Notes on Fourier-Bessel wavelets

Abstract:These notes develop the mathematical foundations and construction of a Fourier-Bessel wavelet family inspired by the disk harmonics of Shaqfa et al.[9]. We begin with the relevant properties of Bessel and modified Bessel functions and introduce the wavelet properties required for the construction. We then derive the Fourier-Bessel disk harmonics as solutions to the Helmholtz equation on the unit disk subject to a Neumann boundary condition. Building on this basis, we construct a wavelet family by applying a Gaussian spatial envelope and introducing a zero-mean correction for the zeroth angular order. We derive the corresponding normalisation constants for $L^2$-based applications and discuss $L^1$-based normalisation for frequency-domain peak consistency. Finally, we derive a closed-form Fourier-domain representation of the resulting wavelets. The main motivation is the approximately linear spacing, which converges to $\pi$ between consecutive radial eigenvalues. Rather than replacing the conventional dyadic organisation of wavelet families, this construction lays out the foundation to explore whether a more uniform radial frequency allocation can be useful for applications in which broad and balanced frequency coverage is desirable.

Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV); Numerical Analysis (math.NA)

arXiv:2609.26537 [cs.LG]

(or arXiv:2609.26537v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.26537 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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