## 3 Experiments and Discussion
### Simulated Experiments  
#### 3.1.1 Sinusoidal Surface  
The first experiment simulated the measurement of a sinusoidal surface with two different sensors. For each sensor, the measurement had different uncertainty values. Figure 6 shows the simulated measurement results and the measurement can be determined as:  
\[z_{1,2}=\sin\left(\frac{\pi}{10}x\right)+\cos\left(\frac{\pi}{10}y\right)+ \varepsilon_{1,2}, \tag{20}\]  
where \(\varepsilon_{1}=30\)\(\mu\)m and \(\varepsilon_{2}=20\)\(\mu\)m are the added normal distributed noises, which represent different measurement uncertainties.  
Modelling of the two measured datasets was undertaken by the Gaussian process method. In the Gaussian process modelling, the mean function was initially chosen to be zero mean function and the covariance function was Squared Exponential function, while the likelihood function was specified to be Gaussian. The parameters of the covariance function corresponding to unit characteristic length-scale and unit signal standard deviation were firstly initiated to be zeros and the likelihood parameter was initiated to be \(\ln\left(0.1\right)\), which denotes the standard deviation of the noise to be 0.1 mm. The parameters of the Gaussian process were then optimized by minimizing the negative log marginal likelihood. The results of Gaussian process modelling are shown in Figure 7. The covariance is shown in a profile view for better illustration. The root mean squared (RMS) values of the covariance surfaces are 29.8 \(\mu\)m and 19.8 \(\mu\)m, and the modelling errors are both 0.2 \(\mu\)m, which clearly demonstrate the effectiveness of modelling the noise for the two measurement datasets.  
Figure 5: The designed and machined workpieces: (**a**) sinusoidal workpiece; and (**b**) f-theta lens workpiece (unit in mm).  
Figure 6: Simulated measurement results of the designed sinusoidal surface: (**a**) with 30 \(\mu\)m measurement noise; and (**b**) with 20 \(\mu\)m measurement noise.  
The fused result with maximum likelihood data fusion algorithm is shown in Figure 8. The result shows that the overall uncertainty value is reduced and the RMS value of the covariance surface is 16.5 mm, which is smaller than both of the original measurement data.  
The deviations from the original measured surfaces and fused surface to the underlying surface were evaluated and the result is shown in Figure 9. Figure 9a,b shows the evenly distributed patterns along the whole surface which demonstrates the effectiveness of the modelling method and choice of zero mean surface. It is interesting to note that the edge area of the measurement results, especially the corner area, has a larger prediction error since there are insufficient measurement data in those areas. Table 1 shows the RMS value of the uncertainties and the deviations for the original measurement datasets and the fused dataset. The result clearly shows that both the uncertainty and the deviation from the reference surface have improvements over the original measurement results.  
Figure 8: Fused result with the maximum likelihood data fusion method.  
Figure 7: Gaussian process modelling results of the two simulated measured datasets: (**a**) with 30 \(\upmu\)m measurement noise; and (**b**) with 20 \(\upmu\)m measurement noise.