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.
Attenuation Shrinkage Convolution
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