Laplacian 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:
- [math]\displaystyle{ \bar{x}_{i}= \frac{1}{N} \sum_{j=1}^{N}\bar{x}_j }[/math]
Where [math]\displaystyle{ N }[/math] is the number of adjacent vertices to node [math]\displaystyle{ i }[/math], [math]\displaystyle{ \bar{x}_{j} }[/math] is the position of the [math]\displaystyle{ j }[/math]-th adjacent vertex and [math]\displaystyle{ \bar{x}_{i} }[/math] is the new position for node [math]\displaystyle{ i }[/math].[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
- ↑ Herrmann, Leonard R. (1976), "Laplacian-isoparametric grid generation scheme", Journal of the Engineering Mechanics Division 102 (5): 749–756, doi:10.1061/JMCEA3.0002158.
- ↑ 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. http://doi.acm.org/10.1145/1057432.1057456. Retrieved 1 December 2013.
- ↑ Hansen, Glen A.; Douglass, R. W; Zardecki, Andrew (2005). Mesh enhancement. Imperial College Press. p. 404. https://archive.org/details/meshenhancements00hans_469.
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