Weighted Spectral Filters for Kernel Interpolation on Spheres: Estimates
  of Prediction Accuracy for Noisy Data
Xiaotong Liu Jinxin Wang Di Wang Shao-Bo Lin
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.