## 3 Signal Processing  
The signal processing is done subsequent to the measurement procedure. The vibration compensating signal processing can be separated into two parts. In the first part, the correction of the distorted CSI signals occurs, whereby the evaluation of the IDS signal is performed to obtain the distance changes during the CSI measurement followed by the correction of the CSI datasets. These corrected CSI interference signals are then analyzed in the second step. A two-beam interference signal can be described by  
\[I_{\mathrm{int}}=I_{\mathrm{m}}+I_{\mathrm{r}}+2\sqrt{I_{\mathrm{m}}I_{ \mathrm{r}}}|\gamma_{12}|\cos(\Phi), \tag{1}\]  
where \(I_{\mathrm{m}}\) and \(I_{\mathrm{r}}\) represent the intensities of reflected beams from the measurement object and the reference mirror, respectively. \(|\gamma_{12}|\) is the absolute value of the complex degree of coherence, which is \(<\)1 in case of the CSI depending on the use of temporally low coherent light. The phase \(\Phi\) is represented by  
\[\Phi_{\mathrm{CSI}}=\frac{4\pi}{\lambda_{\mathrm{LED}}}\Delta z_{\mathrm{m}}( x,y)=\frac{4\pi}{\lambda_{\mathrm{LED}}}(z_{\mathrm{m}}(x,y)+z_{\mathrm{ds}}+z_{ \mathrm{vib}}), \tag{2}\]  
with the center wavelength of the illumination source \(\lambda_{\mathrm{LED}}\) and the path length change \(\Delta z_{\mathrm{m}}\) between the reference mirror and the surface caused by the surface structure \(z_{\mathrm{m}}(x,y)\), the depth scan \(z_{\mathrm{ds}}\) and environmental vibrations \(z_{\mathrm{vib}}\). Assuming vibrations with oscillation frequencies below \(1\,\mathrm{kHz}\) these can be detected by the IDS. Its interference equation equals Equation (1) with \(|\gamma_{12}|\approx 1\) and  
\[\Phi_{\mathrm{IDS}}=\frac{4\pi}{\lambda_{\mathrm{LED}}}[\hat{z}_{\mathrm{a}} \cos(2\pi f_{\mathrm{a}}t)-\Delta z_{\mathrm{m}}(x_{0},y_{0})], \tag{3}\]  
where \(\hat{z}_{\mathrm{a}}\) and \(f_{\mathrm{a}}\) represent, respectively, the amplitude and the frequency of the oscillating reference mirror, leading to an OPLM. \(x_{0}\) and \(y_{0}\) describe a certain point on the surface under investigation. Such an interference signal (blue curve) for an OPLM resulting from the electrical actuator signal \(u(t)\) (green) and the related deflection of the reference mirror \(z(t)\) (brown) is depicted in Figure 3. Assuming a linear deflection regarding the rising and falling flank (see Figure 3) of the sinusoidal reference mirror movement a constant fringe frequency \(f_{\rm e}=4\pi\hat{z}_{\rm a}f_{\rm a}/\lambda_{\rm LD}\) of the interference signal can be assumed. Therefore, the phase values of the corresponding sections of the interference signal are determined using a so-called single-point-DFT also known as lock-in method:  
\[\Phi_{\rm IDS}=\arg\left(\sum_{n=0}^{N_{\rm w}-1}I(n)W(n)\exp\left(-\mathrm{i}2 \pi n\frac{f_{\rm e}}{f_{\rm s}}\right)\right), \tag{4}\]  
with the window function \(W(n)\) to reduce leakage, the number \(N_{\rm w}\) of values covered by \(W(n)\) and the sampling frequency \(f_{\rm s}\). Finally, the relationship between the phase and the distance value \(\Delta z_{\rm m}\) is given by  
\[\Delta z_{\rm m}=\frac{\lambda_{\rm LD}}{4\pi}\Delta\Phi_{\rm IDS}, \tag{5}\]  
where \(\Delta\Phi_{\rm IDS}\) represents the difference of consecutive phase values determined according to Equation (4). This finally results in two height values per actor period [14, 16]. The correction of the CSI data occurs after the evaluation of the IDS signal is finished. For this purpose, the distance changes compared to the desired movement obtained from the IDS during a depth scan is used to rearrange the CSI data according to Figure 4. The rearranging process leads to non-equidistant sampling points of the low coherent interference signals. However, in the processing of CSI signals by commonly utilized algorithms we expect an equidistant sampling of the correlograms. Therefore, an interpolation of the data is performed to obtain sample points at equidistant intervals.