Continuous Spectral Transform and Modulation for Signal Processing on Arbitrary Data

Abstract The Fourier transform (FT) and convolution are fundamental tools for signal analysis and training convolutional neural networks. However, their extension and computation on arbitrary data structures (e.g., graphs, discrete 3D surfaces, or n D point sets) remain an active research area. As an alternative to discrete convolution and FTs, we introduce the continuous spectral modulation and continuous spectral transform (ST), formulated as a linear combination of complex exponentials with Fourier coefficients. The continuous ST exhibits several advantages over the discrete FT, including smoothness, periodicity, and multi-scale representation. It also satisfies standard properties of the FT, such as linearity, continuity, and preservation of angles and distances between signals. The continuous spectral transform provides a compact representation, enabling us to analyse signals in [0, 1] instead of $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> ; derive key properties and upper bounds for the continuous ST’s behaviour using geometric series; efficiently and stably compute the continuous ST using polynomial representations and geometric series. Representing the continuous ST as a geometric series allows us to estimate the minimum number of Laplacian eigenpairs needed to approximate the ST to the desired accuracy. This aspect makes the continuous ST scalable for large datasets, an advantage over the discrete FT, where determining the optimal number of Laplacian eigenpairs in advance is generally not feasible. The continuous ST and modulation are versatile and can be applied to signals defined on various discrete data structures, including graphs, 3D point sets, and n D data.