Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds

TL;DR

This paper introduces S²DF, a new implicit surface representation based on Monge-Ampère regularization, enabling high-quality shape reconstruction from point clouds without ground-truth surface supervision.

Contribution

The authors propose S²DF, a novel implicit surface model that addresses non-differentiability issues of UDF and leverages Monge-Ampère regularization for unsupervised learning from raw point clouds.

Findings

S²DF outperforms state-of-the-art supervised methods in shape reconstruction.

The method effectively models arbitrary surface types.

It does not require ground-truth surface data for training.

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 source code is available at https://github.com/chuanxiang-yang/S2DF.