Fast Kernel Smoothing by a Low-Rank Approximation of the Kernel Toeplitz Matrix

Kernel smoothing methods, including the bilateral filter, are commonly used in data processing/modeling and edge-aware image smoothing. Due to their nonlinear nature, these filters require significant computational time. In this paper, we address this problem by studying a practical case in which the data to be processed are integers. The basic idea is to use eigendecomposition to approximate the kernel matrix which is a real symmetric Toeplitz matrix. This approximation leads to more efficient computation. We study the distribution of its eigenvalues and show that the upper bounds of the eigenvalues can be expressed analytically in terms of the Fourier transform of the kernel function. This result not only captures the relationship between the order of the low-rank approximation of the kernel matrix and the filtering quality, but also shows that among the three kernel functions considered in this work, the Gaussian kernel can be most efficiently approximated. We have applied the proposed fast algorithm to implement the bilateral filter. By taking advantage of a property of the Gaussian kernel, we have also proposed another algorithm with even faster speed. Experimental results show that the performance of the proposed algorithms is competitive with those state-of-the-art algorithms in terms of speed and quality.