Keenan Crane

CARNEGIE MELLON UNIVERSITY

Optimal Cone Singularities for Conformal Flattening

SIGGRAPH 2018 / ACM Transactions on Graphics 2018

Angle-preserving or *conformal* surface parameterization has proven to be a powerful tool across applications ranging from geometry processing, to digital manufacturing, to machine learning, yet conformal maps can still suffer from severe area distortion. *Cone singularities* provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult. This paper develops a strategy that is globally optimal in the sense that it minimizes total area distortion among all possible cone configurations (number, placement, and size) that have no more than a fixed total cone angle. A key insight is that, for the purpose of optimization, one should not work directly with curvature measures (which naturally represent cone configurations), but can instead apply *Fenchel-Rockafellar duality* to obtain a formulation involving only ordinary functions. The result is a convex optimization problem, which can be solved via a sequence of sparse linear systems easily built from the usual cotangent Laplacian. The method supports user-defined notions of importance, constraints on cone angles (*e.g.*, positive, or within a given range), and sophisticated boundary conditions (*e.g.*, convex, or polygonal). We compare our approach to previous techniques on a variety of challenging models, often achieving dramatically lower distortion, and demonstrating that global optimality leads to extreme robustness in the presence of noise or poor discretization.

@article{Soliman:2018:OCS,
author = {Soliman, Yousuf and Slep\v{c}ev, Dejan and Crane, Keenan},
title = {Optimal Cone Singularities for Conformal Flattening},
journal = {ACM Trans. Graph.},
volume = {37},
number = {4},
year = {2018},
publisher = {ACM},
address = {New York, NY, USA},
}

Thanks to Henrik Schumacher for useful references to algorithms from optimal control, to Rohan Sawhney for implementation help with BFF, and to Alex Huth for sharing cortical surface data. This work was sponsored in part by NSF Awards CCF 1717320 and DMS 1516677, a CMU Summer Undergraduate Research Fellowship (SURF), and gifts from Autodesk Research and Adobe Research.

This material is based upon work supported by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Figures