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.

翻译：神经隐式场（例如形状的神经有符号距离场SDF）已成为许多应用（如三维形状编码和碰撞检测）的强大表示方式。通常，隐式场通过配备位置编码（PE）的多层感知机（MLP）进行编码，以捕捉高频几何细节。然而，此类配备PE的MLP的一个显著副作用是学习到的隐式场存在噪声伪影。虽然提高采样率通常能缓解这些伪影，但本文旨在通过傅里叶分析的视角解释这一不良现象。我们设计了一个工具来确定合适的采样率，以学习精确且无不良副作用的神经隐式场。具体而言，我们提出了一种简单有效的方法，基于网络响应的傅里叶分析，估计具有随机权重的给定网络的内在频率。观察到配备PE的MLP的内在频率远高于PE层中的最高频率分量。根据奈奎斯特-香农采样定理，针对这一内在频率进行采样，使我们能够确定合适的训练采样率。在SDF拟合场景中，我们通过实验证明，该推荐采样率足以确保准确的拟合结果，而进一步提高采样率不会显著降低拟合误差。仅使用我们的采样策略训练配备PE的MLP，即可获得优于现有方法的性能。