Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds

2024-10-24

arXiv.org

AI Summary

Reconstructing arbitrary-topology surfaces from unoriented point clouds remains challenging due to the non-differentiability of unsigned distance functions (UDFs) at their zero-level sets, hindering gradient-based optimization. Method: This paper introduces the Scale-Squared Distance Function (S²DF) as an implicit surface representation and derives a Monge–Ampère-type partial differential equation (PDE) that S²DF inherently satisfies. Leveraging this PDE, we propose the first fully unsupervised implicit learning framework—optimizing end-to-end without ground-truth S²DF labels. Our method jointly enforces PDE consistency, Monge–Ampère regularization, and geometric fidelity to the input point cloud. Results: Extensive experiments demonstrate that our approach significantly outperforms state-of-the-art methods relying on ground-truth surface supervision across multiple benchmarks, achieving high-fidelity, topology-agnostic, and robust surface reconstruction.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

As commonly used implicit geometry representations, the signed distance function (SDF) is limited to modeling watertight shapes, while the unsigned distance function (UDF) is capable of representing various surfaces. However, its inherent theoretical shortcoming, i.e., the non-differentiability at the zero level set, would result in sub-optimal reconstruction quality. In this paper, we propose the scaled-squared distance function (S$^{2}$DF), a novel implicit surface representation for modeling arbitrary surface types. S$^{2}$DF 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 S$^{2}$DF 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 S$^{2}$DF from raw unoriented point clouds without supervision from ground-truth S$^{2}$DF values. Extensive experiments across multiple datasets show that our method significantly outperforms state-of-the-art supervised approaches that require ground-truth surface information as supervision for training. The code will be publicly available at https://github.com/chuanxiang-yang/S2DF.

Problem

• Address non-differentiability of UDF at zero level set
• Model arbitrary surface types using S2DF
• Learn S2DF from unoriented point clouds without supervision

Innovation

• Introduced scaled-squared distance function (S2DF)
• Applied Monge-Ampere regularization technique
• Learned S2DF from unoriented point clouds