提出S²DF新方法，从无向点云直接学习任意形状表面。
Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds
- 用S²DF替代UDF，解决零等值面不可导问题。
- 引入闵可夫斯基-阿普雷型正则化，无需真实标签即可训练。
- 在多个数据集上优于有监督的先进方法，适合复杂形状建模。
现有的隐式几何表示中，符号距离函数（SDF）仅能建模封闭形状，而无符号距离函数（UDF）虽可表示多种曲面，但其在零等值面处不可导，导致重建质量不佳。本文提出一种新型隐式曲面表示——缩放平方距离函数（S²DF），不区分内外区域，有效克服了UDF在零等值面处的不可导问题。我们证明S²DF满足一类二阶蒙日-阿普雷型偏微分方程，从而设计了一种基于新蒙日-阿普雷正则化的学习流程，可直接从原始无向点云中学习S²DF，无需依赖真实的S²DF真值作为监督。大量实验表明，该方法在多个数据集上显著优于需真值监督的最新有监督方法。源代码已公开于https://github.com/chuanxiang-yang/S2DF。
原文摘要 · Abstract (English)
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.
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