Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint

Abstract

Implicit planar curve and surface fitting to a set of scattered points plays an important role in solving a wide variety of problems occurring in computer graphics modelling, computer graphics animation, and computer assisted surgery. The fitted implicit surfaces can be either algebraic or non-algebraic. The main problem with most algebraic surface fitting algorithms is that the surface fitted to a given data set is often unbounded, multiple sheeted, and disconnected when a high degree polynomial is used, whereas a low degree polynomial is too simple to represent general shapes. Recently, there has been increasing interest in non-algebraic implicit surface fitting. In these techniques, one popular way of representing an implicit surface has been the use of radial basis functions. This type of implicit surface can represent various shapes to a high level of accuracy. In this paper, we present an implicit surface fitting algorithm using radial basis functions with an ellipsoid constraint. This method does not need to build interior and exterior layers for the given data set or to use information on surface normal but still can fit the data accurately. Furthermore, the fitted shape can still capture the main features of the object when the data sets are extremely sparse. The algorithm involves solving a simple general eigen-system and a computation of the inverse or psedo-inverse of a matrix, which is straightforward to implement.