Implicit Surfaces that Interpolate
Implicit surfaces are often created by summing a collection of radial basis functions. Recently, researchers have begun to create implicit surfaces that exactly interpolate a given set of points by solving a simple linear system to assign weights to each basis function. Due to their ability to interpolate, these implicit surfaces are more easily controllable than traditional “blobby” implicits. There are several additional forms of control over these surfaces that make them attractive for a variety of applications. Surface normals may be directly specified at any location over the surface, and this allows the modeller to pivot the normal while still having the surface pass through the constraints. The degree of smoothness of the surface can be controlled by changing the shape of the basis functions, allowing the surface to be pinched or smooth. On a point-by-point basis the modeller may decide whether a constraint point should be exactly interpolated or approximated. Applications of these implicits include shape transformation, creating surfaces from computer vision data, creation of an implicit surface from a polygonal model, and medical surface reconstruction.
Greg Turk, Huong Quynh Dinh, James F. O'Brien, and Gary Yngve. "Implicit Surfaces that Interpolate". In Shape Modeling International 2001, pages 62–71, May 2001.