ShapeGradientDomain: [SGP 2009]
This code performs gradient domain processing on signals defined on a mesh, where the signal can be either a color-field represented as a color per vertex or is the position of the vertices themselves..
The code supports both sharpening and smoothing of the signals through the solution of a screened-Poisson equation.
Specifically, given an input signal F, it solves for the signal G minimizing:
E(G) = α⋅||F-G||^{2} + ||β⋅∇F - ∇G||^{2}
where α is the value-fitting weight, and β is the gradient scale factor.
The code supports inhomogenous processing by allowing the user to replace the Riemannian metric, g, given by the embedding, with a metric that adjusts to the curvature of the surface in two ways.
- Anisotropic Processing: Given orthonormal principal curvature directions, the (idenity) metric is replaced with:
Id. + ε⋅Κ^{2}
where Id. is the identity matrix and Κ^{2} is the diagonal matrix whose entries are the squares of the principal curature values and ε is the curvature weight.
- Isotropic Processing: Given orthonormal principal curvature directions, the (identity) metric is replaced with:
( 1 + ε⋅tr(Κ^{2})/2)⋅Id.
where tr(Κ^{2}) is the total curvature (the sum of the squares of the principal curvatures) and ε is the curvature weight.
The curvature is estimated using the surface normals. If none are provided, the vertex normals are estimated as the area-weighted sum of adjacent triangle normals.