Spherical radial-basis-based kernel interpolation abounds in image sciences including geophysical image reconstruction, climate trends description and image rendering due to its excellent spatial localization property and perfect approximation performance. However, in dealing with noisy data, kernel interpolation frequently behaves not so well due to the large condition number of the kernel matrix and instability of the interpolation process. In this paper, we introduce a weighted spectral filter approach to reduce the condition number of the kernel matrix and then stabilize kernel interpolation. The main building blocks of the proposed method are the well developed spherical positive quadrature rules and high-pass spectral filters. Using a recently developed integral operator approach for spherical data analysis, we theoretically demonstrate that the proposed weighted spectral filter approach succeeds in breaking through the bottleneck of kernel interpolation, especially in fitting noisy data. We provide optimal approximation rates of the new method to show that our approach does not compromise the predicting accuracy. Furthermore, we conduct both toy simulations and two real-world data experiments with synthetically added noise in geophysical image reconstruction and climate image processing to verify our theoretical assertions and show the feasibility of the weighted spectral filter approach.

翻译：基于球面径向基的核插值因优异空间局部化性质与完美逼近性能，广泛应用于地球物理图像重建、气候趋势描述和图像渲染等图像科学领域。然而在处理含噪数据时，由于核矩阵条件数较大及插值过程不稳定性，核插值方法往往表现欠佳。本文引入加权谱滤波器方法以降低核矩阵条件数，从而稳定核插值过程。该方法的核心构件是成熟的球面正求积法则与高通谱滤波器。借助近期发展的球面数据分析积分算子方法，我们从理论上证明本文提出的加权谱滤波器方法能够突破核插值瓶颈，尤其在拟合含噪数据方面表现优异。我们给出新方法的最优逼近率，表明本方法不会降低预测精度。此外，通过在地球物理图像重建和气候图像处理中开展合成噪声添加的玩具仿真与两项真实数据实验，验证了理论断言并展示了加权谱滤波器方法的可行性。