Quantitative Error Analysis of Bilateral Filtering

One of the fastest acceleration techniques for bilateral image filtering is the real time O(1) quantization method proposed by Yang 2009, which first computes some Principal Bilateral Filtered Image Components (PBFICs) and then applies linear interpolation to estimate the filtered output images. There is a trade-off between accuracy and efficiency in selecting the number of PBFICs: the more PBFICs are used, the higher the accuracy, and the higher the computational cost. A question arises: how many PBFICs are required to achieve a certain level of accuracy? In this letter, we address this question by investigating the properties of bilateral filtering and deriving the linear interpolation error bounds when only a subset of PBFICs is used. The provided theoretical analysis indicates that the necessary number of PBFICs for user-provided precision depends on the range kernel and, for typical Gaussian range kernels, a small percentage (typically less than 4%) of the PBFICs are enough for good approximations.