Implicit Neural Representations and the Algebra of Complex Wavelets

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 couple spatial and spectral features of the represented signal in a way that is not obvious in the usual discrete representation. 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 performed in the first layer of the MLP. This leads to multiple prescriptions for the design of INR architectures, including the use of progressive wavelets, decoupling of low and high-pass approximations, and initialization schemes based on the singularities of the target signal.