Implicit neural representations (INRs) have arisen as useful methods for
representing signals on Euclidean domains. By parameterizing an image as a
multilayer perceptron (MLP) on Euclidean space, INRs effectively represent
signals in a way that couples spatial and spectral features of the signal that
is not obvious in the usual discrete representation, paving the way for
continuous signal processing and machine learning approaches that were not
previously possible. Although INRs using sinusoidal activation functions have
been studied in terms of Fourier theory, recent works have shown the advantage
of using wavelets instead of sinusoids as activation functions, due to their
ability to simultaneously localize in both frequency and space. In this work,
we approach such INRs and demonstrate how they resolve high-frequency features
of signals from coarse approximations done in the first layer of the MLP. This
leads to multiple prescriptions for the design of INR architectures, including
the use of complex wavelets, decoupling of low and band-pass approximations,
and initialization schemes based on the singularities of the desired signal.

利用隐式神经表示 (INRs) 以欧几里得空间的多层感知器 (MLP) 对图像进行参数化，有效地在信号中表示了在常规离散表示中看不到的耦合空间和频谱特征，为以前不可能的连续信号处理和机器学习方法铺平了道路。本文研究了使用正弦激活函数的 INRs 在傅里叶理论方面的工作，并展示了与正弦函数相比，使用小波作为激活函数的优势，因为它们能够同时在频率和空间上进行本地化。我们在这项工作中探讨了这种 INRs，并演示了它们如何从 MLP 的第一层中进行粗糙逼近从而解析出信号的高频特征。这导致了多种 INR 架构设计的建议，包括使用复数小波，解耦低通和带通逼近以及基于所需信号奇点的初始化方案。