# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 4 Results and Discussion
### Analysis of Optimal Parameter for Orthogonal Polynomial

Table 3: Recognition accuracy for noisy environments with different OP parameter \(p\) and order (\(Ord=8\)). at \(p=0.2\). When Gaussian noise is applied to the objects, the recognition accuracy shows a gradual decline as the noise level increases. At a noise level of 1%, the system maintains a recognition accuracy of 85.464 at \(p=0.4\), indicating that it performs well under low noise conditions. However, as the noise level rises to 5%, the accuracy decreases to 82.127. The recognition accuracy for Gaussian noise at different levels demonstrate a consistent downward trend, with accuracy values of 84.615 at \(p=0.4\) for a noise level of 2% and 83.993 at the same parameter for a noise level of 3%. The performance in the salt and pepper noisy environment reveals the same trends observed in the Gaussian noise scenario. The system achieves a maximum recognition accuracy of 85.351 at \(p=0.4\) and a noise density of 0.01. As the noise density increases, the recognition accuracy declines, reaching 80.882 at \(p=0.4\) and a noise density of 0.05. This decline highlights that the system is sensitive to high levels of salt and pepper noise. The recognition accuracy for salt and pepper noise also reflect a gradual decrease, with values of 84.333 at \(p=0.4\) for a noise density of 0.02 and 83.258 at the same parameter for a noise density of 0.03. The analysis of speckle noise further illustrates the challenges associated with maintaining recognition accuracy in the presence of noise. The system achieves a recognition accuracy of 85.238 at \(p=0.4\) and a noise variance of 0.2. However, as the noise variance increases, the recognition accuracy declines, with a score of 83.088 at \(p=0.4\) and a noise variance of 1. In Table 5, we present a thorough evaluation of 3D object recognition performance using the proposed technique for polynomial order (\(Ord=12\)) across various environments with differing noise parameters. In the clean environment, the system demonstrates comparable performance, achieving a peak recognition accuracy of 86.029 at \(p=0.4\). When Gaussian noise is applied to 3D objects, the recognition accuracy exhibits a decline as the noise level increases. At a noise level of 1%, the system maintains a recognition accuracy of 84.898 at \(p=0.4\), indicating that it performs well under low noise conditions. However, as the noise level rises to 5%, the accuracy decreases to 80.769. The accuracy for Gaussian noise at different levels shows a consistent downward trend, with accuracy values of 83.880 \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Environment** & & & & & \(p\) & \\ \cline{3-6}  & & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** \\ \hline Clean & & 81.618 & 83.824 & 86.765 & 85.294 & 82.353 \\ \hline  & 0.01 & 81.109 & 82.862 & 85.464 & 84.446 & 82.070 \\  & 0.02 & 79.921 & 81.505 & 84.615 & 84.050 & 79.016 \\ Gaussian & 0.03 & 78.054 & 80.600 & 83.993 & 83.484 & 78.281 \\  & 0.04 & 76.640 & 79.977 & 82.975 & 81.505 & 77.771 \\  & 0.05 & 75.735 & 77.602 & 82.127 & 80.147 & 77.206 \\ \hline  & 0.01 & 81.335 & 83.145 & 85.351 & 84.106 & 82.353 \\  & 0.02 & 79.808 & 82.014 & 84.333 & 83.484 & 81.505 \\ Salt and Pepper & 0.03 & 77.941 & 80.600 & 83.258 & 82.523 & 80.430 \\  & 0.04 & 76.867 & 79.355 & 81.618 & 81.957 & 79.299 \\  & 0.05 & 75.962 & 78.450 & 80.882 & 81.618 & 78.054 \\ \hline  & 0.2 & 81.165 & 83.428 & 85.238 & 84.389 & 82.127 \\  & 0.4 & 79.751 & 81.957 & 84.276 & 83.993 & 82.014 \\ Speckle & 0.6 & 78.054 & 80.373 & 84.219 & 82.975 & 81.222 \\  & 0.8 & 75.396 & 78.507 & 83.428 & 82.183 & 80.260 \\  & 1 & 74.434 & 76.471 & 83.088 & 81.448 & 79.412 \\ \hline
**Average of both clean and noisy environments** & 78.362 & 80.667 & 83.852 & 82.975 & 80.211 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Recognition accuracy for noisy environments with different OP parameter \(p\) and order (\(Ord=10\)). at \(p=0.4\) for a noise level of 2% and 83.258 at the same parameter for a noise level of 3%.