# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 1 Introduction  
Three-dimensional object recognition is considered an important research area in computer vision and artificial intelligence. It focuses on the identification and classification of objects in three-dimensional space. In contrast to 2D, 3D object recognition makes use of depth data, spatial relationships, and geometric features, which lead to a better understanding of the object in the real world [1]. This capability is considered essential for applications where both perception accuracy and real-world interaction are required [2]. The importance of 3D object recognition arises from its wide range of applications across various industries. In augmented and virtual reality, 3D recognition improves the user experiences by allowing for simple interactions with virtual objects in the real world [3]. In robotics, 3D object recognition enables machines to navigate and process objects efficiently, facilitating several tasks such as autonomous navigation and object grasping [4, 5]. In healthcare, 3D object recognition enhances advanced medical imaging and surgical planning by providing detailed 3D reconstructions of anatomical structures [6]. In addition, in security and surveillance, 3D recognition improves object detection and tracking in complex environments, which enhances the capabilities of safety and monitoring [7]. Moreover, accurate recognition and interpretation of 3D objects is crucial for advancements in autonomous vehicles, where understanding the 3D structure of the environment is important for safe navigation [8]. Three-dimensional object recognition is important for bridging the gap between digital systems and the real world. As this technology continues to develop, 3D object recognition reveals exciting new possibilities, making it significant in the field of modern computer vision and AI research. After capturing visual data, image processing algorithms are utilized to identify 3D objects, and areas of interest. These algorithms play a crucial role in extracting meaningful information from visual inputs. This is essential for tasks such as 3D object detection, classification, and scene understanding. Utilizing advanced algorithms, systems can interpret and analyze visual data efficiently across various domains [9]. Within this context, orthogonal polynomials (OPs) can be used, which serve as powerful mathematical tools that significantly enhance the overall performance of the system [10, 11, 12]. The application of OPs in image preprocessing ensures accurate and efficient processing, making them indispensable for achieving reliable and precise navigation in autonomous robotic systems [13]. A great deal of progress in the OPs field has been made recently in relation to significant analysis branches because OPs are connected to different functions, such as hypergeometric, trigonometric, and elliptic functions. In addition, OPs are related to continued fractions theory, as well as to problems of interpolation, mechanical quadrature, and their occurrence in the theory of integral and differential equations. Moreover, OPs provide instructive illustrations of some situations in the theory of orthogonal systems. In addition, some OPs have recently played important roles in mathematical statistics and quantum mechanics [14]. Moments are considered as the projection of OPs on the signal domain [15]. Discrete moments, also known as discrete orthogonal moments, have been utilized in various signal processing fields such as speech, image, and video processing [16, 17, 18, 19]. They are commonly used as a shape descriptor and are applied in different tasks such as face recognition [20], image classification [21], differential equations [22, 23, 24, 25], speech enhancement [26], and edge detection [27]. Moreover, orthogonal moments have been utilized in other areas such as speech enhancement [28, 29], watermarking [30], medical image analysis [31, 32, 33], and shot boundary detection [34].