COLOR BILATERAL FILTERING USING STRATIFIED FOURIER SAMPLING

Brute force implementation of the bilateral filter is known to be prohibitively slow. Several fast approximations have been proposed in the literature that are able to accelerate the filtering without perceptible loss of filtering quality. In particular, it has been shown that by replacing the range kernel (usually Gaussian) of the bilateral filter with its Fourier approximation, the filtering can be performed using fast convolutions. While an accurate Fourier approximation of a one-dimensional Gaussian (for grayscale filtering) can be obtained using N similar to 10 terms, a comparable approximation in three dimensions (for color filtering) requires N-3 Fourier terms, and proportionate number of convolutions. As shown in prior work, we can overcome this problem using Monte Carlo (MC) sampling. In this paper, we demonstrate that the variance of MC sampling can be reduced using stratified sampling, i.e., by conditionally sampling the low and high frequency terms. Importantly, we are able to cut down the pixelwise fluctuation of the filtered output as a result. We analytically compute the variances of MC and stratified sampling, whereby the variance reduction is evident. The PSNR fluctuation of our approximation is also shown to be smaller than existing Monte-Carlo algorithms.