Spherical framelets from spherical designs

Abstract

In this paper, we investigate in detail the structures of the variational characterization $A_{N,t}$ of the spherical $t$-design, its gradient $\nabla A_{N,t}$, and its Hessian $\mathcal{H}(A_{N,t})$ in terms of fast spherical harmonic transforms. Moreover, we propose solving the minimization problem of $A_{N,t}$ using the trust-region method to provide spherical $t$-designs with large values of $t$. Based on the obtained spherical $t$-designs, we develop (semi-discrete) spherical tight framelets as well as their truncated systems and their fast spherical framelet transforms for the practical spherical signal/image processing. Thanks to the large spherical $t$-designs and localization property of our spherical framelets, we are able to provide signal/image denoising using local thresholding techniques based on a fine-tuned spherical cap restriction. Many numerical experiments are conducted to demonstrate the efficiency and effectiveness of our spherical framelets, including Wendland function approximation, ETOPO data processing, and spherical image denoising.

arXiv comment30 pages, 9 figures

References in corpus (6)

• HEALPix -- a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere
• Localisation of directional scale-discretised wavelets on the sphere
• Complex data processing: fast wavelet analysis on the sphere
• Directional Tensor Product Complex Tight Framelets with Low Redundancy
• Tight framelets and fast framelet filter bank transforms on manifolds
• Frames of directional wavelets on $n$-dimensional spheres

Cited by in corpus (1)

• A framework for diffuseness evaluation using a tight-frame microphone array configuration