  The bilateral and nonlocal means filters are instances of kernel-based
filters that are popularly used in image processing. It was recently shown that
fast and accurate bilateral filtering of grayscale images can be performed
using a low-rank approximation of the kernel matrix. More specifically, based
on the eigendecomposition of the kernel matrix, the overall filtering was
approximated using spatial convolutions, for which efficient algorithms are
available. Unfortunately, this technique cannot be scaled to high-dimensional
data such as color and hyperspectral images. This is simply because one needs
to compute/store a large matrix and perform its eigendecomposition in this
case. We show how this problem can be solved using the Nystr\"om method, which
is generally used for approximating the eigendecomposition of large matrices.
The resulting algorithm can also be used for nonlocal means filtering. We
demonstrate the effectiveness of our proposal for bilateral and nonlocal means
filtering of color and hyperspectral images. In particular, our method is shown
to be competitive with state-of-the-art fast algorithms, and moreover it comes
with a theoretical guarantee on the approximation error.
