On Optimal Sampling for Learning SDF Using MLPs Equipped with Positional Encoding

TL;DR

This work analyzes why sinusoidal positional encoding (PE) in MLPs used for neural signed distance fields (SDFs) produces wavy artifacts and proposes a practical, frequency-based approach to determine training data sampling. By approximating the intrinsic response spectrum of randomly initialized PE-enabled MLPs and applying Nyquist-Shannon sampling theory, the authors define a cut-off frequency $F_c$ from a fitted intrinsic spectrum and set the training density as $\mu = (2F_c)^\ell$, achieving high-fidelity SDF fitting. They demonstrate that sampling at or above this rate suppresses aliasing-induced artifacts and outperforms baselines like SIREN, SPE, and NGLOD across multiple shapes, including high-frequency details and surface geometry. The method also generalizes to other architectures and remains robust under different initializations, offering a practical baseline for neural implicit representations while highlighting the need for theoretical grounding of the cut-off frequency. Overall, the paper provides a principled, data-efficient strategy to select sampling densities that enable PE-equipped networks to accurately capture complex 3D geometry with neural implicit representations.

Abstract

Neural implicit fields, such as the neural signed distance field (SDF) of a shape, have emerged as a powerful representation for many applications, e.g., encoding a 3D shape and performing collision detection. Typically, implicit fields are encoded by Multi-layer Perceptrons (MLP) with positional encoding (PE) to capture high-frequency geometric details. However, a notable side effect of such PE-equipped MLPs is the noisy artifacts present in the learned implicit fields. While increasing the sampling rate could in general mitigate these artifacts, in this paper we aim to explain this adverse phenomenon through the lens of Fourier analysis. We devise a tool to determine the appropriate sampling rate for learning an accurate neural implicit field without undesirable side effects. Specifically, we propose a simple yet effective method to estimate the intrinsic frequency of a given network with randomized weights based on the Fourier analysis of the network's responses. It is observed that a PE-equipped MLP has an intrinsic frequency much higher than the highest frequency component in the PE layer. Sampling against this intrinsic frequency following the Nyquist-Sannon sampling theorem allows us to determine an appropriate training sampling rate. We empirically show in the setting of SDF fitting that this recommended sampling rate is sufficient to secure accurate fitting results, while further increasing the sampling rate would not further noticeably reduce the fitting error. Training PE-equipped MLPs simply with our sampling strategy leads to performances superior to the existing methods.

Figures (20)

• Figure 1: The benefits from positional encoding (PE) and our sampling strategy. Zero-level sets of the learned signed distance field (SDF) on the Bimba model, using (a) vanilla MLP without PE, (b) PE-equipped MLP with insufficient training samples (33k points), and (c) PE-equipped MLP with our recommended sampling strategy, respectively.
• Figure 2: Intrinsic frequency of a network. (a) Consistent response frequency spectra are observed in randomized PE-equipped MLPs with the same architecture (8 layers with 512 neurons per layer); (b) The averaged spectrum of the network over multiple randomized network weights is defined as the intrinsic spectrum of the PE-equipped MLP.
• Figure 3: Intrinsic spectra of PE-equipped MLPs with different $D$. The degree of sinusoidal positional encoding (PE), $D$, increases from 3 to 5. The highest frequency in the sinusoidal PE is marked with the vertical black line. The networks carry high-frequency components with large magnitudes beyond the PE's highest frequency. The cut-off frequency of the networks (vertical dashed lines) is defined to determine the sampling rate and avoid aliasing. The magnitude beyond the cut-off frequency exhibits negligible values.
• Figure 4: Training sample points are from the grid cells intersecting with the surface shape.
• Figure 5: Validation of the recommended sampling rate. (a) to (c) show the spectra of PE-equipped MLPs (8 layers with 512 neurons per layer) using different $D$ (the degree of PE). Black curves are fit to the spectra for determining the respective cut-off frequencies (vertical dashed lines). SDF convergence errors against the sampling rates are shown in (d).