Congpei An

Congpei An, Siyong Chen

This paper studies numerical integration over the unit sphere $ \mathbb{S}^2 \subset \mathbb{R}^{3} $ by using spherical $t$-design, which is an equal positive weights quadrature rule with polynomial precision $t$. We investigate two kinds of spherical $t$-designs with $t$ up to 160. One is well conditioned spherical $t$-design(WSTD), which was proposed by [1] with $ N=(t+1)^{2} $. The other is efficient spherical $t$-design(ESTD), given by Womersley [2], which is made of roughly of half cardinality of WSTD. Consequently, a series of persuasive numerical evidences indicates that WSTD is better than ESTD in the sense of worst-case error in Sobolev space $ \mathbb{H}^{s}(\mathbb{S}^2) $. Furthermore, WSTD is employed to approximate integrals of various of functions, especially including integrand has a point singularity over the unit sphere and a given ellipsoid. In particular, to deal with singularity of integrand, Atkinson's transformation [3] and Sidi's transformation [4] are implemented with the choices of `grading parameters' to obtain new integrand which is much smoother. Finally, the paper presents numerical results on uniform errors for approximating representive integrals over sphere with three quadrature rules: Bivariate trapezoidal rule, Equal area points and WSTD.

Congpei An, Hao-Ning Wu, Xiaoming Yuan

The total variation (TV) regularization has phenomenally boosted various variational models for image processing tasks. We propose to combine the backward diffusion process in the earlier literature of image enhancement with the TV regularization, and show that the resulting enhanced TV minimization model is particularly effective for reducing the loss of contrast. The main purpose of this paper is to establish stable reconstruction guarantees for the enhanced TV model from noisy subsampled measurements with two sampling strategies, non-adaptive sampling for general linear measurements and variable-density sampling for Fourier measurements. In particular, under some weaker restricted isometry property conditions, the enhanced TV minimization model is shown to have tighter reconstruction error bounds than various TV-based models for the scenario where the level of noise is significant and the amount of measurements is limited. Advantages of the enhanced TV model are also numerically validated by preliminary experiments on the reconstruction of some synthetic, natural, and medical images.

Yuchen Xiao, Congpei An

A point set $\mathrm X_N$ on the unit sphere is a spherical $t$-design is equivalent to the nonnegative quantity $A_{N,t+1}$ vanished. We show that if $\mathrm X_N$ is a stationary point set of $A_{N,t+1}$ and the minimal singular value of basis matrix is positive, then $\mathrm X_N$ is a spherical $t$-design. Moreover, the numerical construction of spherical $t$-designs is valid by using Barzilai-Borwein method. We obtain numerical spherical $t$-designs with $t+1$ up to $127$ at $N=(t+2)^2$.