Generalized Savitzky-Golay filter for smoothing triangular meshes

In this paper, the well-known Savitzky-Golay filter is generalized for three-dimensional triangular meshes. Savitzky-Golay filter is designed for smoothing noisy measurement signals, initially, a locally polynomial model is fitted to approximate the discrete measure-ment values. We will show, that the original idea can be naturally adapted for smoothing functions defined on an irregular two-dimensional triangular mesh. Primarily surfaces rep-resented by three-dimensional triangular meshes are in the focus of this paper, which are by definition locally homeomorphic to the two-dimensional topological disk. As the generalized filter is meaningful on the plane, we will define local embeddings that send vertices and their neighborhoods to the disk. The points of the two-dimensional disk and the three-dimensional surface can be paired to each other using these embeddings, there-after low-degree multidimensional polynomials can be fitted to the point pairs obtaining a similar continuous model, just like in the one-dimensional case. Finally, we show an ap-plication of our method for mesh smoothing, and a comparison to other mesh smoothing methods, e.g. the Laplacian smoothing and the Taubin smoothing.(c) 2022 Elsevier B.V. All rights reserved.