Robust Deformation-based 3D Bijective Mapping through Differential Boundary Shape Matching
Materials
Abstract
We introduce a novel framework to generate piecewise linear tetrahedral maps, by expressing the map-finding problem as a constrained mesh deformation problem, where constraints satisfaction implies both injectivity and target shape conformity. Deformation is supported by the two core components of our approach: First, an augmentation of the mesh by boundary-incident virtual elements, encoding the target shape in a differential manner via its first and second fundamental form; in contrast to position-driven approaches this prevents optimisation failure due to improper winding. Second, a remeshing scheme maintaining mesh element quality and preventing blocking configurations. A first implementation is empirically shown to have higher robustness and/or higher efficiency than state-of-the-art methods on challenging, yet practically relevant inputs. We illustrate our method's versatility in a range of applications, demonstrating its value as a new item in the algorithmic toolbox for volumetric mapping.
Selected figures
(𝑎) Given a solid shape (with tetrahedralisation M) and a boundary map 𝜙𝜕 onto a target space boundary 𝜕C, our method completes this to a volumetric map 𝜙, in a piecewise linear manner. It is based on performing a deformation towards the target shape. (𝑏 ) As its key to success, this deformation is not driven by explicit positional objectives; instead our approach compatibly augments the mesh, in its input shape and in its target shape, with a layer of boundary-incident virtual tetrahedral elements (green). (𝑐 ) Continuously deforming each virtual element into its target shape while dragging M along, yields a global deformation of M into the target shape mesh MC . (𝑑, 𝑒 ) For a different instance (rod to spiral): trajectories of boundary vertices under the deformation successfully driven by our technique (𝑑 ) or by a classical positional objective (𝑒 ) where near-linear movement leads into a blocking state (red).
BibTeX
@article{Nigolian:2026:boundary_deformation_maps,
author = {Nigolian, Valentin Z. and Campen, Marcel and Bommes, David},
title = {Robust Deformation-based 3D Bijective Mapping through Differential Boundary Shape Matching},
journal = {ACM Transactions on Graphics},
volume = {45},
number = {6},
year = {2026},
publisher = {ACM},
address = {New York, NY, USA},
doi = {10.1145/3842573}
}