Surface representation and processing

Surface representation and processing is one of the key topics in computer graphics, since it greatly affects the range of possible applications. This paper reviews the Laplacian operator (differential coordinates) which represent surface´s differential geometric properties , discuss the mesh deformation and smoothing based on Laplacian operator. Laplacian operator denotes a vector that is the difference between the absolute coordinates of a vertex and the center of mass of its 1-ring neighbors in the mesh. In other word, Laplacian operator shows surface concave or convex information. The deformation techniques based on interpolation and forward reconstruction schemes, such as skeletal subspace deformation, free form deformation, and multiresolution deformation, may lead to serious artifacts, when applied to large deformation. While using Laplacian operator, the high quality of deformation effects can be obtained, that means geometric details to be preserved. The deformation operation: interactive free form deformation in a region of interest based on the transformation of a handle, transfer and mixing of geometric details between two surfaces, and transplanting of a partial surface mesh onto another surface. The core of the mesh deformation is editing Laplacian operator´s direction and magnitude. Mesh smoothing algorithm based on Laplacian operator minimizes the energy function, that is a quadratic optimization problem. The smooth.ed solution is a combination of the Laplacian eigenvectors with modified coefficients. This analysis shows this mesh smoothing algorithm are very similar to classic spectral smoothing.