# Divergence-Free Shape Interpolation and Correspondence

arXiv: https://arxiv.org/abs/1806.10417 · HTML: https://mlsanity.com/paper/1806.10417/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2018.

## Built on by (most cited first, 6 of 6)

- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Dynamic Point Fields](https://mlsanity.com/paper/2304.02626/) (2023)
- [4Deform: Neural Surface Deformation for Robust Shape Interpolation](https://mlsanity.com/paper/2502.20208/) (2025)
- [Spline Deformation Field](https://mlsanity.com/paper/2507.07521/) (2025)
- [Simulating Refractive Distortions and Weather-Induced Artifacts for Resource-Constrained Autonomous Perception](https://mlsanity.com/paper/2507.05536/) (2025)
- [On the Structural Failure of Chamfer Distance in 3D Shape Optimization](https://mlsanity.com/paper/2603.09925/) (2026)

## Abstract (arXiv)

We present a novel method to model and calculate deformation fields between shapes embedded in ℝD. Our framework combines naturally interpolating the two input shapes and calculating correspondences at the same time. The key idea is to compute a divergence-free deformation field represented in a coarse-to-fine basis using the Karhunen-Lo`eve expansion. The advantages are that there is no need to discretize the embedding space and the deformation is volume-preserving. Furthermore, the optimization is done on downsampled versions of the shapes but the morphing can be applied to any resolution without a heavy increase in complexity. We show results for shape correspondence, registration, inter- and extrapolation on the TOSCA and FAUST data sets.