On Fast Bilateral Filtering Using Fourier Kernels

Abstract

It was demonstrated in earlier work that, by approximating its range kernel using shiftable functions, the non-linear bilateral filter can be computed using a series of fast convolutions. Previous approaches based on shiftable approximation have, however, been restricted to Gaussian range kernels. In this work, we propose a novel approximation that can be applied to any range kernel, provided it has a pointwise-convergent Fourier series. More specifically, we propose to approximate the Gaussian range kernel of the bilateral filter using a Fourier basis, where the coefficients of the basis are obtained by solving a series of leastsquares problems. The coefficients can be efficiently computed using a recursive form of the QR decomposition. By controlling the cardinality of the Fourier basis, we can obtain a good tradeoff between the run-time and the filtering accuracy. In particular, we are able to guarantee sub-pixel accuracy for the overall filtering, which is not provided by most existing methods for fast bilateral filtering. We present simulation results to demonstrate the speed and accuracy of the proposed algorithm.Index Terms-bilateral filter, shiftability, Fourier basis, fast algorithm, accuracy.

A Fast Approximation of the Bilateral Filter using the Discrete Fourier Transform

“…Similarly to [4,11,2,7] the present idea is to use trigonometric sums for approximating the range kernel. The key difference is that instead of approximating the continuous kernel, we propose to approximate the discrete kernel samples.…”

“…The key difference is that instead of approximating the continuous kernel, we propose to approximate the discrete kernel samples. This was motivated by the recent work in [7], where the continuous kernel was first approximated using a Fourier series and then sampled for optimization purpose. We turn this idea 3 around and directly consider the sequence of kernel samples that appear in (1) and (2), which we then approximate using the discrete Fourier transform.…”

“…which is then used in place of T in (6) and (7). Notice that when σ r is sufficiently small, T max = T .…”

“…In fact, one would expect ω to scale with the image size in most applications of the bilateral filter [9]. Several fast algorithms have been proposed that offer various trade-offs between speed and accuracy [6,8,10,13,4,11,3,1,2,7]. We refer the interested reader to these recent papers [11,3] for a survey and comparison of various fast algorithms.…”

“…The present work was motivated by a series of recent papers [4,1,2,7], where it was demonstrated that a fast O(1) algorithm can be derived by approximating the Gaussian range kernel using polynomial and trigonometric functions. These functions have the so-called shiftability property [1] that can be used to derive an O(1) algorithm.…”