# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 2 Preliminaries
### Computation of 3D Moments
#### Research Design  
In this section, the design of the methodology is provided. The feature extraction process plays a crucial role in object recognition systems. Features extracted using orthogonal polynomials are considered robust. However, to improve the representation of feature, the selection of orthogonal polynomials is substantial. Therefore, in the proposed recognition system, we have utilized the squared Krawtchouk-Tchebichef polynomial (SKTP) due to its superior performance in signal representation. The SKTP is formed of Krawtchouk and Tchebichef polynomials using (5), which also can be written as follows:  
\[R_{S}=(R_{K}\times R_{T})^{2}, \tag{18}\]  
where \(R_{S}\), \(R_{K}\), and \(R_{T}\) represent the SKTP, Krawtchouk polynomials, and the Tchebichef polynomials, respectively. To improve the robustness of the features, we embedded kernels in the orthogonal polynomials to perform feature extraction. This will speed up the feature extraction process compared with a traditional feature extraction process. The traditional process starts with processing the object using the image kernels, then the process of feature extraction is performed. However, for the proposed technique, the feature extraction process is performed directly on the object, i.e., the proposed technique eliminates extra process while maintaining its effect. To embed the kernel in the orthogonal polynomials, we follow Equation (12). For more illustration, Figures 1 and 2 show the effect of gradient kernels on 3D object samples using conventional convolution and OP-based convolution.  
To generate the kernel, first, the length of the kernel (\(l_{kernel}\)) is set to generate 1D kernel. Moreover, the object information is extracted from the object dataset, where the object information includes the size of the object \(N_{x}\), \(N_{y}\), and \(N_{z}\). According to the extracted  
Figure 1: Samples showing the effect of kernels on ant 3D objects using conventional convolution and OP-based convolution.  
Figure 2: Samples showing the effect of kernels on plane 3D objects using conventional convolution and OP-based convolution.  
object information, the orthogonal polynomials (SKTP) are generated. After generating the orthogonal polynomials, the generated kernel is embedded into the orthogonal polynomials. The resulting polynomials \(P_{x}\), \(P_{y}\), \(P_{z}\) are then used to compute the moments. Figure 3 shows the proposed technique described above.  
After computing the moments, the feature vector is generated and it is used for classification. The features are set to a specific size, which is performed by setting up the order of the orthogonal polynomials. In other words, the moments size is limited to the order of the orthogonal polynomials. These orders are \(Ord_{x}\), \(Ord_{y}\), and \(Ord_{z}\) for the polynomials  
Figure 3: Flow diagram of the proposed embedded kernel.  
\(P_{x}\), \(P_{y}\), and \(P_{z}\), respectively. Therefore, the moment size will be \(Ord_{x}\times Ord_{y}\times Ord_{z}\). After computing the moment, it is flattened to form the feature vector. Thereafter, the feature vector is normalized and used for classification. It is noteworthy that after obtaining the normalized feature vector, each normalized feature vector is associated with a label (ID) for each object. The labels are used for training and testing processes. In this paper, support vector machine (SVM) is employed for its ability to optimize the margin between two hyperplanes that separate the classes [52]. Moreover, SVM is widely used in recognition for its robustness against signal fluctuations compared to other classifiers [53]. Figure 3 shows the presented technique for embedding kernels. Moreover, Figure 4 shows the methodology using the proposed technique for 3D object recognition.  
Figure 4: Flow diagram of object recognition using the proposed technique.