# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 4 Results and Discussion
### Analysis of Optimal Parameter for Orthogonal Polynomial  
In this section, first, we performed an experimental analysis to evaluate the performance of the proposed system for different values of the orthogonal polynomial parameter (\(p\)). The goal of this analysis is to identify the optimal value that enhances the performance of the proposed system. Support vector machine (SVM) is utilized as a classifier for this analysis. SVM is a powerful tool in the field of machine learning which is widely employed for classification and regression tasks. Compared to other classifiers, SVM's robustness against signal fluctuations makes it particularly effective for recognition applications [53]. In this paper, SKTP is applied to 3D objects that are chosen from the McGill dataset. Various values of the polynomial parameter (\(p\)) are tested for the \(n\)th order of the polynomial (\(R_{n}^{p}(x;N)\)), where (\(p\)) affects the polynomial's shift. The maximum order of the polynomial,  
Figure 5: Samples of 3D objects extracted from McGill dataset. denoted as (_Ord_), is set to 8, 10, and 12, while the values of the polynomial parameter (_p_) include 0.2, 0.3, 0.4, 0.5, and 0.6. The SVM model was trained using 4134 3D objects from the McGill dataset, and the testing phase involved 1768 3D objects. The details of the dataset are given in Table 1 We examined three different cases of the orthogonal parameter to determine which configuration yields the best fitting effect of SKTP, ultimately aiming to achieve the most accurate signal representation based on the information provided by the 3D objects. First, the SVM model is trained and tested in clean environment with the results reported in Table 2.  
The results shows that the highest performance score is 86.765, which is achieved with (_p_ = 0.4) and (_Ord_ = 10). This combination indicates that a moderate value of (_p_) paired with a polynomial order (_Ord_) of 10 yields optimal results. Furthermore, the results demonstrate that the accuracy increases as the polynomial parameter \(p\) increases from 0.2 to 0.4, and the recognition accuracy declines after the polynomial parameter \(p\) becomes greater than 0.4 (0.5 and 0.6). Moreover, the analysis of the average recognition accuracy across different polynomial orders (_Ord_) is important to providing the effect of the polynomial order. For (_Ord_ = 8), the average recognition accuracy is 79.593, indicating a good ability to capture a relevant representation of the data. This recognition accuracy increases to 83.971 for (_Ord_ = 10), which shows that the model benefits from increasing the number of features by a higher polynomial order, which enhances the fitting capabilities of data distribution. However, as the polynomial order increases to (_Ord_ = 12), the average performance score decreases to 81.912. The decrease in average performance shows that as the order of the polynomial increases, the complexity is also increases, potentially resulting in overfitting. These findings show the importance of selecting an appropriate polynomial order, as the results indicate that while higher orders can improve performance, there exists an optimal threshold. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Class ID**} & \multirow{2}{*}{**Class Name**} & **Number of** & \multirow{2}{*}{**Class ID**} & \multirow{2}{*}{**Class Name**} & **Number of** \\  & & **Samples** & & & **Samples** \\ \hline
1 & airplanes & 26 & 11 & hands & 20 \\
2 & ants & 30 & 12 & humans & 26 \\
3 & birds & 21 & 13 & octopus & 25 \\
4 & chairs & 23 & 14 & pliers & 20 \\
5 & crabs & 30 & 15 & snakes & 25 \\
6 & cups & 25 & 16 & spectacles & 25 \\
7 & dinosaurs & 19 & 17 & spiders & 31 \\
8 & dolphins & 12 & 18 & tables & 22 \\
9 & fishes & 23 & 19 & teddy & 20 \\
10 & four & 31 & & & \\ \hline \multicolumn{5}{c}{Total number of samples} & \multicolumn{5}{c}{454} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of the McGill dataset.