Monge-Ampere Regularization for Learning Arbitrary Shapes From Point Clouds.

Area of Science:

• Computer Vision
• Computer Graphics
• Geometric Deep Learning

Background:

• Signed Distance Functions (SDF) are limited to watertight shapes.
• Unsigned Distance Functions (UDF) handle diverse surfaces but suffer from non-differentiability issues, impacting reconstruction quality.

Purpose of the Study:

• Introduce Scaled-Squared Distance Function (S²DF) for arbitrary surface modeling.
• Address UDF's non-differentiability problem for improved implicit surface representation.
• Develop a learning pipeline for S²DF from unoriented point clouds.

Main Methods:

• Propose S²DF, an implicit surface representation that avoids inside/outside distinctions and resolves non-differentiability.
• Demonstrate S²DF satisfies a Monge-Ampere-type partial differential equation.
• Develop a novel Monge-Ampere regularization for unsupervised S²DF learning from point clouds.

Main Results:

• The proposed unsupervised learning pipeline effectively learns S²DF from raw, unoriented point clouds.
• S²DF-based reconstruction achieves significantly higher quality compared to state-of-the-art supervised methods.
• Experimental validation across multiple datasets confirms the method's robustness and superiority.

Conclusions:

• S²DF offers a robust and effective implicit surface representation for diverse geometric modeling tasks.
• Unsupervised learning of S²DF via Monge-Ampere regularization is a viable and high-performing alternative to supervised methods.
• The proposed approach advances the state-of-the-art in implicit neural representations and 3D shape reconstruction.