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Imaging Sci."],"published-print":{"date-parts":[[2023,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this paper, we investigate in detail the structures of the variational characterization [Formula: see text] of the spherical [Formula: see text]-design, its gradient [Formula: see text], and its Hessian [Formula: see text] in terms of fast spherical harmonic transforms. Moreover, we propose solving the minimization problem of [Formula: see text] using the trust-region method to provide spherical [Formula: see text]-designs with large values of [Formula: see text]. Based on the obtained spherical [Formula: see text]-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 [Formula: see text]-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.<\/jats:p>","DOI":"10.1137\/22m1542362","type":"journal-article","created":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T12:47:50Z","timestamp":1699966070000},"page":"2072-2104","source":"Crossref","is-referenced-by-count":5,"title":["Spherical Framelets from Spherical Designs"],"prefix":"10.1137","volume":"16","author":[{"given":"Yuchen","family":"Xiao","sequence":"first","affiliation":[{"name":"Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7238-0143","authenticated-orcid":true,"given":"Xiaosheng","family":"Zhuang","sequence":"additional","affiliation":[{"name":"Corresponding author. 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