## 3 Experiments and Discussion
### Measurement Experiment Using a Multi-Sensor CMM
#### 3.4.2 Measurement of an f-Theta Lens Surface  
The f-theta lens freeform surface was measured in the present study. Instead of showing the improvement of the final result, this experimental result shows that when one measurement has a large systematic error, the performance of the final result will be affected. Similar to the previous experiment, the surface was measured using a touch trigger probe with a dense measurement of 0.5-mm pitch and the result was determined as the reference data. In this experiment, there existed a large systematic error in the measurement of the laser sensor which may have been caused by the outliers in the measurement result, as shown in Figure 23. It is noted that the outliers were removed in the data processing with a statistical analysis method [26]. However, there was still residual error in the dataset which may affect the final result. The outliers may be caused by the characteristics of the measured surface such as specular reflection of the reflectance light [27] which is considered to influence the optical sensor. After the outliers were removed, the raw data were processed with Gaussian process modelling and the mean surface was obtained together with the associated uncertainty. The deviation of the mean surface to the reference surface was also determined as shown in Figure 24. The RMS of the uncertainty was 18.9 \(\upmu\)m and the RMS of the deviation map was 24.9 \(\upmu\)m. The result shows a larger uncertainty  
Figure 23: Measurement result with outliers of the laser sensor. Figure 21: Mean surface, associated uncertainty and deviation from the reference data after data fusion: (a) mean surface; (b) associated uncertainty; and (c) deviation from the reference data. Figure 22: Repeated measurement results for: (a) uncertainty; and (b) deviation from the reference surface. and even larger deviation than the previous measurement experiment which may introduce a larger systematic error for this measurement. A subset of 50 points of the measurement data from the touch trigger probe was used as the second dataset for the measurement. The data were modelled using the Gaussian process and the mean surface and the associated uncertainty were determined. The deviation from the mean surface to the reference surface was also determined as shown in Figure 25. The RMS of the uncertainty was 5.7 \(\upmu\)m while the RMS of the deviation map was 2.9 \(\upmu\)m.  
Figure 24: Measurement data and Gaussian process result: (**a**) raw data of measurement with the laser sensor; (**b**) mean surface; (**c**) associated uncertainty for the laser sensor; and (**d**) deviation of the mean surface from the reference data. Figure 25: Measurement data and Gaussian process result: (**a**) raw data of measurement with the touch trigger probe; (**b**) mean surface; (**c**) associated uncertainty for the laser sensor; and (**d**) deviation of the mean surface from the reference data. The fused results of the two datasets are shown in Figure 26. The RMS of fused uncertainty was 5.2 \(\upmu\)m while the RMS of the deviation map was 4.8 \(\upmu\)m. It is found that the fused uncertainty was improved and smaller than the original two datasets. However, the deviation is larger than the one only considering the touch trigger probe. Another five measurement results were obtained at different sampling locations and the results are shown in Figure 27. All the results show that the uncertainty was improved while the deviations were worse. This is due to the fact that the measurement data from the laser sensor have large bias from the true value which affects the final fused result. This experiment demonstrated the case similar to the situation as discussed in Figure 3d, where the true value is larger than the mean values of both original measurements. This is the limitation of the data fusion method in that when one of the sensors has large systematic error, the fused result will be poorer than when only considering the data from another sensor.