# The Study of Tooth Erosion Tested by the Color Digital Holography (CDH) Detection System
## 2 Materials and Methods  
### Color Digital Holography  
In the process of light propagation, each spatial coordinate has its corresponding phase. DH can record the phase information of the illuminated object, and the object deformation information can be calculated by the phase difference [32]. Single wavelength DH can only detect the deformation and displacement in the direction of the light wave. The relationship between the phase difference and deformation can be expressed as Equation (1) [36].  
\[\Delta\phi=\frac{2\pi}{\lambda}\Delta I\left(\overset{\rightarrow}{K_{L}}- \overset{\rightarrow}{K_{O}}\right)=\frac{2\pi}{\lambda}\overset{ \rightarrow}{DU} \tag{1}\]  
where \(\Delta\phi\) is the phase difference, \(\lambda\) is the wavelength, \(\Delta I\) is the light path difference, \(\overset{\rightarrow}{K_{L}}\) is the illumination vector from the light source to the object, \(\overset{\rightarrow}{K_{O}}\) is the observation vector from the object to the charge-coupled device (CCD ) direction, \(\overset{\rightarrow}{D}\) is the induction vector corresponding to the deformation direction, and \(\overset{\rightarrow}{U}\) is the object deformation.  
CDH uses three different wavelengths of the laser to irradiate an object. The three-dimensional deformation or displacement field can be reconstructed by vector synthesis using Equation (1). The obtained equations can be written as [34]:\[\left\{\begin{array}{l}\Delta\phi_{\lambda_{R}}=-\frac{2\pi}{\lambda_{R}}\left( \cos\theta_{xy}^{\lambda_{R}}\sin\theta_{xz}^{\lambda_{R}}u_{x}+\sin\theta_{xy}^ {\lambda_{R}}u_{y}+\left(1+\cos\theta_{xy}^{\lambda_{R}}\cos\theta_{xz}^{ \lambda_{R}}\right)u_{z}\right)\\ \Delta\phi_{\lambda_{G}}=-\frac{2\pi}{\lambda_{G}}\left(\cos\theta_{xy}^{ \lambda_{G}}\sin\theta_{xz}^{\lambda_{G}}u_{x}+\sin\theta_{xy}^{\lambda_{G}}u _{y}+\left(1+\cos\theta_{xy}^{\lambda_{G}}\cos\theta_{xz}^{\lambda_{G}}\right) u_{z}\right)\\ \Delta\phi_{\lambda_{B}}=-\frac{2\pi}{\lambda_{B}}\left(\cos\theta_{xy}^{ \lambda_{B}}\sin\theta_{xz}^{\lambda_{B}}u_{x}+\sin\theta_{xy}^{\lambda_{B}}u _{y}+\left(1+\cos\theta_{xy}^{\lambda_{B}}\cos\theta_{xz}^{\lambda_{B}}\right) u_{z}\right)\end{array}\right\} \tag{2}\]  
Equation (2) can also be expressed as matrix form:  
\[\left(\begin{array}{l}\lambda_{R}\Delta\varphi_{\lambda_{R}}\\ \lambda_{G}\Delta\varphi_{\lambda_{G}}\\ \lambda_{B}\Delta\varphi_{\lambda_{B}}\end{array}\right)=2\pi\left[\begin{array} []{ccc}\cos\theta_{xy}^{\lambda_{R}}\sin\theta_{xz}^{\lambda_{R}}&\sin \theta_{xy}^{\lambda_{R}}&1+\cos\theta_{xy}^{\lambda_{R}}\cos\theta_{xz}^{ \lambda_{R}}\\ \cos\theta_{xy}^{\lambda_{B}}\sin\theta_{xz}^{\lambda_{B}}&\sin\theta_{xy}^{ \lambda_{B}}&1+\cos\theta_{xy}^{\lambda_{B}}\cos\theta_{xz}^{\lambda_{B}}\\ \cos\theta_{xy}^{\lambda_{B}}\sin\theta_{xz}^{\lambda_{B}}&\sin\theta_{xy}^{ \lambda_{B}}&1+\cos\theta_{xy}^{\lambda_{B}}\cos\theta_{xz}^{\lambda_{B}}\\ \end{array}\right] \tag{3}\]  
Matrix A in Equation (3) can be shown as:  
\[A=\left[\begin{array}{ccc}\cos\theta^{\lambda_{R}}\sin\theta^{\lambda_{R}}& \sin\theta^{\lambda_{R}}&1+\cos\theta^{\lambda_{R}}\cos\theta^{\lambda_{R}}\\ \cos\theta^{\lambda_{G}}\sin\theta^{\lambda_{G}}&\sin\theta^{\lambda_{G}}&1+ \cos\theta^{\lambda_{G}}\cos\theta^{\lambda_{G}}\\ \cos\theta^{\lambda_{B}}\sin\theta^{\lambda_{B}}&\sin\theta^{\lambda_{B}}&1+ \cos\theta^{\lambda_{B}}\cos\theta^{\lambda_{B}}\end{array}\right] \tag{4}\]  
Matrix A is reversible when the three-color light is not coplanar; then, Equation (3) can be expressed as:  
\[\left(\begin{array}{l}u_{x}\\ u_{y}\\ u_{z}\end{array}\right)=\frac{1}{2\pi}A^{-1}\left(\begin{array}{l}\lambda_{R} \Delta\varphi_{\lambda_{R}}\\ \lambda_{G}\Delta\varphi_{\lambda_{G}}\\ \lambda_{B}\Delta\varphi_{\lambda_{B}}\end{array}\right) \tag{5}\]  
where \(\Delta\phi_{\lambda_{R}},\Delta\phi_{\lambda_{G}},\Delta\phi_{\lambda_{B}}\) can be measured by CDH, and the three-dimensional deformation field of the object can be calculated numerically based on the Equation (5).