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

Implicit Neural Representations and the Algebra of Complex Wavelets