The homotopy model: a generalized model for smooth surface generation from cross sectional data

Abstract

A generalized model, called the homotopy model, is presented to reconstruct surfaces from cross-sectional data of objects using a homotopy to generate surfaces connecting consecutive contours. The homotopy model consists of continuous toroidal graph representation and homotopic generation of surfaces from the representation. It is shown that the homotopy model includes triangulation as a special case and generates smooth parametric surfaces from contour-line definitions using homotopy. The model can be applied to contours represented by parametric curves as well as linear line segments. First, a heuristic method that finds the optimal path on the toroidal graph is presented. Then the toroidal graph is expanded to a continuous version. Finally, homotopy is used for reconstructing parametric surfaces from the toroidal graph representation. A loft surface is also a special case of homotopy, a straight-line homotopy. Homotopy that corresponds to the cardinal spline surface is also introduced. Three-dimensional surface reconstruction of human auditory surface reconstruction of human auditory ossicles illustrates the advantages of the homotopy model over the others.