Tracing ridges on B-Spline surfaces

Abstract

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 either of the principal curvatures attain a critical value in its respective principal direction. These curves have 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 algorithm for numerically tracing ridges on B-Spline surfaces 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.

Ridge Extraction from Isosurfaces of Volumetric Data Using Implicit B-Splines

“…The proposed approach uses trivariate implicit BSpline representations of isosurfaces, created by filtering the grid data using 3D B-Spline filters of appropriate degree, in conjunction with a generalization of a previously presented approach for robustly tracing ridges on parametric surfaces [6], to extract ridges directly from the smooth representation. 3D data grids are abundantly available in the form of medical images (MRI, CT scans), simulation data where the resulting grids approximate solutions of discretized partial differential equations and in the field of graphics and visualization in the form of level set models [7].…”

“…So ridges can terminate abruptly as they get closer to such regions. Other types of ridges including elliptic and hyperbolic ridges are distinguished based on [6] as demonstrated in Figure 6 (c).…”

“…Sampling based methods are presented in [17], [16], [12], [13], [37] where the ridge function is evaluated on various tessellations of the parametric domain and ridges are identified on the tessellation. A robust tracing method computationally suitable for complicated B-spline surfaces is presented in [6]. In this paper, the tracing approach of [6] is extended for extracting ridges from level sets of implicit trivariate B-splines.…”

“…A robust tracing method computationally suitable for complicated B-spline surfaces is presented in [6]. In this paper, the tracing approach of [6] is extended for extracting ridges from level sets of implicit trivariate B-splines. (4) trivariate function that maps a point (x 1 , x 2 , x 3 ) ∈ Ω to a scalar value.…”

“…In addition, given any algebraic function a(x, y, z), there exists a set of coefficients c i such that (6) defined over the parallelepiped Ω ∈ R 3 , where the Bspline basis matches the highest degree of a(x, y, z). The set of coefficients c i can be derived by a multivariate version of Marsden's identity [41].…”