Fast high-dimensional filtering using clustering

Several useful algorithms for image filtering involve non-linear processing of high-dimensional data. Instances of these so-called high-dimensional filters are the bilateral, joint-bilateral, and non-local means filters. Real-time implementation of high-dimensional filters can be challenging. In this paper, we present a simple and fast algorithm for generic high-dimensional filtering. The algorithm is based on a linearization mechanism, which allows us to approximate the high-dimensional filtering using a series of spatial convolutions. We use clustering for the linearization, whereby we are able to exploit the strong correlation between the components of the high-dimensional image. The highlight of our method is that we can prove that the approximation error (the gap between the fast approximation and the exact filtering) vanishes with the increase in the number of clusters. To the best of our knowledge, this is the first algorithm for high-dimensional filtering that enjoys this theoretical guarantee. In fact, we provide empirical evidence which suggests that this basic requirement is not met by the state-of-the-art Adaptive Manifolds algorithm. We use the proposed algorithm for edge-preserving smoothing and denoising of color and hyperspectral images. The results demonstrate that our algorithm is competitive with existing fast algorithms.