Other approaches to obtain the fast and Piecewise-linear Bilateral Filter in 3D space

Bilateral Filter has demonstrated great effectiveness for a variety of applications in image process, computer vision and computer graphic due to its property of edge preserving while removing noise. Since it is not as fast as linear filter because of its nonlinear nature, lots of researchers have proposed their own fast versions for Bilateral Filter. In this paper, we focus on the fast versions of Fast and Piecewise-linear Bilateral Filter. With the idea of weight averaging the 3D function which is defined on the image surface, we obtained the Fast Bilateral Filter through a more intuitionistic way than the pure mathematic derivation method which is used by Paris. By varying the 3D function form, we derived another 3D expression of Bilateral Filter, then demonstrated that the Piecewise-linear version is a downsampled version of ours, and proposed some theoretical interpretation for this downsampling strategy.