# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 4 Results and Discussion
### Comparison between the proposed system and existing works
The performance of the proposed approach is further evaluated by benchmarking the proposed approach with the traditional method in terms of computational cost. In the traditional method, the 3D object is convolved by the kernel and then the moments are computed. It is noteworthy that the kernel utilized in the experiment is the averaging kernel with a length of 7. For this purpose, the computation cost is evaluated by performing five runs for all 3D objects across different classes. Then, the average computation time is recorded for each 3D object. The comparison is conducted for three different polynomial orders (\(Ord=8\), 10, and 12), and the results are summarized in Table 7. The table includes the computation time for both the proposed approach (Time (ours)) and the traditional method (Time (Trad)), \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Average** & **Average of** \\
**Method Name** & **Recognition** & **Recognition** \\  & **Accuracy** & **Accuracy for Noisy** \\ \hline DKMI [56] & 62.01\% & 57.92\% \\ HMI [56] & 60.89\% & 56.69\% \\ KMI [56] & 60.32\% & 56.44\% \\ TMI [56] & 60.54\% & 56.40\% \\ TTTMI [57] & 71.57\% & 67.30\% \\ KKKMI [57] & 71.11\% & 66.97\% \\ TKKMI [57] & 72.19\% & 67.64\% \\ TTKMI [57] & 72.87\% & 68.20\% \\ GMI [57] & 70.26\% & 65.49\% \\ OBP [50] & 80.21\% & 80.22\% \\ Proposed System (\(p=0.4\) and \(Ord=8\)) & 82.73\% & 82.56\% \\ Proposed System (\(p=0.4\) and \(Ord=10\)) & 83.85\% & 83.66\% \\ Proposed System (\(p=0.4\) and \(Ord=12\)) & 83.33\% & 83.16\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison between the presented recognition system and existing methods using the McGill database. as well as the percentage improvement (Imp) achieved by the proposed method over the traditional method. The percentage improvement is calculated using the following formula:  
\[\textit{Percentage Improvement}\left(Imp\right)=\left(\frac{T_{\textit{ Trad}}-T_{\textit{Ours}}}{T_{\textit{ Trad}}}\right)\times 100\% \tag{19}\]  
The minimum percentage improvement is observed for Class 16, with improvements of 47.0% at \(\textit{Ord}=8\), 47.7% at \(\textit{Ord}=10\), and 46.1% at \(\textit{Ord}=12\). In contrast, the highest percentage improvement is achieved for Class 19, where the proposed method attains improvements of 64.2% at \(\textit{Ord}=8\), 64.1% at \(\textit{Ord}=10\), and 64.0% at \(\textit{Ord}=12\). The results demonstrate that the proposed algorithm significantly outperforms the traditional algorithm across all polynomial orders and classes. For instance, at \(\textit{Ord}=8\), the average percentage improvement across all classes is 54.7%, while at \(\textit{Ord}=10\) and \(\textit{Ord}=12\), the average improvements are 54.8% and 53.4%, respectively. This indicates that the proposed method consistently reduces computation time by more than half compared to the traditional approach. This clearly signifies the robustness and efficiency of the proposed algorithm, particularly when dealing with higher polynomial orders and complex 3D objects. The last row of the table provides the average computation time and improvement per 3D object, further confirming the superiority of the proposed method in terms of computational efficiency.