ResearchPod Summary
Traditional wavelet filter banks, such as those used in scattering networks, typically rely on dyadic scaling, where each scale doubles the previous one. While this logarithmic frequency tiling is mathematically elegant and provides stability, it may not be optimal for all signal processing tasks. This paper explores an alternative construction inspired by disk harmonics—solutions to the Helmholtz equation on a unit disk—to determine if a more uniform radial frequency allocation can be achieved. The authors derive a wavelet family by applying a Gaussian spatial envelope to these disk harmonics, incorporating a zero-mean correction for the zeroth angular order, and providing closed-form expressions for their normalization and Fourier-domain representations.
The primary contribution is the mathematical framework for a Fourier-Bessel wavelet family. By using the Neumann eigenvalues of the Helmholtz equation as the radial frequency parameter, the authors demonstrate that the spacing between consecutive radial eigenvalues converges to pi. This linear spacing contrasts with the dyadic structure of conventional wavelets. The authors provide explicit normalization constants for both L2-based applications and frequency-domain peak consistency. Furthermore, they derive a closed-form Fourier-domain representation, which allows for efficient filter implementation without numerical transforms. Preliminary numerical experiments suggest that these wavelets provide a flatter frequency coverage profile with lower ripple compared to solid harmonic wavelets.
This work offers a flexible alternative to standard wavelet filter banks. By moving away from strict dyadic scaling, researchers can explore frequency organizations that provide more uniform coverage, which may be particularly beneficial for reconstruction-oriented tasks where balanced frequency representation is preferred over the low-frequency emphasis inherent in dyadic systems. The authors have also released the fbscatnet library to facilitate the integration of these wavelets into existing scattering network frameworks.
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