Adaptive Volumetric Parameterization of Simply Connected 3-Manifolds with Applications

A new framework for volumetric parameterization is pushing beyond the standard “map everything to a ball” approach used in many 3D processing pipelines. Instead of forcing every shape into the same fixed target, the method adapts the destination domain while optimizing the mapping, with the goal of reducing both local shape distortion and mass distortion.

The framework supports three target-domain modes: a prescribed solid ellipsoid, a volume-normalized adaptive ellipsoid with variable radii, and a sea-embedded free-boundary domain. Each mode combines a 3D quasi-conformality shape update, a diffusion-based density-equalizing update, and a geometric correction step to eliminate foldings. That makes it more than a single mapping trick; it is a full pipeline for keeping volumetric parameterizations usable in practice.

For developers working in graphics, simulation, or content tools, the practical appeal is cleaner volume representations for downstream tasks. The method is positioned for multiresolution and localized adaptive volumetric remeshing, volumetric registration, and volumetric morphing, all of which benefit when the parameterization preserves structure instead of introducing unnecessary warping.

The work is also a reminder that volumetric workflows are still constrained by domain choice, not just solver quality. Adapting the target shape during optimization could be a useful lever for pipelines that need better fidelity on irregular anatomy, scanned assets, or procedural volumes, especially when later stages depend on stable correspondence across the mesh.