Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds

Chuanxiang Yang, Yuanfeng Zhou*, Guangshun Wei, Long Ma, Junhui Hou, Yuan Liu, Wenping Wang

*Corresponding author for this work

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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 (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. 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. © 2025 IEEE.
Original languageEnglish
Pages (from-to)6809-6822
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Volume47
Issue number8
Online published23 Apr 2025
DOIs
Publication statusOnline published - 23 Apr 2025

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 62172257, in part by the Natural Science Foundation of Shandong Province under Grant ZR2024ZD12, in part by the NSFC Excellent Young Scientists Fund under Grant 62422118, and in part by the Hong Kong RGC under Grant 11219324 and Grant 11219422.

Research Keywords

  • Implicit neural representation
  • distance function
  • surface reconstruction

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