Mat\'ern Kernels for Tunable Implicit Surface Reconstruction

TL;DR

This paper introduces the use of Matérn kernels for implicit surface reconstruction, demonstrating their advantages over existing methods in terms of performance, simplicity, and scalability, with theoretical analysis and practical results.

Contribution

It presents the novel application of Matérn kernels for surface reconstruction, including theoretical insights and data-dependent variants, outperforming state-of-the-art methods in efficiency and competitiveness.

Findings

Matérn kernels outperform arc-cosine kernels in surface reconstruction.

Laplace kernel achieves near state-of-the-art accuracy with significantly reduced training time.

Matérn kernels offer tunable, scalable, and easier-to-implement solutions for 3D reconstruction.

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.