A Triangulation-Invariant Method for Anisotropic Geodesic Map Computation on Surface Meshes

Abstract

This paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner.

Practical Anisotropic Geodesy

“…For this, M needs to be an acute triangulation, no obtuse inner angles are allowed. As this is hardly ever the case for unstructured meshes in practice, techniques that add additional virtual edges/triangles to be considered during the computation have been presented as a remedy [KS98, SV04, YSS*12]. Alternative non‐linear propagation rules have been proposed [NK02, TWZZ07], which can typically increase accuracy in practice – although at the expense of losing consistency [WDB*08].…”

“…While such tensor based metrics are widely used, it bears noting that also more general metrics, based on non‐elliptic norms, are of interest. Examples are terrain steepness profiles [LMS99], curvature (variation) minimizing metrics [YSS*12], or high angular resolution diffusion imaging (HARDI) metrics [PWT05]. We will hence keep the exposition general instead of restricting to Riemannian metrics.…”

Anisotropic geodesics for live‐wire mesh segmentation

“…We will briefly review methods in each strategy, with an eye towards their balance between efficiency and accuracy. Note that while many of these methods return only a distance field, smooth geodesic paths can be traced from the gradient of the field as in [YSS*12].…”

“…While more accuracy can be gained by adding Steiner points and edges [LMS99], the addition comes at the cost of significantly increased computational overhead [CHK13]. Similar performance issue can be found with the ordered upwind method (OUM) [SV03, YSS*12], which extends the Fast Marching method [KS98] to handle general anisotropy but at much higher computational cost.…”

“…Non‐tensor based anisotropic metrics have also been studied for anisotropic heat diffusion [WHSQ11, HLS*13], quad meshing [TPP*11], and computing curvature‐aware geodesic maps [YSS*12]. These metrics are either not directly related to principle curvature directions [WHSQ11, HLS*13] or have similar limitations as the tensor‐based metrics discussed above.…”