Notes on Fourier-Bessel wavelets

The paper considers the construction of two new orthogonal multiwavelets with supercompact support by using the Fast Bauer's method for matrix spectral factorization on the matrix product filter of the orthogonal CL multiwavelet filter. The new multiwavelets possess orthogonality, symmetry/antisymmetry, and one of them provides better coding and smoothness than other supercompact multiwavelets. The performance of the new multiwavelet filters in subband-based edge detection, grayscale and color image compression and 1D and 2D signal denoising is compared with the GHM, SA4, CL, Integer Haar and Alpert multifilters. The comparative analysis shows that new multiwavelets can provides better human visual measures, SSIM and MS-SSIM in image compression and denoising applications.

We introduce PHAST-Net, an attention-guided, physics-informed network for unified estimation of Ideal Time-Frequency Representations (ITFRs), spanning spectral, tempo-based, metrical, and harmonic representations such as Spectrograms, Tempograms, and Metrograms. PHAST-Net learns an application-general mapping from a constellation of wavelet transforms, the proposed Continuous Log-frequency Adaptive Wavelet Transform (CLAWT), to high-resolution, cross-term-suppressed time-frequency (T-F) representations. The proposed constellation of CLAWTs is selected through Cohen's class kernel analysis to maximise curvature coverage in a logarithmic-frequency T-F plane tailored to harmonic signal structure. PHAST-Net further incorporates a proposed physics-informed auxiliary reprojection loss designed to reconstruct the idealised observed CLAWT constellation from the predicted ITFR and the corresponding Cohen's class kernels during training. This auxiliary objective promotes transform consistency and energy conservation, mitigates pathological target sparsity, and enhances optimisation stability. Attention layers further promote effective cross-term suppression across the input constellation. The log-frequency formulation also enables Harmonic PHAST-Net, which estimates a Harmonic ITFR that isolates fundamental structure, supporting robust fundamental-only representations for speech and music, such as derived fundamental Tempograms and Metrograms. We further introduce Spline-PHAST-Net, which parameterises detected and associated T-F ridges as continuous spline trajectories, enabling arbitrary-grid re-rendering and signal reconstruction. Trained on an effectively unbounded procedurally generated dataset, PHAST-Net demonstrates improved accuracy over established approaches, providing a unified framework for high-resolution, cross-term-robust analysis of speech, music, and broader nonstationary signals.

Reconstructing oscillatory wave fields from scattered sensors is a severely underdetermined inverse problem. Beyond the challenges of general physical-field reconstruction, wave responses are complex-valued, frequency-sensitive, and highly oscillatory, while costly simulation and sensing often leave only extreme-sparse observations. Existing low-rank, operator, and diffusion approaches are largely designed for real-valued, smoother fields; dense pixel-space diffusion is particularly inefficient for oscillatory complex fields and difficult to scale to 3D. We propose HarmoCore, which places a generative prior in a compact, continuous, and structured wave-field latent. HarmoCore represents joint real--imaginary channels with Functional Tucker cores over shared continuous spatial bases, learns a frequency-conditioned core diffusion prior, and performs Diffusion Posterior Sampling directly in core space. At fixed sensor coordinates, the multilinear decoder induces an explicit likelihood guidance operator, avoiding dense pixel-space correction. Optional target-equation residual guidance further promotes physical consistency. Experiments on 2D Helmholtz, 2D synthetic wave fields, and 3D Helmholtz show substantial gains under 1%--2% sensing while remaining practical in three dimensions.