Continuous Spectral Transform and Modulation for Signal Processing on Arbitrary Data

Giuseppe Patanè

doi:10.1007/s10915-025-02856-7

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Continuous Spectral Transform and Modulation for Signal Processing on Arbitrary Data

2025-04-03

Giuseppe Patanè *

DOI：10.1007/s10915-025-02856-7

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Abstract

En 中文

The Fourier transform (FT) and convolution are fundamental tools for signal analysis andtraining convolutional neural networks. However, their extension and computation on arbi-trary data structures (e.g., graphs, discrete 3D surfaces, ornD point sets) remain an activeresearch area. As an alternative to discrete convolution and FTs, we introduce thecontin-uous spectral modulationandcontinuous spectral transform(ST), formulated as a linearcombination of complex exponentials with Fourier coefficients. The continuous ST exhibitsseveral advantages over the discrete FT, including smoothness, periodicity, and multi-scalerepresentation. It also satisfies standard properties of the FT, such as linearity, continuity,and preservation of angles and distances between signals. The continuous spectral transformprovides a compact representation, enabling us to analyse signals in[0,1]instead ofR;derivekey properties and upper bounds for the continuous ST's behaviour using geometric series;efficiently and stably compute the continuous ST using polynomial representations and geo-metric series. Representing the continuous ST as a geometric series allows us to estimatethe minimum number of Laplacian eigenpairs needed to approximate the ST to the desiredaccuracy. This aspect makes the continuous ST scalable for large datasets, an advantage overthe discrete FT, where determining the optimal number of Laplacian eigenpairs in advance isgenerally not feasible. The continuous ST and modulation are versatile and can be applied tosignals defined on various discrete data structures, including graphs, 3D point sets, andnDdata

Keywords:

Fourier transform

Convolution

Signal processing

Spectral geometry processings