Laplacian smoothing

This article is about the mesh smoothing algorithm. For the multinomial shrinkage estimator, also called Laplace smoothing or add-one smoothing, see additive smoothing.

Laplacian smoothing is an algorithm to smooth a polygonal mesh.^[1][2] For each vertex in a mesh, a new position is chosen based on local information (such as the position of neighbours) and the vertex is moved there. In the case that a mesh is topologically a rectangular grid (that is, each internal vertex is connected to four neighbours) then this operation produces the Laplacian of the mesh.

More formally, the smoothing operation may be described per-vertex as:

${\displaystyle {\bar {x}}_{i}={\frac {1}{N}}\sum _{j=1}^{N}{\bar {x}}_{j}}$

Where ${\displaystyle N}$ is the number of adjacent vertices to node ${\displaystyle i}$, ${\displaystyle {\bar {x}}_{j}}$ is the position of the ${\displaystyle j}$-th adjacent vertex and ${\displaystyle {\bar {x}}_{i}}$ is the new position for node ${\displaystyle i}$.^[3]

See also

• Tutte embedding, an embedding of a planar mesh in which each vertex is already at the average of its neighbours' positions

References

1. Herrmann, Leonard R. (1976), "Laplacian-isoparametric grid generation scheme", Journal of the Engineering Mechanics Division, 102 (5): 749–756, doi:10.1061/JMCEA3.0002158.
2. Sorkine, O., Cohen-Or, D., Lipman, Y., Alexa, M., Rössl, C., Seidel, H.-P. (2004). "Laplacian Surface Editing". Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. SGP '04. Nice, France: ACM. pp. 175–184. doi:10.1145/1057432.1057456. ISBN 3-905673-13-4. S2CID 1980978. Retrieved 1 December 2013.
3. Hansen, Glen A.; Douglass, R. W; Zardecki, Andrew (2005). Mesh enhancement. Imperial College Press. p. 404.