In this paper, we investigate in detail the structures of the variational characterization $A_{N,t}$ of the spherical $t$-design, its gradient $\nabla A_{N,t}$, and its Hessian $\mathcal{H}(A_{N,t})$ 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.

翻译：本文详细研究了球面t-设计变分特征$A_{N,t}$的结构、其梯度$\nabla A_{N,t}$以及Hessian矩阵$\mathcal{H}(A_{N,t})$，并利用快速球面调和变换进行表述。此外，我们提出采用信赖域方法求解$A_{N,t}$的极小化问题，以生成大参数t的球面t-设计。基于所获得的球面t-设计，我们构建了（半离散）球面紧框架小波及其截断系统，并开发了适用于实际球面信号/图像处理的快速球面框架小波变换。得益于大参数球面t-设计以及球面框架小波的局域化特性，我们能够基于精细调制的球冠约束，通过局部阈值技术实现信号/图像去噪。通过包括Wendland函数逼近、ETOPO数据处理和球面图像去噪在内的多项数值实验，验证了我们所构建球面框架小波的效率与有效性。