Injective and bounded distortion mappings in 3D

Abstract

We introduce an efficient algorithm for producing provably injective mappings of tetrahedral meshes with strict bounds on their tetrahedra aspect-ratio distortion.

The algorithm takes as input a simplicial map (e.g., produced by some common deformation or volumetric parameterization technique) and projects it on the space of injective and bounded-distortion simplicial maps. Namely, finds a similar map that is both bijective and bounded-distortion. As far as we are aware, this is the first algorithm to produce injective or bounded-distortion simplicial maps of tetrahedral meshes. The construction of the algorithm was made possible due to a novel closed-form solution to the problem of finding the closest orientation-preserving bounded-distortion matrix to an arbitrary matrix in three (and higher) dimensions.

The algorithm is shown to have quadratic convergence, usually not requiring more than a handful of iterations to converge. Furthermore, it is readily generalized to simplicial maps of any dimension, including mixed dimensions. Finally, it can deal with different distortion spaces, such as bounded isometric distortion. During experiments we found the algorithm useful for producing bijective and bounded-distortion volume parameterizations and deformations of tetrahedral meshes, and improving tetrahedral meshes, increasing the tetrahedra quality produced by state-of-the-art techniques.

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Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures

    • Computational geometry