Fast and High-Quality Bilateral Filtering Using Gauss-Chebyshev\n Approximation

The bilateral filter is an edge-preserving smoother that has diverse\napplications in image processing, computer vision, computer graphics, and\ncomputational photography. The filter uses a spatial kernel along with a range\nkernel to perform edge-preserving smoothing. In this paper, we consider the\nGaussian bilateral filter where both the kernels are Gaussian. A direct\nimplementation of the Gaussian bilateral filter requires $O(\\sigma_s^2)$\noperations per pixel, where $\\sigma_s$ is the standard deviation of the spatial\nGaussian. In fact, it is well-known that the direct implementation is slow in\npractice. We present an approximation of the Gaussian bilateral filter, whereby\nwe can cut down the number of operations to $O(1)$ per pixel for any arbitrary\n$\\sigma_s$, and yet achieve very high-quality filtering that is almost\nindistinguishable from the output of the original filter. We demonstrate that\nthe proposed approximation is few orders faster in practice compared to the\ndirect implementation. We also demonstrate that the approximation is\ncompetitive with existing fast algorithms in terms of speed and accuracy.\n