## 4 Results
### Machine Learning Results
#### 4.2.2 Computational Cost  
Tables 7 and 10 show that kFFNN and mFFNN achieve the best FPS and number of variables, and Table 11 presents the relation of NPam of CNNG, CNNP, and rFFNN against NPam of kFFNN and mFFNN. The differences in the computational costs are because \(I_{\alpha,\beta}(x,y)^{G}\) and \(P_{\alpha,\beta}(x,y)\) have 307,200 elements and many abstract features that the CNNs need to codify. The rFFNN processes the signals \(r_{\alpha,\beta}(n)\), which have 600 to 640 elements and are time-domain signals with shape, form deviation, and resolution error features. On the other hand, \(M_{\Delta,\beta}(k)\) and \(D_{\Delta,\beta}(k)\) have 150 or 170 elements that represent the shape and form deviation features.  
The computational performance of renowned networks, such as AlexNet [45], VGG-16, and ResNet [46], comes at a higher cost, resulting in accuracy levels below 85% for shape and quality classification. The results are attributed to the networks' tendency to learn redundant features, which may not contribute significantly to the classification task. Moreover, their computational demands render them impractical for real-time processing, as the frames per second (FPS) achieved are not conducive to swift and efficient analysis.  
According to the machine learning experiments, any ANN with \(r_{\alpha,\beta}(n)\), \(M_{\alpha,\beta}(k)\), or \(I_{\alpha,\beta}(x,y)^{G}\) and \(P_{\alpha,\beta}(x,y)\) generate good performance in shape classification, but regular results in quality classification. The results are obtained because these inputs concentrate the energy in shape information and have little energy in form deviation information. Then, the ANN cannot learn from the form deviation features.  
On the other hand, \(D_{\Delta,\beta}(k)\) and the FFNN generate the best computational cost and good performance in shape and quality classification. These results are obtained because \(D_{\Delta,\beta}(k)\) have only deviation features, and the tolerance zone is represented with  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Metric** & **Diffnn** & **Mffnn** & **Rffnn** & **Icnn** & **Pcnn** \\ \hline TAcc & 95.2\% & 73.8\% & 71.1\% & 100\% & 95.1\% \\ PAcc & 94.6\% & 74.5\% & 69.1\% & 80.1\% & 78.5\% \\ F1 & 97\% & 86\% & 83\% & 87\% & 88\% \\ FPS & 9000 & 8000 & 421 & 30 & 32 \\ NPam & 8066 & 8067 & 24,386 & 8,256,104 & 8,256,104 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Average results of the quality contour profile classification.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Folder** & **1** & **2** & **3** & **4** & **5** & **Average** \\ \hline Shape Classification & 100.0\% & 98.0\% & 99.0\% & 100.0\% & 99.0\% & 99.2\% \\ Quality Classification & 95.0\% & 94.0\% & 96.0\% & 96.0\% & 93.0\% & 94.8\% \\ \hline \hline \end{tabular}
\end{table}
Table 9: Cross-validation of kFFNN based on Acc.  
100 frequency elements; it can be seen in Section 2.3.3. The obtained performance of \(D_{\Delta,\beta}(k)\) is because the form deviation feature vector is a signature of the CNC milling process that generates the profile of the calibration piece.  
Moreover, according to the discussion section, the \(D_{\Delta,\beta}(k)\) feature vector generates high energy when the piece has a shape deformation and can be used to identify if the profile machine process accomplishes the geometric standard tolerances.