TY - JOUR
T1 - Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds
AU - Yang, Chuanxiang
AU - Zhou, Yuanfeng
AU - Wei, Guangshun
AU - Ma, Long
AU - Hou, Junhui
AU - Liu, Yuan
AU - Wang, Wenping
PY - 2025/4/23
Y1 - 2025/4/23
N2 - 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. The code will be publicly available at https://github.com/chuanxiang-yang/S2DF. © 2025 IEEE.
AB - 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. The code will be publicly available at https://github.com/chuanxiang-yang/S2DF. © 2025 IEEE.
KW - Implicit neural representation
KW - distance function
KW - surface reconstruction
U2 - 10.1109/TPAMI.2025.3563601
DO - 10.1109/TPAMI.2025.3563601
M3 - RGC 21 - Publication in refereed journal
SN - 0162-8828
SP - 1
EP - 15
JO - IEEE Transactions on Pattern Analysis and Machine Intelligence
JF - IEEE Transactions on Pattern Analysis and Machine Intelligence
ER -