Computing Inversion-Free Mappings by Simplex Assembly

We present a novel method, called Simplex Assembly, to compute inversion-free mappings with low or bounded distortion on simplicial meshes. Our method involves two steps: simplex disassembly and simplex assembly. Given a simplicial mesh and its initial piecewise affine mapping, we project the affine transformation associated with each simplex into the inversion-free and distortion-bounded space. The projection disassembles the input mesh into disjoint simplices. The disjoint simplices are then assembled to recover the original connectivity by minimizing the mapping distortion and the difference of the disjoint vertices with respect to the piecewise affine transformations, while the piecewise affine mapping is restricted inside the feasible space. Due to the use of affine transformations as variables, our method explicitly guarantees that no inverted simplex occurs, and that the mapping distortion is below the bound during the optimization. Compared with existing methods, our method is robust to an initialization with many inverted elements and positional constraints. We demonstrate the efficiency and robustness of our method through a variety of geometric processing tasks.