Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds

TPAMI 2025

Paper | Project Page | arXiv

In this paper, we propose the scaled-squared distance function (S²DF), a novel implicit surface representation for modeling arbitrary surface types. S²DF does not distinguish between inside and outside regions while effectively addressing the non-differentiability issue of UDF at the zero level set. We demonstrate that S²DF satisfies a second-order partial differential equation of Monge-Ampere-type, allowing us to develop a learning pipeline that leverages a novel Monge-Ampere regularization to directly learn S²DF from raw unoriented point clouds without supervision from ground-truth S²DF values.

Install

conda create -n S2DF python=3.8
conda activate S2DF
pip install -r requirements.txt
conda install -c plotly plotly plotly-orca # conda only 
# Install with instructions from https://pytorch.org/get-started/locally/
conda install pytorch==2.0.1 torchvision==0.15.2 torchaudio==2.0.2 pytorch-cuda=11.7 -c pytorch -c nvidia

Dataset

Please organize the data according to the following structure. We provide the input point clouds of the 3D scene.

│data/
├──shapenet/
│  ├── input
│  ├── ground_truth
├──3dscene/
│  ├── input
│  ├── ground_truth
...

Running Surface Reconstruction

To train and test on 3D scene run

cd ./surface_reconstruction
./scripts/run_3dscene_recon.sh

If you want to reconstruct other data, please refer to run_3dscene_recon.sh and write a script accordingly.

CAP-UDF provides excellent evaluation code.

Thanks

Thanks to the DiGS codebase, StEik codebase and DCUDF codebase off whom we built upon.

Citation

If you find our work useful in your research, please cite our paper:

@ARTICLE{yang2025monge,
    author={Yang, Chuanxiang and Zhou, Yuanfeng and Wei, Guangshun and Ma, Long and Hou, Junhui and Liu, Yuan and Wang, Wenping},
    journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, 
    title={Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds}, 
    year={2025},
    volume={},
    number={},
    pages={1-15},
    keywords={Surface reconstruction;Point cloud compression;Shape;Neural networks;Training;Mathematical models;Three-dimensional displays;Decoding;Topology;Surface fitting;Implicit neural representation;distance function;surface reconstruction},
    doi={10.1109/TPAMI.2025.3563601}
}