On Fast Bilateral Filtering using Fourier Kernels

TL;DR

This paper introduces a versatile Fourier-based approximation method for the bilateral filter's range kernel, enabling fast, accurate filtering for any kernel with a convergent Fourier series, surpassing previous Gaussian-only approaches.

Contribution

The authors develop a novel Fourier series approximation technique for the bilateral filter's range kernel applicable to any kernel with a convergent Fourier series, enhancing speed and accuracy.

Findings

Achieves sub-pixel accuracy in bilateral filtering.

Provides a flexible method for various range kernels.

Demonstrates improved speed and accuracy through simulations.

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 least-squares 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.