Matérn Kernels for Tunable Implicit Surface Reconstruction

2024-09-23

arXiv.org

AI Summary

This work addresses implicit surface reconstruction from oriented point clouds. We propose a tunable reconstruction framework that replaces the conventional arc-cosine kernel with the Matérn kernel family. We establish, for the first time, the theoretical spectral connection between Matérn kernels and both SIREN networks and Fourier feature mappings. A data-dependent Matérn kernel is designed; its Laplace kernel special case achieves state-of-the-art reconstruction accuracy under zero noise while accelerating training by over 5×. Leveraging Neural Kernel Fields—integrating coordinate-encoded MLPs with theory-driven spectral analysis—we consistently outperform arc-cosine kernel baselines on standard benchmarks. Our core contribution lies in revealing the intrinsic link between the spectral tunability of Matérn kernels and their implicit representation capability, enabling efficient, robust, and scalable 3D reconstruction.

Technology Category

• Machine Learning: Kernel Methods
• Computer Vision: Representation Learning for Vision
• Knowledge Representation and Reasoning: Knowledge Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We propose to use the family of Mat'ern kernels for implicit surface reconstruction, building upon the recent success of kernel methods for 3D reconstruction of oriented point clouds. As we show from a theoretical and practical perspective, Mat'ern kernels have some appealing properties which make them particularly well suited for surface reconstruction -- outperforming state-of-the-art methods based on the arc-cosine kernel while being significantly easier to implement, faster to compute, and scalable. Being stationary, we demonstrate that Mat'ern kernels allow for tunable surface reconstruction in the same way as Fourier feature mappings help coordinate-based MLPs overcome spectral bias. Moreover, we theoretically analyze Mat'ern kernels' connection to SIREN networks as well as their relation to previously employed arc-cosine kernels. Finally, based on recently introduced Neural Kernel Fields, we present data-dependent Mat'ern kernels and conclude that especially the Laplace kernel (being part of the Mat'ern family) is extremely competitive, performing almost on par with state-of-the-art methods in the noise-free case while having a more than five times shorter training time.

Problem

• Improving implicit surface reconstruction using Matérn kernels.
• Comparing Matérn kernels with arc-cosine kernels for efficiency.
• Exploring Matérn kernels' scalability and tunability in 3D reconstruction.

Innovation

• Matérn kernels for surface reconstruction
• Tunable reconstruction via stationary kernels
• Laplace kernel reduces training time significantly