Spherical Framelets from Spherical Designs

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)2072-2104
Number of pages33
Journal / PublicationSIAM Journal on Imaging Sciences
Issue number4
Online published14 Nov 2023
Publication statusPublished - Dec 2023



In this paper, we investigate in detail the structures of the variational characterization AN,t of the spherical t-design, its gradient ∇AN,t, and its Hessian ℋ(AN,t}) in terms of fast spherical harmonic transforms. Moreover, we propose solving the minimization problem of AN,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 (semidiscrete) 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 and spherical designs, including Wendland function approximation, ETOPO data processing, and spherical image denoising. © 2023 Society for Industrial and Applied Mathematics,

Research Area(s)

  • tight framelets, spherical framelets, spherical t-designs, fast spherical harmonic transforms, fast spherical framelet transforms, trust-region method, Wendland functions, ETOPO1, spherical signals/images, image/signal denoising

Citation Format(s)

Spherical Framelets from Spherical Designs. / Xiao, Yuchen; Zhuang, Xiaosheng.
In: SIAM Journal on Imaging Sciences, Vol. 16, No. 4, 12.2023, p. 2072-2104.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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