Fourier-Bessel wavelets offer evenly spaced radial frequency analysis
Notes on Fourier-Bessel wavelets
Machine LearningComputer Vision and Pattern Recognition
Summary
The paper explores a new kind of wavelet that uses math based on circles and vibrations to analyze signals or images. Instead of the usual way wavelets space frequencies, these wavelets space them more evenly in the radial direction. The authors built the wavelets using special mathematical functions and made formulas for how to use them efficiently. This could help in situations where an even spread of frequencies is more useful than traditional spacing.
What this means in practice
- •For signal processing engineers: Build signal analysis tools that need balanced and broad radial frequency coverage using a new wavelet basis.
- •For image processing developers: Develop image filters and feature extractors benefiting from uniformly spaced radial frequency bands for better frequency coverage.
A theory result. No direct application yet.
Authors
Marcel Venturotti, Georgios Exarchakis
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 $π$ 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.