Optimal fitting polynomial for linear time bilateral filters

The bilateral filter (BF) has showed great effectiveness for a variety of problems. However, its brute-force implementation is time consuming. One way of accelerating a BF is to approximate the nonlinear range kernel of the BF by a set of linear time shiftable kernels. To achieve this goal, only finite values of the kernel of the BF have been used to perform smoothing due to the quantisation of digital images. Thus, the filtering results are not changed by substituting the range kernel with the function having the same values at finite discrete points. The Lagrange interpolation polynomial can exactly pass through predefined points and therefore can be employed to replace original kernels for accurate by accelerating the BF. To speed up the BF at the cost of small approximation error, two approximation methods are proposed to obtain the optimal fitting polynomial. The performance of the proposed method is validated by extensive experiments.