Measure controllable volumetric mesh parameterization

Volumetric parameterization is a fundamental problem in solid and physical modeling. In practice, it is highly desirable to control the volumes of the regions of interest in the parameter domain. This work introduces a novel volumetric parameterization method, which allows users to prescribe the target volumetric measure of the input solid. Given a simply connected tetrahedral mesh with a single boundary surface, we first compute a volumetric harmonic map to parameterize the solid onto the unit solid ball; then we compute an optimal mass transportation map from the unit solid ball with the push-forward volume element induced by the harmonic map onto the parameter domain with the user prescribed volumetric measure. The composition of the volumetric harmonic map and the optimal mass transportation map gives a measure controllable volumetric parameterization. Furthermore, this method can handle solids with empty voids inside. The method has solid theoretic foundation, and is based on conventional algorithms in computational geometry, and easy to implement. The experimental results demonstrate the efficiency and efficacy of the proposed method.