Representing surface curvature discontinuities on curved surfaces

A representation scheme is presented for shapes of three-dimensional curved objects whose surface curvatures are not continuous at all points, i.e., surfaces with C¹ continuity only. A class of surface curves and points called surface structure descriptors for this representation are defined. These descriptors consist of subsets of lines of curvature and their intersections and are based on local descriptions of smooth surfaces. The surface structure curves are derived from the zero-crossings, asymptotes and gradient directions of the Gaussian and mean curvatures of the surface. A smooth surface sketch in terms of the geometrical structures of enclosing boundaries of bumps and dents, ridge and valley lines, and peaks and pits is developed. A hierarchical description of curved surfaces that leads to a global shape representation of the object is obtained from these descriptors. To handle the discontinuities of surface curvatures, a computational scheme is introduced in which gradient directions of surface curvatures at the discontinuities can be inferred. Simulation results on the computation of these descriptors from curved surfaces are shown. Potential applications of these representation schemes in computer vision tasks are discussed