# Notes on Fourier-Bessel wavelets

arXiv: https://arxiv.org/abs/2609.26537 · HTML: https://mlsanity.com/paper/2609.26537/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 11 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 2 of 2)

- [Rigid-Motion Scattering for Texture Classification](https://mlsanity.com/paper/1403.1687/) (2014)
- [Solid Harmonic Wavelet Scattering for Predictions of Molecule Properties](https://mlsanity.com/paper/1805.00571/) (2018)

## Abstract (arXiv)

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 L2-based applications and discuss L1-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.