Spherical framelets from spherical designs
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In this paper, we investigate in detail the structures of the variational characterization A_N,t of the spherical t-design, its gradient ∇ A_N,t, and its Hessian 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.
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