Fast and Faithful Geometric Algorithm for Detecting Crest Lines on Meshes

Abstract

A new geometry-based finite difference method for a fast and reliable detection of perceptually salient curvature extrema on surfaces approximated by dense triangle meshes is proposed. The foundations of the method are two simple curvature and curvature derivative formulas overlooked in modern differential geometry textbooks and seemingly new observation about inversion-invariant local surface-based differential forms. Problem setting and solutionThis paper is inspired by two simple but beautiful formulas of classical differential geometry and a seemingly new observation about inversion-invariant local surface-based differential forms. The formulas and observation are the key ingredients of our approach to fast and reliable detection of the so-called crest lines [30], salient subsets of the extrema of the surface principal curvatures along their corresponding curvature lines.Related work. The full set of such extrema, frequently called ridges [25], corresponds to the edges of regression of the envelopes of the surface normals and was first studied by A. Gullstrand in connection with his work in ophthalmology (1911 Nobel Prize in Physiology and Medicine). Since then the ridges and their subsets were frequently used for shape interrogation purposes (see, for example, [14, Sect. 7.4], [11, Chap. 6], [26, Chap. 11], and references therein). The ridges possess many interesting properties. In particular, they can be defined as the loci of surface points where the osculating spheres have high-order contacts with the surface and, therefore, the ridges are invariant under inversion of the surface w.r.t. any sphere [25].The principal difficulty in detecting the crest lines and similar features on discrete surfaces consists of achieving an accurate estimation of surface curvatures and their derivatives. The main approaches here are local polynomial fitting [2,3,16], the use of discrete differential operators [12], and various combinations of continuous and discrete techniques [23,37]. While estimating surface curvatures and their derivatives with geometrically inspired discrete differential operators [12,27] is much more elegant than using fitting methods, the former usually requires noise elimination and the latter seems more robust. On the other hand, as pointed out in [27], fitting methods incorporate a certain amount of smoothing in the curvature and curvature derivative estimation processes and that amount is very difficult to control. Another limitation of fitting schemes consists of their relatively low speed to compare with discrete differential operators. Further, since predefined local primitives are used for fitting, one cannot expect a truly faithful estimation of surface differential properties.Our approach to estimating the surface's principal curvatures and their derivatives is pure geometrical.Two forgotten formulas. The first formula [33, §119] 1 states that for a smooth surface r oriented by its unit normal nwhere ∆ S is the Laplace-Beltrami operator (the surface Laplacian), ∇ S is the surface tangential g...

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

“…Consequently, the quality of ridges extracted depends on the quality and consistency with which these quantities are estimated. In addition, fragmentation of the ridges can occur due to inconsistent choices of principal direction vector orientations [24] and when ridges are near parallel to mesh edges [25]. The aim of the method presented in this paper is to overcome the aforementioned problems with the discrete techniques and to extract all types of ridges on discrete surfaces for which generic conditions hold.…”

“…Curvatures and their derivatives are estimated at mesh vertices by fitting smooth surfaces locally or over the entire mesh such as compactly supported radial basis functions [32], polynomials [25], [28], [33], MLS based implicit functions [34] or using discrete methods [24], [25]. Ridges are traced by detecting zero crossings of the ridge function on the vertices and edges of the meshes.…”

“…Smoothing of the ridge function (as opposed to the surface) as well as smoothing crest space curves themselves was proposed in [24] to obtain crests with fewer undulations. Local angle measures between end points of crest segments have been used for connecting disjoint segments [25].…”

“…of[25],Figure 2of[32] and Figures 2 and 6 of[24] respectively. The crests extracted using the proposed method may vary from the results Ridges extracted using the proposed approach on algebraic functions represented with a single trivariate Bézier patch: (a) Ellipsoid; (b) Tangle:x 4 − 5 x 2 + y 4 − 5 y 2 + z 4 − 5 z 2 + 11.8 = 0.…”

Data-driven curvature for real-time line drawing of dynamic scenes

“…However, even the fastest curvature estimation algorithms (e.g., Taubin [1995], Meyer et al [2002], Cohen-Steiner and Morvan [2003]) suffer from degenerate cases and noisy estimates, and do not compute third-order surface derivatives [Rusinkiewicz 2004]. Other fast methods based on focal surface approximations [Yoshizawa et al 2007] are also affected by degeneracies and do not apply in parabolic regions (unless refined by slow nonlinear optimization techniques [Yu et al 2007]). In general, in order to maintain robustness to noise, irregular tessellation, and also to fully compute third-order derivatives, more expensive computations are necessary.…”