Optimal cone singularities for conformal flattening - Robotics Institute Carnegie Mellon University

Optimal cone singularities for conformal flattening

Yousuf Soliman, Dejan Slepčev, and Keenan Crane
Journal Article, ACM Transactions on Graphics (TOG), Vol. 37, No. 4, August, 2018

Abstract

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.

BibTeX

@article{Soliman-2018-121353,
author = {Yousuf Soliman and Dejan Slepčev and Keenan Crane},
title = {Optimal cone singularities for conformal flattening},
journal = {ACM Transactions on Graphics (TOG)},
year = {2018},
month = {August},
volume = {37},
number = {4},
}