  Existing fast algorithms for bilateral and nonlocal means filtering mostly
work with grayscale images. They cannot easily be extended to high-dimensional
data such as color and hyperspectral images, patch-based data, flow-fields,
etc. In this paper, we propose a fast algorithm for high-dimensional bilateral
and nonlocal means filtering. Unlike existing approaches, where the focus is on
approximating the data (using quantization) or the filter kernel (via analytic
expansions), we locally approximate the kernel using weighted and shifted
copies of a Gaussian, where the weights and shifts are inferred from the data.
The algorithm emerging from the proposed approximation essentially involves
clustering and fast convolutions, and is easy to implement. Moreover, a variant
of our algorithm comes with a guarantee (bound) on the approximation error,
which is not enjoyed by existing algorithms. We present some results for
high-dimensional bilateral and nonlocal means filtering to demonstrate the
speed and accuracy of our proposal. Moreover, we also show that our algorithm
can outperform state-of-the-art fast approximations in terms of accuracy and
timing.
