Estimating the principal curvatures and the Darboux frame from real 3-D range data

Principal curvatures and the local Darboux frame are natural tools to be used during processes which involve extraction of geometric properties from three-dimensional (3-D) range data. As second-order features their estimations are highly sensitive to noise and therefore, until recent years, it was almost impractical to extract reliable results from real 3-D data. Since the use of more accurate 3-D range imaging equipment has become more popular, as well as the use of polyhedral meshes to approximate surfaces, evaluation of existing algorithms for curvature estimation is again relevant. The work presented here, makes some subtle but very important modifications to two such algorithms, originally suggested by Taubin [17] and Chen and Schmitt [2]. The algorithms have been adjusted to deal with real discrete noise range data, given as a cloud of sampled points, lying on surfaces of free-form objects. The results of this linear time (and space) complexity implementation were evaluated in a series of tests on synthetic and real input. We also present one of many possible uses for these extracted features in an efficient and robust application for the recovery of 3-D geometric primitives from range data of complex scenes. The application combines the segmentation, classification and fitting processes in a single process which advances monotonously through the recovery procedure. It is also very robust and does not use any least-squares fittings. The conclusion of this study is that with current scanning technology and the algorithms presented here, reliable estimates of the principal curvatures and Darboux frame can be extracted from real data and used in a large variety of tasks.