Subdivision Schemes for Thin Plate Splines

Thin plate splines are a well known entity of geometric design. They are defined as the minimizer of a variational problem whose differential operators approximate a simple notion of bending energy. Therefore, thin plate splines approximate surfaces with minimal bending energy and they are widely considered as the standard “fair” surface model. Such surfaces are desired for many modeling and design applications. Traditionally, the way to construct such surfaces is to solve the associated variational problem using finite elements or by using analytic solutions based on radial basis functions. This paper presents a novel approach for defining and computing thin plate splines using subdivision methods. We present two methods for the construction of thin plate splines based on subdivision: A globally supported subdivision scheme which exactly minimizes the energy functional as well as a family of strictly local subdivision schemes which only utilize a small, finite number of distinct subdivision rules and approximately solve the variational problem. A tradeoff between the accuracy 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 scattered data interpolation and the design of fair surfaces. In particular we suggest an efficient methodology for finding control points for the local subdivision scheme that will lead to an interpolating limit surface and demonstrate how the schemes can be used for the effective and efficient design of fair surfaces.