• Source: Laplacian smoothing

Laplacian smoothing is an algorithm to smooth a polygonal mesh. 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:








x
¯




i


=


1
N



∑

j
=
1


N






x
¯




j




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


Where



N


{\displaystyle N}

is the number of adjacent vertices to node



i


{\displaystyle i}

,







x
¯




j




{\displaystyle {\bar {x}}_{j}}

is the position of the



j


{\displaystyle j}

-th adjacent vertex and







x
¯




i




{\displaystyle {\bar {x}}_{i}}

is the new position for node



i


{\displaystyle i}

.




See also


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




References

Kata Kunci Pencarian:

• Laplacian smoothing
• Smoothing
• Laplace operator
• Laplacian matrix
• Pyramid (image processing)
• Lloyd's algorithm
• Laplace operators in differential geometry
• Blob detection
• Trace (linear algebra)
• Gaussian blur