Subdivision schemes for thin plate splines

Thin plate splines are a well known entity of geometric design. They are de fined as the minimizer of a variational problem whose differential operator s approximate a simple notion of bending energy. Therefore, thin plate spli nes approximate surfaces with minimal bending energy and they are widely co nsidered as the standard "fair" surface model. Such surfaces are desired fo r many modeling and design applications. Traditionally, the way to construct such surfaces is to solve the associate d variational problem using finite elements or by using analytic solutions based on radial basis functions. This paper presents a novel approach for d efining and computing thin plate splines using subdivision methods. We pres ent two methods for the construction of thin thin plate splines based on su bdivision: A globally supported subdivision scheme which exactly minimizes the energy functional as well as a family of strictly local subdivision sch emes which only utilize a small, finite number of distinct subdivision rule s and approximately solve the variational problem. A tradeoff between the a ccuracy of the approximation and the locality of the subdivision scheme is used to pick a particular member of this family of subdivision schemes. Later, we show applications of these approximating subdivision schemes to s cattered data interpolation and the design of fair surfaces. In particular we suggest an efficient methodology for finding control points for the loca l subdivision scheme that will lead to an interpolating limit surface and d emonstrate how the schemes can be used for the effective and efficient desi gn of fair surfaces.