Laplacian Mesh Processing

Laplacian Mesh Processing Olga Sorkine, EuroGraphics '05, State of The Art Report

用Laplacian operator 来 differential surface representation of a mesh:

vertex position vi=(xi, yi, zi) is Cartesian coordinate defined Cartesian space,

we can define a differential coordinate of vi in differential space:

将所有顶点放一个n*1的向量中，将这个Laplace operator放到一个n * n的Ls matrix中，就可以表示整个mesh的Laplacian coordinate.

从两个角度来看这个Ls matrix。

角度 1. 可以called the topological (or graph) Laplacian of the mesh. 它具有的一些properties例如：

Ls matrix is singular, we can't restore the global coordinates only given a set of laplacian-coordinates. 为了解决这个问题，在Ls这个symmetric matrix中加入constraints，成为一个least-squares equations.











角度 2. from a differential geometry perspective this coordinates can be views a discretization of the continuous Laplace-Beltrami operator[dC76], if we assume that our mesh is a piecewise-linear approximation of a smooth surface. 见Fig 1.
文中还提到geometric discretizations fo the Laplacian have better approximation qualities. 这里所谓的逼近是对假设的那个smooth surface来说的。列举的两种geometric discretizations包括:

reference:

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, Mark Meyer1, Mathieu Desbrun1,2, Peter Schr¨oder1, and Alan H. Barr1,

Mean Value Coordinates, Mean value coordinates in 3D,
M. S. Floater. 里面很多跟MVC有关的paper.

MVC 在‘05被用作deformation了, Mean Value Coordinates for Closed Triangular Meshes.

其实跟Laplacian Operation相关的工作挺多了，还有Laplacian Mesh Optimization, Andrew Nealen, Olga Sorkine and Macr Alexa. 详细的请找Olgo Sorkine的主页.