Stretch-minimizing Volumetric Parameterization

Not many methods for parameterization guarantee bijectivity or local injectivity, which is essential for foldover-free mappings. Stretch-minimizing parameterization which is widely used for surface parameterization, provides foldover-free mappings and is capable of trading off between angle and area distortions. We extend its usage to volumetric parameterization in this paper by deriving a 3D version of stretch-distortion energy and incorporating fixed boundary conditions. Our energy definition includes a natural barrier term which effectively prevents elements from collapsing and folding over. It saves the effort in other methods of formulating additional energy or constrains to ensure free of foldover.
We propose to minimize the overall energy integrated over the whole mesh with a relaxation-enhanced solver, which optimizes the parameterization globally. This is different from the conventional approach of surface parameterization where mesh nodes are optimized individually. Comparing to other volumetric parameterization, our approach bears the advantages of stretch-minimizing method, being foldover-free and offering a good trade-off between angle and volume distortions.