Spherical Framelets from Spherical Designs

In this paper, we investigate in detail the structures of the variational characterization A(N,t) of the spherical t-design, its gradient del 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 (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.