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.

通过多个二维网格片段线性映射的深度组合，本研究提出了一种新的代表性注射变形的三维空间表达方式，克服了注射方法的现有限制：不准确性、缺乏鲁棒性以及与通用学习和优化框架的不兼容性。该方法能够高效准确地优化和学习复杂变形，并在注射方法中表现出色。作为主要应用，我们生成了复杂且无伪影的 NeRF 和 SDF 变形。