Conformal Factor Persistence for Fast Hierarchical Cone Extraction

Abstract

This paper presents a new algorithm for the fast extraction of hierarchies of cone singularities for conformal surface parameterization. Cone singularities have been shown to greatly improve the distortion of such parameterizations since they locally absorb the area distortion. Therefore, existing automatic approaches aim at inserting cones where large area distortion can be predicted. However, such approaches are iterative, which results in slow computations, even often slower than the actual subsequent parameterization procedure. This becomes even more problematic as often the user does not know in advance the right number of needed cones and thus needs to explore cone hierarchies to obtain a satisfying result. Our algorithm relies on the key observation that the local extrema of the conformal factor already provide a good approximation of the cone singularities extracted with previous techniques, while needing only one linear solving where previous approaches needed one solving per hierarchy level. We apply concepts from persistent homology to organize very efficiently such local extrema into a global hierarchy. Experiments demonstrate the approximation quality of our approach quantitatively and report time-performance improvements of one order of magnitude, which makes our technique well suited for interactive contexts.