Robust Deformation-based 3D Bijective Mapping through Differential Boundary Shape Matching

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.