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TutteNet: Injective 3D Deformations by Composition of 2D Mesh Deformations
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Proposes TutteNet, an injective 3D deformation representation built by composing differentiable 2D Tutte-embedding mesh deformations over different planes, applied to NeRF deformation and learned SMPL human deformations.
Problem
Injective (non-self-intersecting) 3D deformations are needed for applications like NeRF deformation, but existing methods are inaccurate, unstable, or hard to combine with learning and optimization frameworks.
Approach
Reduce 3D injectivity to 2D: each layer lifts a 2D Tutte-embedding piecewise-linear mesh map (injective in 2D, parameterized by learnable Laplacian weights and convex boundary conditions) into a prismatic 3D map over a rotated plane, and composing several such layers yields an expressive injective 3D deformation with an explicit inverse and fast Jacobians.
Key results
- On SMPL learning experiment, TutteNet achieves the lowest fitting error among RealNVP, i-ResNet, and NeuralODE, with near-identical deformations, per Table 2 and Figure 4.
- In the overfitting experiment, TutteNet forward time is 0.09 s and Jacobian time 0.02 s, versus 0.11 s and 0.19 s for NeuralODE.
- Compared with NeRF deformation methods, only TutteNet and NeuralODE are injective; non-injective methods (NeRF-Editing, Deforming-NeRF) show rendering artifacts.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use TutteNet instead of NeuralODE for human deformation learning | Human deformation learning · Human deformation learning | vertex error (×10^3): 1.24 -> 0.16 | Helped |
| Use TutteNet instead of NeuralODE for human deformation learning | Human deformation learning · Human deformation learning | mesh-gradient error (×10^3): 25.4 -> 7.7 | Helped |
| Use TutteNet instead of RealNVP for human deformation learning | Human deformation learning · Human deformation learning | vertex error (×10^3): 4.21 -> 0.16 | Helped |
| Use TutteNet instead of i-ResNet for human deformation learning | Human deformation learning · Human deformation learning | mesh-gradient error (×10^3): 92.3 -> 7.7 | Helped |
| Use TutteNet instead of NeuralODE for human deformation fitting | Human deformation fitting · Human deformation fitting | vertex error (×10^3): 0.19 -> 0.15 | Helped |
| Use TutteNet instead of NeuralODE for human deformation fitting | Human deformation fitting · Human deformation fitting | mesh-gradient error (×10^3): 2.6 -> 4.4 | Hurt |
| Use TutteNet instead of RealNVP for human deformation fitting | Human deformation fitting · Human deformation fitting | vertex error (×10^3): 1.7 -> 0.15 | Helped |
| Use TutteNet instead of i-ResNet for human deformation fitting | Human deformation fitting · Human deformation fitting | mesh-gradient error (×10^3): 40.2 -> 4.4 | Helped |
Surprising findings
- NeuralODE attains slightly lower mesh-gradient error than TutteNet in the overfitting experiment, despite much higher vertex error.
- NeuralODE's accuracy degrades markedly in the learning setting, while it is competitive when fitting a single deformation.
Limitations
- Map evaluation is not interactive-time, preventing real-time rendering and interaction.
- Deforming one region affects other regions; localizing deformations is non-trivial.
Experimental setup
Datasets: SMPL-generated human meshes AMASS
Benchmarks: SMPL human deformation fitting and learning NeRF deformation qualitative comparison
Metrics: vertex L2 error intrinsic gradient L2 error timing (sec)
Baselines: RealNVP i-ResNet NeuralODE SPIDR NeRF-Editing Deforming-NeRF
Builds on: Tutte's embedding Bozic et al. injective 2D mesh deformation (neural Tutte) RealNVP NeuralODE i-ResNet
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