# Accelerated Correction of Reflection Artifacts by Deep Neural Networks in Photo-Acoustic Tomography
## 2 Data Generation
### The Forward Model  
We describe the forward model, which is used to generate ultrasound signals for training purpose. Sound hard reflectors, which can bounce the incident ultrasound back with the opposite speed without absorbing any energy, are placed around the tissue to simulate the acoustic reflections. As such reflectors do not dissipate energy, the resultant reflection artifacts are the strongest, hence can be used as a benchmark to test the artifact-correction capability of different algorithms.  
Denote by \(\Omega\) the biological tissue, which is illuminated by rapid pulses of laser, and afterwards, emits the ultrasound. The boundary of the tissue is represented using the conventional notation \(\partial\Omega\). The forward ultrasound propagation model, in the presence of sound hard reflectors, reads in 2D as follows:  
\[\left\{\begin{array}{rcl}\left(\partial_{t}^{2}-c^{2}(r)\Delta\right)p&=&0& \text{in }(0,T)\times\Omega,\\ p|_{t=0}&=&p_{0}(r),\\ \partial_{t}p|_{t=0}&=&0,\\ \partial_{v}p|_{[0,T]\times\partial\Omega}&=&0,\end{array}\right. \tag{1}\]where \(\Delta=\frac{\hat{\partial}^{2}}{\partial x^{2}}+\frac{\hat{\partial}^{2}}{\partial y ^{2}}\) is the Laplace operator, \(T\) is the stoppage time, \(p_{0}(r)\) is the initial ultrasound pressure, \(p(t,r)\) is the ultrasound pressure of the spatial location \(r\) at time \(t\), \(c(r)\) is the wave speed, and \(\partial_{v}\) is the normal boundary derivative. The main difference of this model with the conventional photo-acoustic models (see, for instance [2; 36]) is the boundary condition \(\partial_{v}p|_{[0,T]\times\partial\Omega}=0\), which is the standard mathematical model to describe sound hard reflectors. Please note that the Neumann boundary condition corresponds to the sample/water interface, and the Dirichlet boundary condition, which corresponds to the sample/air interface, can be handled similarly. We remark that modeling PAT using the Neumann and Dirichlet boundary conditions was reported in the literature [37; 38]. The ultrasound is recorded at the boundary of \(\Omega\) using 508 sensors deployed evenly around the tissue. This measurement amounts to the temporal boundary value of \(p\), i.e., \(p|_{[0,T]\times\partial\Omega}\). The image formation in PAT aims to recover the distribution of the deposited energy that is the local optical fluence. As the initial ultrasound pressure \(p_{0}(r)\) is proportional to the optical fluence, it remains to reconstruct the initial ultrasound pressure \(p_{0}(r)\) from the recorded ultrasound signals.  
We simulate numerous ultrasound signals to train the neural network. This is accomplished by solving the forward model (1) using the second order finite difference time domain (FDTD) method with the central difference formula. The simulation is implemented using the MATLAB code written by the authors. The computational domain is a \(128\times 128\) grid with uniform spacing. The time step is chosen according to the spatial step to fulfill the Courant-Friedrichs-Lewy (CFL) condition. The sound speed \(c(r)\) is either a constant or a spatially varying function, with the former modeling the propagation in homogeneous media and the latter in heterogeneous media. The stoppage time is \(T=4\), which is chosen in such a way that the ultrasound generated from any interior source can reach the boundary.