Computing Inversion-Free Mappings by Simplex Assembly - Fu, Xiao-Ming; Liu, Yang (ACM Transactions on Graphics (TOG) - Proceedings of ACM SIGGRAPH Asia 2016, 2016)
We present a novel method, called Simplex Assembly, to computeinversion-free mappings with low or bounded distortion on simpli-cial meshes. Our method involves two steps: simplex disassemblyand simplex assembly. Given a simplicial mesh and its initial piece-wise afﬁne mapping, we project the afﬁne transformation associatedwith each simplex into the inversion-free and distortion-boundedspace. The projection disassembles the input mesh into disjointsimplices. The disjoint simplices are then assembled to recover theoriginal connectivity by minimizing the mapping distortion and thedifference of the disjoint vertices with respect to the piecewise afﬁnetransformations, while the piecewise afﬁne mapping is restrictedinside the feasible space. Due to the use of afﬁne transformations asvariables, our method explicitly guarantees that no inverted simplexoccurs, and that the mapping distortion is below the bound duringthe optimization. Compared with existing methods, our method isrobust to an initialization with many inverted elements and positionalconstraints. We demonstrate the efﬁciency and robustness of ourmethod through a variety of geometric processing tasks