G-VR: INVERTIBLE GEOMETRIC VOLUME REPRESEN- TATION FOR LEARNING

Investigating geometric structure representation is key to explainable and controllable learning. We introduce G-VR, an invertible geometric representation for 3D volume data that enables lossless reversible encoding and exact restoration. Our framework operates on a standard 3D uniform grid, where the associated scalar field, whether voxel intensities from medical scans or occupancy ratios from solid geometries, serves as the target measure. Grounded in optimal transport and quasiconformal mappings on three-manifolds, G-VR encodes each volume into a canonical geometric signature: a quasiconformal deformation field that captures the intrinsic geometric structure of the data. Together with the extrinsic total mass of the volume, this representation guarantees exact restoration of both the geometry and the original scalar field. The representation naturally preserves geometric structure under interpolation and manipulation, while ensuring full invertibility. Experiments on both image-based and geometry-based volumes validate its fidelity, reversibility, and effectiveness in interpolation.