Monge-Ampere Regularization for Learning Arbitrary Shapes from Point Clouds
This work solves the problem of reconstructing arbitrary surfaces from unoriented point clouds for applications in computer graphics and 3D modeling, representing a novel method rather than an incremental improvement.
The paper tackles the problem of learning arbitrary shapes from point clouds by proposing a novel implicit surface representation called scaled-squared distance function (S$^{2}$DF), which addresses non-differentiability issues in unsigned distance functions and significantly outperforms state-of-the-art supervised methods in reconstruction quality.
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.