Ridges and Umbilics of Polynomial Parametric Surfaces

Abstract

Abstract. Given a smooth surface, a blue (red) ridge is a curve along which the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. State of the art methods for ridge extraction either report red and blue ridges simultaneously or separately -in which case a local orientation procedure of principal directions is needed, but no method developed so far topologically certifies the curves reported. In this context, we make two contributions.First, for any smooth parametric surface, we exhibit the implicit equation P = 0 of the singular curve P encoding all ridges of the surface (blue and red), we analyze its singularities and we explain how colors can be recovered. Second, we instantiate to the algebraic setting the implicit equation P = 0. For a polynomial surface, this equation defines an algebraic curve, and we develop the first certified algorithm to produce a topologically certified approximation of it. The algorithm exploits the singular structure of P -umbilics and purple points, and reduces the problem to solving zero dimensional systems using Rational Univariate Representations and isolate roots of univariate rational polynomials. An experimental section illustrates the efficiency of the algorithm on a Bezier patch.

Tracing ridges on B-Spline surfaces

“…The research presented in Cazals et al [4,5,6] falls into the first category. In [5], a system of equations that encodes all the ridges and umbilics of a parametric surface represented by a single polynomial is presented.…”

“…In addition, to the best of our knowledge, there is no technique in the existing literature that can accurately compute the topology of ridges of NURBS surfaces at all locations from a disjoint set of ridge points. The authors of [4,6] note that at the time of writing (2007), their technique for processing single polynomial surfaces was too slow to compute results in reasonable time for a bi-quintic Bézier patch. A domain tessellation or sampling-based method, albeit computationally fast, has other disadvantages.…”

“…Various aspects of the behavior of ridges on surfaces are summarized in this section (See [4,5,6,24,37] for discussions and proofs). In this paper, only the generic case is considered.…”

“…This system is essentially the product of the ridge conditions for both the principal curvatures. An algorithm is presented to solve the resulting polynomial equations using algebraic techniques [4,6], and to compute topologically correct ridges. Their research presents the first technique to compute the topology of ridges exactly at umbilics.…”

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

“…Three approaches have been presented to extract ridges curves from smooth parametric surfaces. A lattice method for single patch polynomial (Bézier) surfaces is presented [35], [36] where solutions of the ridge equations are computed on a dense grid of isoparametric lines. Ridge curves are obtained by connecting ridge points on the grid.…”