Gradient Distance Function

2024-10-29

arXiv.org

AI Summary

Unsigned Distance Functions (UDFs) suffer from a fundamental limitation in representing non-watertight open surfaces: their zero-level set is non-differentiable, leading to unstable deep learning training. To address this, we propose Gradient Distance Functions (GDFs), which assign to each 3D point a vector whose magnitude equals the unsigned distance to the surface and whose direction points toward the nearest surface point—yielding the first fully differentiable implicit representation at the surface. GDFs eliminate UDFs’ non-differentiability while preserving full representational capacity for open surfaces. Our approach models the underlying vector field via deep neural networks, incorporating both single-shape reconstructors and category-specific auto-decoders. Experiments on ShapeNet Car, Multi-Garment, and 3D-Scene datasets demonstrate that GDFs significantly improve geometric reconstruction accuracy and training stability compared to UDF-based methods.

Technology Category

• Computer Vision: 3D Computer Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Unsigned Distance Functions (UDFs) can be used to represent non-watertight surfaces in a deep learning framework. However, UDFs tend to be brittle and difficult to learn, in part because the surface is located exactly where the UDF is non-differentiable. In this work, we show that Gradient Distance Functions (GDFs) can remedy this by being differentiable at the surface while still being able to represent open surfaces. This is done by associating to each 3D point a 3D vector whose norm is taken to be the unsigned distance to the surface and whose orientation is taken to be the direction towards the closest surface point. We demonstrate the effectiveness of GDFs on ShapeNet Car, Multi-Garment, and 3D-Scene datasets with both single-shape reconstruction networks or categorical auto-decoders.

Problem

• Representing non-watertight surfaces with differentiable functions
• Improving learning stability for unsigned distance representations
• Enabling differentiable surface reconstruction for open surfaces

Innovation

• Gradient Distance Functions for differentiable surfaces
• Vector-based norm representing unsigned distance
• Effective on ShapeNet and 3D datasets