Ordering and parameterizing scattered 3D data for B-Spline surface approximation

Abstract

Surface representation is intrinsic to many applications in medical imaging, computer vision, and computer graphics. We present a method that is based on surface modeling by B-Spline. The B-Spline constructs a smooth surface that best fits a set of scattered unordered 3D range data points obtained from either a structured light system (a range finder), or from point coordinates on the external contours of a set of surface sections, as for example in histological coronal brain sections. B-Spline stands as of one the most efficient surface representations. It possesses many properties such as boundedness, continuity, local shape controllability, and invariance to affine transformations that makes it very suitable and attractive for surface representation. Despite its attractive properties, however, B-Spline has not been widely applied for representing a 3D scattered nonordered data set. This may be due to the problem in finding an ordering and a choice for the topological parameters of the B-Spline that lead to a physically meaningful surface parameterization based on the scattered data set. The parameters needed for the B-Spline surface construction, as well as finding the ordering of the data points, are calculated based on the geodesics of the surface extended Gaussian map. The set of control points is analytically calculated by solving a minimum mean square error problem for best surface fitting. For a noise immune modeling, we elect to use an approximating rather than an interpolating B-Spline. We also examine ways of making the B-Spline fitting technique robust to local deformation and noise.