A direct implementation of the bilateral filter [1] requires O(\sigma_s^2) operations per pixel, where \sigma_s is the (effective) width of the spatial kernel. A fast implementation of the bilateral filter was recently proposed in [2] that required O(1) operations per pixel with respect to \sigma_s. This was done by using trigonometric functions for the range kernel of the bilateral filter, and by exploiting their so-called shiftability property. In particular, a fast implementation of the Gaussian bilateral filter was realized by approximating the Gaussian range kernel using raised cosines. Later, it was demonstrated in [3] that this idea could be extended to a larger class of filters, including the popular non-local means filter [4]. As already observed in [2], a flip side of this approach was that the run time depended on the width \sigma_r of the range kernel. For an image with (local) intensity variations in the range [0,T], the run time scaled as O(T^2/\sigma^2_r) with \sigma_r. This made it difficult to implement narrow range kernels, particularly for images with large dynamic range. We discuss this problem in this note, and propose some simple steps to accelerate the implementation in general, and for small \sigma_r in particular. [1] C. Tomasi and R. Manduchi, "Bilateral filtering for gray and color images", Proc. IEEE International Conference on Computer Vision, 1998. [2] K.N. Chaudhury, Daniel Sage, and M. Unser, "Fast O(1) bilateral filtering using trigonometric range kernels", IEEE Transactions on Image Processing, 2011. [3] K.N. Chaudhury, "Constant-time filtering using shiftable kernels", IEEE Signal Processing Letters, 2011. [4] A. Buades, B. Coll, and J.M. Morel, "A review of image denoising algorithms, with a new one", Multiscale Modeling and Simulation, 2005.

本文探讨了双边滤波器运行时间依赖于其空域核宽度的问题，提出了用 raised cosines 近似高斯范围核的方法，及其用于扩展到更广泛类别的滤波器，从而减少运行时间。

双边滤波和非局部均值可转移O(1)算法的加速