Spherical Designs for Function Approximation and Beyond

TL;DR

The paper addresses the challenge of constructing and applying spherical $t$-designs on $\mathbb{S}^2$ by leveraging a variational criterion $A_{N,t}(X_N)$ and two optimization strategies, LS-RCG and TR-PCG, with full and approximate Hessians. It demonstrates how design-ruled point sets enable accurate projection of functions onto $\Pi_t$ via weighted least-squares, and analyzes the approximation of Wendland RBFs as well as non-smooth functions on the sphere. A semi-discrete spherical framelet framework built from spherical designs is developed to denoise spherical data using four thresholding schemes and spherical-cap neighborhoods, improving reconstruction quality for Wendland signals. Overall, the work shows that spherical designs facilitate accurate function approximation and robust, cap-assisted denoising, highlighting their potential for spherical data processing in diverse applications such as geophysics and computer vision.

Abstract

In this paper, we compare two optimization algorithms using full Hessian and approximation Hessian to obtain numerical spherical designs through their variational characterization. Based on the obtained spherical design point sets, we investigate the approximation of smooth and non-smooth functions by spherical harmonics with spherical designs. Finally, we use spherical framelets for denoising Wendland functions as an application, which shows the great potential of spherical designs in spherical data processing.

Figures (9)

• Figure 1: Numerical simulation of spherical initial point sets (left column) as input point sets and related spherical $t$-design point sets (right column) as output point sets based on TR-PCG for $t=50$ and $N=(t+1)^2$ on $\mathbb S^2$. (a) on sphere. (b) equirectangular projection.
• Figure 2: Numerical simulation of spherical initial point sets (left column) as input point sets and resulted spherical $t$-design point sets (right column) as output point sets based on TR-PCG for $t=50$ and $N=(t+1)^2$ on $\mathbb S^2$. (a): on sphere. (b): equirectangular projection.
• Figure 3: Numerical simulation of real part of Projection term (first row), residual term (middle row), and the equirectangular projection of the residual (last row) for RBF $f_4$ under the setting $T=\frac{t}{2}$ and $W=\mathbf w$ on Algorithm \ref{['alg:projCG']} for $t=200,N=(t+1)^2$.
• Figure 4: Numerical simulation of the real part of Projection term (first row), the residual term (middle row), and the equirectangular projection of the residual (last row) for RBF $f_4$ under the setting $T=\frac{t}{2}$ and $W=\mathbf w$ on Algorithm \ref{['alg:projCG']} for $t\approx 200,N\approx (t+1)^2$.
• Figure 5: Relative projection $L_2$-error with $err(f,f_k)$ for $k=0,1,2,3,4$ in different noise level $\sigma$. (a)-(d) are the extracted view from $\sigma=0.05,0.1,0.15,0.2$, $(e)-(i)$ are focusing on Wendland function $f_k$ for $k=0,1,2,3,4$.