On Optimal Sampling for Learning SDF Using MLPs Equipped with Positional
  Encoding
Guying Lin Lei Yang Yuan Liu Congyi Zhang Junhui Hou Xiaogang Jin Taku Komura John Keyser Wenping Wang
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.