  This work proposes a novel representation of injective deformations of 3D
space, which overcomes existing limitations of injective methods: inaccuracy,
lack of robustness, and incompatibility with general learning and optimization
frameworks. The core idea is to reduce the problem to a deep composition of
multiple 2D mesh-based piecewise-linear maps. Namely, we build differentiable
layers that produce mesh deformations through Tutte's embedding (guaranteed to
be injective in 2D), and compose these layers over different planes to create
complex 3D injective deformations of the 3D volume. We show our method provides
the ability to efficiently and accurately optimize and learn complex
deformations, outperforming other injective approaches. As a main application,
we produce complex and artifact-free NeRF and SDF deformations.
