## 5 Results
### WAI Reconstruction  
#### 5.2.1 WAI Simulation  
The motion of ocean waves is a complex random process. Using a spectrum to describe ocean waves is one of the most effective means to study ocean waves, since the modeling process of the spectrum is based on a large number of actual observation data [40; 41; 42; 43]. According to the Longuet-Higgins model [40], the height distribution of ocean waves can be expressed as  
\[\eta(x,y,t)=\sum_{i=1}^{M}\sum_{j=1}^{N}a_{ij}\cos[\omega_{i}t-k_{i}(x\cos \theta_{j}+y\sin\theta_{j})+\epsilon_{ij}], \tag{38}\]  
where \(\eta(x,y,t)\) represents the height distribution of a point \((x,y)\) on the WAI at time \(t\), \(a_{ij}\) is the amplitude of each harmonic, \(\omega_{i}\) is the harmonic frequency, \(k_{i}\) is the wave number of the harmonic, \(\theta_{j}\) is the azimuth of the harmonic, \(\epsilon_{ij}\) is the initial phase of the harmonic, and \(M\), \(N\) represent the sampling number of the frequency range of the wave spectrum and the sampling number of the azimuth angle, respectively.  
The amplitude of the harmonics \(a_{ij}\) can be expressed in the wave spectrum as  
\[a_{ij}=\sqrt{2S(\omega_{i},\theta_{j})d\omega d\theta}=\sqrt{2S(\omega_{i}) \varphi(\theta_{j})\Delta\omega\Delta\theta}, \tag{39}\]  
where \(S(\omega,\theta)\) is the direction spectrum. \(S(\omega)\) and \(\varphi(\theta)\) denote the spectrum and directional distribution functions of ocean waves, respectively. \(\Delta\omega\) and \(\Delta\theta\) are the sampling interval for frequency and direction angle, respectively. According to the linear wave theory [44], \(k_{i}\) and \(\omega_{i}\) meet the dispersion equation  
\[k_{i}=\frac{{\omega_{i}}^{2}}{\text{g}}, \tag{40}\]  
where \(g\) is the acceleration of gravity.  
There are various versions of the directional spectrum. We use the P-M spectrum and the directional distribution function suggested by ITTC (International Towing Tank Conference), whose expression is as follows [45]:  
\[S(\omega,\theta)=S(\omega)\varphi(\theta), \tag{41}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**System Parameters** & **Projector** & **Camera v** & **Camera s** \\ \hline CCD/LCD size & \(4.1\times 4.1\) mm & \(5.0\times 4.0\) mm & \(5.0\times 4.0\) mm \\ Image resolution & \(1000\times 1000\) & \(800\times 600\) & \(800\times 600\) \\ \(f\) & \(4.2\) mm & \(3.0\) mm & \(2.0\) mm \\ Rotation matrix & \(\begin{bmatrix}0.6428&0&-0.7660\\ 0&1&0\\ 0.7660&0&0.6428\\ \end{bmatrix}\)\(\begin{bmatrix}0.8988&0&-0.4384\\ 0&1&0\\ 0.4384&0&8.988\\ \end{bmatrix}\)\(\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\\ -286.40&0&190\\ \end{bmatrix}\)\(\begin{bmatrix}0&0\\ 0&1&0\\ 0&0&1\\ -286.40&0&190\\ \end{bmatrix}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: System parameters of the projector, camera v and camera s.  
\[S(\omega)=\frac{8.1\times 10^{-3}\times g^{2}}{\omega^{5}}\exp\biggl{[}-0.74 \times\Bigl{(}\frac{g}{U\omega}\Bigr{)}^{4}\biggr{]}, \tag{42}\]  
\[\varphi(\theta)=\frac{2}{\pi}\cos^{2}(\theta), \tag{43}\]  
where \(U\) is the average wind speed at a height of 19.5 m above the sea surface. In Figure 10, we show the simulated ocean wave when \(U=1.0\) m/s, \(t=10\) s.