Spherical Framelets from Spherical Designs

Abstract

Spherical t-designs, which are point sets that satisfy the equal weight polynomial exact quadrature rule in the space of spherical polynomials on the unit sphere with degree at most t, hold immense potential in a wide range of real-world applications. These applications include satellite signals and global navigation, climate change estimation, virus analysis, planetary studies, cosmic microwave background (CMB) data analysis, and 360º panoramic images and videos in virtual reality. In this thesis, our primary focus is to address two key topics within this field of study. Firstly, we aim to explore efficient methods for constructing large-scale spherical designs in the realm of numerical analysis. Secondly, we will investigate the application of these large-scale spherical designs in signal and image processing.

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 Nonequispaced fast spherical Fourier transforms (NFSFT). Moreover, we explore different optimization methods in solving the minimization problem of A_N,t such as restart conjugate gradient method with Newton-Raphson line search strategy (LS-RCG) and trust-region method with preconditioned conjugate gradient method (TR-PCG). Through extensive experiments, we successfully utilize the TR-PCG method to achieve large spherical t-designs with high values of t (up to 3200). Additionally, we investigate the approximation of smooth and non-smooth functions using spherical harmonics with spherical designs.

Based on the obtained spherical t-designs, we develop (semi-discrete) spherical tight framelets and their truncated systems as well as 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 and non-smooth function approximation, ETOPO data processing, and spherical image denoising.

Date of Award	6 Aug 2024
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Xiaosheng ZHUANG (Supervisor)