# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 4 Results and Discussion
### Analysis of Optimal Parameter for Orthogonal Polynomial
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline _Ord_ & \(p\) & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** & **Average** \\ \hline
8 & 69.853 & 83.258 & 85.294 & 83.088 & 76.471 & 79.593 \\
10 & 81.109 & 83.824 & 86.765 & 85.249 & 82.353 & 83.971 \\
12 & 77.941 & 82.353 & 86.029 & 84.559 & 78.676 & 81.912 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Recognition accuracy for clean environment with different OP parameter \(p\) and order. To investigate the performance of the proposed approach, an experiment is performed in different noisy environments, namely Gaussian, salt-and-pepper, and speckle noise. This experiment aims to examine the effects of these various noise types on the ability of the presented technique to generalize across different configurations of the orthogonal polynomial parameter (\(p\)) and polynomial orders (\(Ord\)). The obtained results are reported in Tables 3-5. In Table 3, a comprehensive evaluation of 3D object recognition based on the proposed technique for polynomial order (\(Ord=8\)) is presented across various environments with different noise parameters. The Gaussian noisy environment reveals a decline in recognition accuracy as the noise level increases. At a Gaussian noise level of 1%, the recognition accuracy is 84.842 at \(p=0.4\), indicating that the performance can be considered good under low noise conditions. However, as the noise level increases to 5%, the recognition accuracy drops to 78.676. The recognition accuracy in the salt and pepper noisy environment follows a comparable trend when compared to the Gaussian noisy environment, with a maximum recognition accuracy of 84.785 at \(p=0.4\) and a noise density of 1%. As the noise level increases, the recognition accuracy declines, reaching a score of 80.147 at \(p=0.4\) and a noise density of 5%. The analysis of speckle noise further emphasizes the challenges posed by noise in maintaining recognition accuracy. The results show that the recognition accuracy is 84.615 at \(p=0.4\) and a noise variance of 0.2, but recognition accuracy decreases as the noise level increases, with a score of 80.882 at \(p=0.4\) and a noise variance of 1. In Table 4, we present a detailed assessment of 3D object recognition performance using the proposed technique for polynomial order (\(Ord=10\)) across various environments with differing noise parameters. In the clean environment, the system exhibits a high recognition accuracy, achieving a peak score of 86.765 at \(p=0.4\). This high accuracy indicates that the system effectively captures the underlying features of the data without the interference of noise. The performance remains commendable across other parameter settings, with scores of 85.294 at \(p=0.5\) and 83.824 at \(p=0.3\). However, the lowest score of 81.618 is observed \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Environment** & & & & \(p\) & & \\ \cline{3-7}  & & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** \\ \hline Clean & & 69.853 & 83.258 & 85.294 & 83.088 & 76.471 \\ \hline  & 0.01 & 69.570 & 82.692 & 84.842 & 82.636 & 76.357 \\  & 0.02 & 67.986 & 80.826 & 83.937 & 81.844 & 75.452 \\ Gaussian & 0.03 & 66.516 & 78.281 & 82.692 & 80.995 & 75.170 \\  & 0.04 & 64.649 & 77.149 & 80.713 & 80.373 & 72.455 \\  & 0.05 & 63.462 & 77.093 & 78.676 & 78.959 & 69.514 \\ \hline  & 0.01 & 69.344 & 82.523 & 84.785 & 82.692 & 75.848 \\  & 0.02 & 67.364 & 81.505 & 83.767 & 81.674 & 75.226 \\ Salt and Pepper & 0.03 & 66.346 & 80.769 & 82.862 & 80.486 & 74.717 \\  & 0.04 & 65.667 & 79.016 & 81.505 & 79.808 & 73.756 \\  & 0.05 & 64.593 & 78.281 & 80.147 & 79.412 & 72.794 \\ \hline  & 0.2 & 69.627 & 82.296 & 84.615 & 82.692 & 76.018 \\  & 0.4 & 69.061 & 80.939 & 83.767 & 82.523 & 75.622 \\ Speckle & 0.6 & 68.722 & 79.299 & 83.258 & 81.844 & 75.057 \\  & 0.8 & 68.382 & 77.771 & 81.957 & 81.222 & 73.416 \\  & 1 & 67.590 & 75.905 & 80.882 & 80.882 & 72.059 \\ \hline
**Average of both clean and noisy environments** & 67.421 & 79.850 & 82.731 & 81.321 & 74.371 \\ \hline \hline \end{tabular}
\end{table}