## 4 Limitations
### Resolution Analysis  
In this paper, we perform quasi-periodic sampling for the WAI, utilizing the position information of the feature points of the structured light pattern, while the neighboring sampling interval is approximately equal to \(D\) (Figure 9). Reducing \(D\) enables the recovery of shorter WAI wavelengths. However, as we describe below, a shorter D decreases the angular resolution of \(\hat{\mathbf{N}}\). The angular resolution of \(\hat{\mathbf{N}}\) increases with \(z_{h}\). Here, we further analyze the relationship between the angular resolution of \(\hat{\mathbf{N}}\) with the interval \(D\) and \(z_{h}\) in the meridional plane, based on geometric optics. Figure 8: (a) Deflection of the reflection angle \(\psi\) as a function of the inclination of the WAI when \(\theta_{\text{pro}}=30^{\circ}\). (b) Deflection of the reflection angle \(\psi\) as a function of the inclination of the WAI when \(\theta_{\text{pro}}=45^{\circ}\). Assume that the \(x\) coordinate of the sampled point \(\mathbf{q}_{k}\) is \(h\)\({}_{k}\), while \(h\)\({}_{k+1}-h\)\({}_{k}\)\(\approx\)\(D\). The projected ray \(\hat{\mathbf{v}}_{k}\) reflects by the WAI, and then irradiates a spot on the diffuser plane, at location \({}^{S}k\). The reflected angle can be given by  
\[\theta_{k}^{\mathrm{w}}=\arccos\ (\hat{\mathbf{v}}_{k}\cdot\ -\hat{\mathbf{z}})= \theta_{k}^{\mathrm{flat}}+\psi_{k}=\arccos\ \left(\hat{\mathbf{v}}_{k}^{\prime}\cdot\ -\hat{\mathbf{z}}\right)+\psi_{k}, \tag{25}\]  
where \(\psi_{k}\) is the angular deviation of the reflected ray (relative to the angle in flat water, \(\theta_{k}^{\mathrm{flat}}\)). \({}^{S}k\) can be calculated as  
\[s_{k}=h_{k}+\left(h_{0}-z_{h}\right)\ \tan\theta_{k}^{\mathrm{w}}, \tag{26}\]  
where \(h\)\({}_{0}\) is the average underwater depth of the projector. As the projector follows a central projection, the reflected angles of the adjacent sampling points \(h\)\({}_{k}\) and \(h\)\({}_{k+1}\) can be expressed as  
\[\theta_{k}^{\mathrm{flat}}=\arctan\frac{h\ _{k}}{h\ _{0}}, \tag{27}\]  
\[\theta_{k+1}^{\mathrm{flat}}=\arctan\frac{h\ _{k}+D}{h\ _{0}}. \tag{28}\]  
Suppose that the projected ray \(\hat{\mathbf{v}}_{k+1}\) corresponding to \(h\)\({}_{k+1}\) is parallel to the projected ray \(\hat{\mathbf{v}}_{k}\) corresponding to \(h\)\({}_{k}\), and the corresponding location of the spot \(s_{k+1}^{0}\) is  
\[s_{k+1}^{0}=h\ _{k+1}+\frac{h\ _{0}-z_{h}}{h\ _{0}}\ h\ _{k}. \tag{29}\]  
Actually, the location of the spot \(s_{k+1}\) on the diffuser plane corresponding to \(h_{k+1}\), according to the central projection, is given by  
\[s_{k+1}=h\ _{k+1}+\frac{h\ _{0}-z_{h}}{h\ _{0}}h\ _{k+1}. \tag{30}\]  
From Equations (29) and (30), we obtain  
\[\left\|s_{k+1}^{0}\overset{\rightarrow}{s}_{k+1}\right\|=\frac{h\ _{0}-z_{h}}{h\ _{0}}D. \tag{31}\]  
Figure 9: Geometry of the structured light projection system \(S\). Let the sensor \(S\) determine \(s_{k}\) with spatial uncertainty \(\Delta p\). This uncertainty may be due to the camera pixel size, diffuser characteristics and so on. This spatial uncertainty converts to uncertainties in the measured angular deviation \(\Delta\psi\) and the measured angular deviation of the inclination angle of the WAI normal \(\Delta\theta\),  
\[\Delta p=\frac{h\,_{0}-z_{h}}{\cos^{2}\theta_{k}^{\mathrm{flat}}}\Delta\psi \approx\frac{h\,_{0}-z_{h}}{\cos^{2}\theta_{k}^{\mathrm{flat}}}\frac{d\psi}{d \theta}\Delta\theta, \tag{32}\]  
Given \(\Delta p\), \(\mathrm{z_{h}}\), \(h\,_{0}\) and \(\Delta\theta\), when the WAI is flat, it is easy to establish the correspondence of \(h_{k}\) with \(s_{k}\), since \(s_{k}<s_{k+1}\).