# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 4 Results and Discussion
### Comparison between the proposed system and existing works  
To assess the effectiveness of the presented recognition system based on the proposed technique, we conducted a comparative analysis of its recognition accuracy against several existing methods. The algorithms included in this comparison are direct Krawtchouk moment invariants (DKMI), Tchebichef moment invariants (TMI), Krawtchouk moment invariants (KMI), Hahn moment invariants (HMI), Tchebichef-Tchebichef-Tchebichef moment invariants (TTTIM), Krawtchouk-Krawtchouk-Krawtchouk moment invariants (KKKMI), Tchebichef-Krawtchouk moment invariants (TKKMI), Tchebichef-Tchebichef-Krawtchouk moment invariants (TTKMI), geometric moment invariants (GMI), and overlapped block processing (OBP). The average recognition accuracy for both the presented recognition system and the existing methods are shown in Table 6. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Environment** & & & & \(p\) & & \\ \cline{2-7}  & & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** \\ \hline Clean & & 77.941 & 82.353 & 86.029 & 84.559 & 78.676 \\ \hline  & 0.01 & 77.432 & 81.900 & 84.898 & 84.276 & 77.941 \\  & 0.02 & 75.396 & 80.826 & 83.880 & 83.484 & 75.905 \\ Gaussian & 0.03 & 74.095 & 78.507 & 83.258 & 82.127 & 75.622 \\  & 0.04 & 73.586 & 77.771 & 82.523 & 81.052 & 75.113 \\  & 0.05 & 72.738 & 76.980 & 80.769 & 78.959 & 74.717 \\ \hline  & 0.01 & 77.489 & 82.127 & 84.842 & 84.333 & 78.563 \\ Salt and Pepper & 0.02 & 76.131 & 81.165 & 84.219 & 83.428 & 77.602 \\  & 0.03 & 75.057 & 79.242 & 83.032 & 82.070 & 76.414 \\  & 0.04 & 74.095 & 78.394 & 82.579 & 81.448 & 74.038 \\  & 0.05 & 73.190 & 77.206 & 81.957 & 79.412 & 73.529 \\ \hline  & 0.2 & 77.206 & 81.165 & 84.729 & 84.106 & 77.941 \\  & 0.4 & 74.717 & 79.129 & 83.767 & 83.258 & 77.262 \\ Speckle & 0.6 & 73.020 & 76.867 & 82.975 & 82.975 & 75.962 \\  & 0.8 & 71.946 & 74.943 & 82.353 & 81.731 & 75.170 \\  & 1 & 70.928 & 74.887 & 81.618 & 80.995 & 74.265 \\ \hline
**Average of both clean and noisy environments** & 74.685 & 78.966 & 83.339 & 82.388 & 76.170 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Recognition accuracy for noisy environments with different OP parameter \(p\) and order (\(Ord=12\)). The comparison presented in Table 6 highlights the performance of the proposed recognition system against several existing methods using the McGill database. The average recognition accuracy of each method is reported, providing a clear perspective on the effectiveness of the proposed approach. The existing methods, including DKMI, HMI, KMI, and TMI, exhibit relatively low recognition accuracy, ranging from 60.32% to 62.01%. These results indicate that they are not too robust for the complexities presented in the McGill database. Moreover, the Tchebichef-based methods, which are TTTMI, KKKMI, TKKMI, and TTKMI, show improved performance, with an accuracy between 71.11% and 72.87%. This indicates that the incorporation of Tchebichef moments enhances the recognition capabilities compared to the earlier methods. However, even the best-performing Tchebichef-based method, TTKMI, achieves only 72.87%, which still falls short of optimal performance. In contrast, the proposed system demonstrates significantly higher recognition accuracy across all tested configurations. The system achieves an accuracy of 82.73% at polynomial order 8, which surpasses the best existing methods by a notable margin. When the polynomial order is increased to 10, the accuracy improves further to 83.85%, indicating that the proposed system benefits from higher polynomial orders, likely due to its enhanced ability to capture complex features in the data. In addition, for polynomial order of 12, the proposed system shows robust performance with an accuracy of 83.33% This stability of performance across different polynomials orders demonstrates that the proposed approach is robust and adaptable. thus, the variations in the dataset are effectively handled by the proposed approach.