# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 1 Introduction
In general, moments are quantities that are used to represent 1D, 2D, and 3D data without redundancy, making them an effective and efficient data descriptor. This meansthat moments have the capability to describe data while avoiding duplication [35; 36; 37]. Signal preprocessing techniques play a crucial role in enhancing the performance of computer vision algorithms by improving the quality of input data [38]. For instance, noise is removed from signals by utilizing smoothing kernels, such as averaging kernels. This is performed to obtain a more clean and reliable data for the remaining processing steps [19; 39]. Furthermore, gradient kernels are usually used to calculate the gradients in the \(x\)-direction for 1D signals, \(x\)- and \(y\)-directions for 2D signals, and \(x\)-, \(y\)- and \(z\)-directions for 3D signals to determine intensity changes in the aforementioned directions, respectively. Extracting moments from images following preprocessing with smoothing and/or gradient image kernels is an effective approach to represent signals [40; 41]. However, some existing works are required to reduce the spectral data space and, consequently, the computational load. More improvement is required to obtain more robust and accurate methods. In order to obtain a smoothed and gradient version of a 3D signal \(f\), kernels \(h_{s}\) and \(h_{g}\) are convolved with the signal. The convolution process is carried out separately for each required version of the 3D signal. Once the smoothed and gradient versions of the 3D signal are obtained, moments can be extracted from them. However, performing these operations for each signal significantly increases the computational cost of a feature extraction algorithm, such as for 3D object recognition. Recently, 3D object recognition has emerged as a transformative technology with significant potential in fields such as virtual reality, autonomous driving, and commercial manufacturing. While advanced feature extraction methods, including deep learning models, have achieved impressive results in this domain, their high computational cost poses a substantial challenge, particularly for resource-constrained mobile applications [42]. Three-dimensional object recognition is a complex process because 3D objects provide a large variety of projected images based on viewing direction, illumination, distance, and other viewing conditions [43]. To deal with viewing direction effects, in [43], a computational work is demonstrated using sets of a small number of object viewpoints. However, this approach leaves many problems unanswered. The main problems include the use of abstractions, class-based recognition, and the classification problem, which requires a large experimental and computational research. Basically, computer vision is an active task and plays a crucial role in making the computer system see and identify the visual world automatically based on simulating the basic biological ability of human visual perception. Therefore, to achieve a high potential, the focus is on having an accurate algorithm with a lower computational cost [44; 45]. Therefore, the aim of this paper is to obtain an accurate algorithm with a lower computational cost by addressing the aforementioned issues. To this end, we propose a new mathematical model for 3D object recognition that computes preprocessed signal moments by combining two operations into one, thereby reducing the cost of clustering. This model can be effectively used in mobile robot applications. To achieve this, we have embedded a smoothing kernel operator into OP functions to extract features quickly and efficiently. Our contributions are as follows:  
1. We presented a mathematical model that integrates image kernels directly into orthogonal polynomial (OP) functions, eliminating the need for separate convolution and orthogonal moment (OM) computation processes. This innovation significantly reduces computational overhead while maintaining accuracy. 2.