# Quaternion Processing Techniques for Color Synthesized NDT Thermography
## 4 Results
### Results in the Quaternion Processing  
The results obtained after the dimensionality reduction process applied to the sequence of polynomial approximations have been processed with Quaternion Fourier Transform and Quaternion Principal Component Analysis. Subsequently, the results obtained with the application of QFT and QPCA have been processed using quaternionic filtering to optimize the visualization. The quaternionic processing algorithms have been programmed in MATLAB with the additional support of the quaternion package [44]. Some of the results obtained for each of the quaternionic processing techniques are shown below. Quaternion Fourier analysis requires a quaternion to which apply the processing and, in addition, it also requires a unit quaternion that defines an analysis direction. Depending on the analysis direction, the QFT result varies. Without going into theoretical details about the effect of the analysis direction, which is beyond the scope of the current study, an experimental analysis of the effect it produces on the visualization of defects has been carried out. For this purpose, 32 different directions defined by the radii of a unit sphere have been analyzed, as shown in Figure 7a. The values required for the calculation of the evaluation metrics, i.e., the mean values and standard deviations of the areas occupied by the defects and the healthy reference areas close to them, are extracted from the results obtained with the QFT processing. These values are calculated for each of the existing defects and for each of the 3 color channels defined by the quaternion axes. An efficient way to collect these data has consisted of storing them using analysis directions spheres, where the calculated values are stored as color data expressed in RGB space for each of the analyzed directions (Figure 7b). This way, each point of the resulting sphere indicates the QFT analysis direction by its spatial coordinates, and the corresponding calculated values for the metrics by its RGB color. After the application of the QFT processing with the direction analysis to the different reduced sequences, the data measured for each defect are recovered and the metrics are calculated. The results obtained in the calculation of the metrics can be analyzed graphically through spheres of directions where the radius of the sphere at each point coincides with the value of the metric calculated for each analysis direction defined by the point, being the optimal direction the one with a larger radius (Figure 7c). Figure 8 shows some representative results obtained with the QFT technique. It was observed that the defects that are detected are represented in different color depending on the depth. Defects of the same type located at different depths produce different color tones, while defects of the same type located at the same depth produce similar tones. This behavior is observed in the results obtained with the synthetic data, where the induced defects are of the same type and are located at different depths (Figure 8a-c, and with the experimental specimen, which contains defects of the same type and located at the  
Figure 7: Supporting spheres used in the optimization of the analysis directions in QFT: (**a**) sphere of analysis directions, (**b**) sphere of calculated values for the analysis directions shown in (**a**), (**c**) sphere of SNR values for the analysis directions shown in (**a**). same depth but of different sizes (Figure 8d-f). Frequently, the QFT results that produce the highest levels of defect detection are in the first 10% of the results in the generated sequence. However, numerous cases have been found in which this rule is not followed. The application of QPCA processing requires only the quaternion being processed. In Figure 9 some results obtained from the application of QPCA to the sequence reduced by Gaussian KPCA are shown.