Laplacian mesh smoothing with bilateral weights for characterization of freeform surfaces

Freeform surfaces are widely used in advanced manufacturing due to their versatility, yet their complex geometries make characterization challenging. To do a surface texture analysis, it is often necessary to first remove the form of the surface. When the nominal form is unknown, the reference form can be approximated by filtration. If a triangular mesh represents the freeform surface, mesh smoothing is a type of filtration. Among various methods, Laplacian smoothing is the most employed for mesh smoothing due to its linear simplicity. Traditional Laplacian smoothing techniques use uniform, distance, or cotangent weights to approximate the discrete Laplace-Beltrami operator for each vertex. However, Laplacian smoothing has limitations, such as accuracy, shrinkage effects, and edge blurring. This study first investigates and compares Gaussian convolution and Laplacian smoothing techniques, aiming to integrate their advantages. Based on these findings, a novel approach is proposed to enhance Laplacian mesh smoothing by introducing a bilateral weighting scheme for each vertex. This enhancement keeps the simplicity of the Laplacian smoothing structure while mitigating the limitations of traditional weighting functions, offering improved performance in terms of accuracy and efficiency. The proposed method demonstrates potential for better estimating reference forms for freeform surfaces, which is the first step for surface characterization in advanced manufacturing.