In this chapter, we cover how meshes are smoothed. This is an important topic in geometry processing since acquired meshes are always subject to noise. The basic principles of signal processing are discussed. Then we present the principles behind Laplacian smoothing and Taubin smoothing, which is based on Laplacian smoothing but suffers less from shrinkage. However, Taubin smoothing causes distortion if the mesh is not highly regular in its structure. Mean curvature flow is much better in this regard.
Using spectral smoothing, it is possible to create filters that more accurately manipulate features of a certain scale. Spectral smoothing works by observing that the Laplace–Beltrami operator, which is central to many smoothing schemes, can be written as a linear operator whose eigenvectors form a function space upon which the vertex positions can be projected—very much like the Fourier basis on a regular grid.
Smoothing generally smooths sharp edges and corners. Fortunately, there are several schemes which remove noise while preserving features such as corners and edges.
Finally, we can see smoothing as an energy minimization and the chapter concludes with a discussion of variational subdivision which is based on repeated refinement and smoothing of meshes.
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.
Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: SIGGRAPH’99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, pp. 317–324. ACM Press, New York (1999). doi:10.1145/311535.311576CrossRefGoogle Scholar
Taubin, G.: A signal processing approach to fair surface design. In: ACM SIGGRAPH’95 Proceedings (1995)
Vallet, B., Lévy, B.: Spectral geometry processing with manifold harmonics. Comput. Graph. Forum 27(2), 251–260 (2008)
Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.-C., Polthier, K. (eds.) Visualization and Mathematics III, pp. 35–57. Springer, Heidelberg (2003)