Matérn Kernels for Tunable Implicit Surface Reconstruction

TL;DR

The paper tackles implicit surface reconstruction from sparse data and introduces Matérn kernels with tunable smoothness and bandwidth as a stationary alternative to arc-cosine. It establishes theoretical links between Matérn kernels, Fourier feature mappings, NTK/SIREN networks, and arc-cosine kernels, and provides a practical bound to guide bandwidth selection. Empirically, Matérn 1/2 and 3/2 deliver competitive or superior geometry and texture reconstructions while offering faster training and inference than arc-cosine-based methods and NKSR. The work also demonstrates the benefits of data-dependent Matérn kernels via the NKF framework, highlighting both performance gains and training efficiency, with broad applicability to 3D reconstruction tasks.

Abstract

We propose to use the family of Matérn 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, Matérn 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 Matérn 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 Matérn 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 Matérn kernels and conclude that especially the Laplace kernel (being part of the Matérn 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.

Figures (12)

• Figure 1: Matérn kernels with associated spectral densities. We propose to use the family of Matérn kernels for tunable implicit surface reconstruction, parametrized by a smoothness parameter, $\nu>0$, that controls the differentiability of the kernel, and a bandwidth parameter $h>0$. Both parameters allow explicit manipulation of the kernel and its spectrum. Importantly, Matérn kernels recover the Laplace kernel for $\nu=1/2$ and the Gaussian kernel as $\nu\rightarrow\infty$.
• Figure 2: Tunable Matérn kernels lead to better surface reconstructions than the previously employed arc-cosine kernel. Here, we show surface reconstructions from sparse point clouds of just 1,000 points.
• Figure 3: Eigenvalue decay of Matérn kernels. While we fix $h=1$ and vary $\nu$ on the left, the EDR for $\nu=1/2$ and different values of $h$ is shown on the right. Larger values of $\nu$ and $h$, i.e., smoother kernels, lead to faster eigenvalue decay; hence, slow convergence to high-frequency details.
• Figure 4: Matérn kernels are tunable. Surface reconstructions can be improved in practice by varying the bandwidth parameter, $h$, effectively tuning the kernels' EDR (see also Figure \ref{['fig:eigenvalue_decay']}). However, setting $h$ too small ($<0.5$) results in overfitting, while setting $h$ too big ($>10$) oversmooths (i.e., underfits) the true surface. This is also reflected in the reconstruction error (measured using Chamfer distance), plotted as a function of $h$ on the right.
• Figure 5: RKHS norm as a function of $h$. We plot the bound from Proposition \ref{['lemma:bound_norm']} as a function of $h$.

Theorems & Definitions (13)

• Proposition 1
• Theorem 2
• Theorem 3: chen2021, Theorem 1
• Corollary 3
• Proposition 3
• proof
• Theorem 4: Bochner
• Theorem 4
• proof
• Remark 5