## 1 Introduction  
### Problem Statement  
Magnetic tiles, as essential components of permanent magnet motors, play a pivotal role in generating stable magnetic fields [1]. However, their surfaces are susceptible to various defects, including cracks, dents, scratches, and microscopic irregularities [2]. These defects can arise from manufacturing imperfections, mechanical stress, thermal expansion, or prolonged wear and aging. If left undetected, they may lead to non-uniform magnetic fields, increased vibration and noise, localized overheating, and ultimately, motor performance degradation or failure [3]. Therefore, the development of automated defect detection algorithms is crucial. Such algorithms enable precise and timely identification of surface anomalies, facilitating early intervention to prevent irreversible damage. Moreover, they contribute to sustainability and circular economy practices by extending product lifespans and minimizing material waste, while also improving the reliability and efficiency of permanent magnet motors. Traditional defect detection techniques, including visual inspection, ultrasonic testing [4], eddy current testing [5], magnetic particle testing [6], and penetrant testing [7], have been widely employed but exhibit inherent limitations. Visual inspection is highly subjective, relies on human expertise, and is prone to inconsistencies. Ultrasonic testing, while effective for detecting subsurface defects, may lack the resolution required for identifying minute surface irregularities. Eddy current testing, limited to conductive materials, struggles with complex surface geometries and is less effective for detecting non-metallic inclusions or fine cracks. Magnetic particle testing, commonly used for ferromagnetic materials, requires surface preparation and is ineffective for non-magnetic regions or defects that do not significantly disrupt the local magnetic field. Similarly, penetrant testing, which relies on liquid dyes seeping into surface-breaking defects, is limited to identifying open cracks and cannot detect subsurface flaws. Furthermore, these methods often require extensive preprocessing and post-processing, making them time-consuming and labor-intensive. With advancements in non-destructive testing (NDT) techniques [8], vision-based surface topography measurement has gained significant attention due to its ease of implementation and cost-effectiveness. This approach utilizes digital cameras to capture surface images and applies image processing techniques to detect and assess defects. While it offers advantages such as ease of implementation and cost-effective hardware, it also presents significant challenges [9]. Image-based methods are highly sensitive to external factors such as lighting conditions, viewing angles, and surface reflectivity, which can compromise detection accuracy. More critically, conventional 2D imaging is inherently limited in its ability to capture complex three-dimensional surface structures. As a result, it struggles to detect microscopic defects that lack significant texture or contrast variations, making them virtually imperceptible to the human eye. These tiny defects, although visually subtle, can have profound implications for product quality, performance, and safety [10]. Furthermore, 2D imaging lacks depth information, making it difficult to quantitatively assess defect size, depth, and volume--key parameters for evaluating structural integrity and predicting failure risks. To overcome these limitations, 3D point cloud data present a compelling alternative. High-resolution 3D laser scanning technology enables the precise acquisition of geometric and topological information, facilitating the accurate localization and quantification of defects on the magnetic tile surface [11]. Unlike traditional image-based methods, point cloud data capture complete three-dimensional spatial information, allowing for a detailed analysis of defect depth, volume, and curvature variations.