial pressure distribution \(p_{0}(\mathbf{r})\) into a finite sum of pre-defined basis functions with decomposition coefficients [75; 76]. This decomposition usually results in a system of linear equations, which have to be solved to find the vector of decomposition coefficients.