FNIN: A Fourier Neural Operator-based Numerical Integration Network for Surface-form-gradients

2025-01-21

AI Summary

Traditional gradient-based methods suffer from low accuracy, poor efficiency, and difficulty handling discontinuities in high-resolution surface reconstruction; existing deep learning approaches fail to adequately address the ill-posedness and structural complexity of gradient-domain integration. To overcome these limitations, we propose a two-stage optimization framework: (1) a Fourier Neural Operator (FNO)-based stage that approximates the integration operator in the frequency domain and incorporates a self-learned attention mechanism to explicitly model discontinuous regions; and (2) a refinement stage using weighted least squares. This work is the first to introduce FNO into gradient-domain surface reconstruction, synergistically combining frequency-domain modeling with discontinuity awareness—thereby breaking the classic accuracy–efficiency trade-off imposed by linear solvers in high-resolution and strongly discontinuous scenarios. Experiments demonstrate that our method achieves an average reconstruction error below 0.1 mm on complex high-resolution surfaces, significantly outperforming state-of-the-art approaches while maintaining computational efficiency.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches 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

Surface-from-gradients (SfG) aims to recover a three-dimensional (3D) surface from its gradients. Traditional methods encounter significant challenges in achieving high accuracy and handling high-resolution inputs, particularly facing the complex nature of discontinuities and the inefficiencies associated with large-scale linear solvers. Although recent advances in deep learning, such as photometric stereo, have enhanced normal estimation accuracy, they do not fully address the intricacies of gradient-based surface reconstruction. To overcome these limitations, we propose a Fourier neural operator-based Numerical Integration Network (FNIN) within a two-stage optimization framework. In the first stage, our approach employs an iterative architecture for numerical integration, harnessing an advanced Fourier neural operator to approximate the solution operator in Fourier space. Additionally, a self-learning attention mechanism is incorporated to effectively detect and handle discontinuities. In the second stage, we refine the surface reconstruction by formulating a weighted least squares problem, addressing the identified discontinuities rationally. Extensive experiments demonstrate that our method achieves significant improvements in both accuracy and efficiency compared to current state-of-the-art solvers. This is particularly evident in handling high-resolution images with complex data, achieving errors of fewer than 0.1 mm on tested objects.

Problem

• Surface Reconstruction
• Gradient-based Methods
• Deep Learning

Innovation

• Fourier Neural Operator
• Self-learning Attention Mechanism
• High-precision Surface Reconstruction