Variational Subdivision for Laplacian Splines
The fundamental problem of geometric design is the representation of curved shapes. Traditionally such shapes are represented by parametric spline curves, e.g. NURBS, which are defined as the minimizers of variational problems. For example, cubic B-splines minimize the bending energy functional. More recently subdivision curves and surfaces emerged as an alternative means for representing curved shapes. In this framework a curve or surface is defined as the limit of a repeated averaging process of control points. The subdivision scheme is determined by the subdivision mask S which specifies the weights used during the averaging. A methodology for deriving subdivision schemes from homogeneous variational problems has been outlined in [Warren97]. This report shows the computational details of this approach and presents a method for deriving a subdivision scheme with predefined local support which produces limit surfaces close to the minimizer of the variational problem. In particular this document contains the detailed derivation of Laplacian Splines discussed in section 4.2 of [Warren97]. The result will be a local stationary subdivision scheme for bounded rectangular grids which produces a limit surface that is close to the minimizer of the original variational problem. By increasing the support of the subdivision basis functions the resulting surface will come arbitrarily close to the real variationally optimal surface. The Mathematica source code for this report can be found here and an HTML version here.