Laplacian smoothing

{{Short description|Algorithm to smooth a polygonal mesh}}

{{about|the mesh smoothing algorithm|the multinomial shrinkage estimator, also called Laplace smoothing or add-one smoothing|additive smoothing}}

Laplacian smoothing is an algorithm to smooth a polygonal mesh.{{citation

| last = Herrmann | first = Leonard R.

| issue = 5

| journal = Journal of the Engineering Mechanics Division

| pages = 749–756

| title = Laplacian-isoparametric grid generation scheme

| volume = 102

| year = 1976| doi = 10.1061/JMCEA3.0002158

}}.

{{cite book| author=Sorkine, O., Cohen-Or, D., Lipman, Y., Alexa, M., Rössl, C., Seidel, H.-P.| chapter=Laplacian Surface Editing| title=Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing| year=2004| pages=175–184| publisher=ACM| location=Nice, France| series=SGP '04| doi=10.1145/1057432.1057456| isbn=3-905673-13-4| s2cid=1980978| url=http://doi.acm.org/10.1145/1057432.1057456| accessdate=1 December 2013}}

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:

:$\backslash bar\{x\}\_\{i\}=\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{j=1\}^\{N\}\backslash bar\{x\}\_j$

Where $N$ is the number of adjacent vertices to node $i$, $\backslash bar\{x\}\_\{j\}$ is the position of the $j$-th adjacent vertex and $\backslash bar\{x\}\_\{i\}$ is the new position for node $i$.{{cite book |title=Mesh enhancement |url=https://archive.org/details/meshenhancements00hans_469 |url-access=limited |last1=Hansen |first1=Glen A. |last2=Douglass |first2=R. W |first3=Andrew |last3=Zardecki |year=2005 |publisher=Imperial College Press |page=[https://archive.org/details/meshenhancements00hans_469/page/n417 404] |ref=Glen05}}