Principal curvature ridges and geometrically salient regions of parametric BB-spline surfaces

Ridges are characteristic curves of a surface that mark salient intrinsic features of its shape and are therefore valuable for shape matching, surface quality control, visualization and various other applications. Ridges are loci of points on a surface where one of the principal curvatures attain a critical value in its respective principal direction. We present a new algorithm for accurately extracting ridges on BB-spline surfaces and define a new type of salient region corresponding to major ridges that characterize geometrically significant regions on surfaces. Ridges exhibit complex behavior near umbilics on a surface, and may also pass through certain turning points causing added complexity for ridge computation. We present a new numerical tracing algorithm for extracting ridges that also accurately captures ridge behavior at umbilics and ridge turning points. The algorithm traverses ridge segments by detecting ridge points while advancing and sliding in principal directions on a surface in a novel manner, thereby computing connected curves of ridge points. The output of the algorithm is a set of curve segments, some or all of which may be selected for other applications such as those mentioned above. The results of our technique are validated by comparison with results from previous research and with a brute-force domain sampling technique.

Research highlights
► This paper addresses the computationally challenging problem of accurate extraction of ridge curves on parametric B-spline surfaces. - Ridge curves highlight intrinsic surface features and are used widely for shape analysis applications. Crests are a special type of ridges. - Unlike other methods, the proposed method extracts actual curves.
► This paper also introduces a new definition of and method for computing geometrically salient regions related to major ridges - Salient regions enable evaluation of geometric significance in the neighborhood of ridge segments - This new definition of salient regions enables new approaches for measuring geometric variation among similar objects.