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

Abstract

The bilateral filter is a popular non-linear smoother that has applications in image processing, computer vision, and computational photography. The novelty of the filter is that a range kernel is used in tandem with a spatial kernel for performing edge-preserving smoothing, where both kernels are usually Gaussian. A direct implementation of the bilateral filter is computationally expensive, and several fast approximations have been proposed to address this problem. In particular, it was recently demonstrated in a series of papers that a fast and accurate approximation of the bilateral filter can be obtained by approximating the Gaussian range kernel using polynomials and trigonometric functions. By adopting some of the ideas from this line of work, we propose a fast algorithm based on the discrete Fourier transform of the samples of the range kernel. We develop a parallel C implementation of the resulting algorithm for Gaussian kernels, and analyze the effect of various extrinsic and intrinsic parameters on the approximation quality and the run time. A key component of the implementation are the recursive Gaussian filters of Deriche and Young. Source CodeThe ANSI C source code used in the demo can be downloaded from the web page of this article 1 . Compilation and usage instruction are included in the README.txt of the archive.

Fast Piecewise Polynomial Fitting of Time-Series Data for Streaming Computing

“…They used the DFT to map time sequences to the frequency domain and then had thus mapped sequences to a lower-dimensionality space by using only the first few Fourier coefficients. Besides, Nair et al [10] applied DFT to approximate the bilateral filter, Kithulgoda et al [11] presented a method for incremental maintenance of the Fourier spectrum to changes in concept that take place in streaming computing, also the schemes for feature selection and synopsis generation that enable the coefficient array to be refreshed efficiently on a periodic basis. In DFT, each coefficient represents a sine wave that is added along the entire length of the query, so it is not possible to utilize DFT methods for queries with ''don't care'' subsections [9] or the more general weighted Euclidean distance [16], [40].…”

Linear Subspaces of the Appearance Space

“…Bilateral filters [36,44] are among the most popular tools for edge-aware image processing. One recent approach gives a linear approximation of a bilateral filter [30]. He et al suggest to improve the edge-preservation property of filters by the use of a guidance map [14].…”

Optimized Fourier Bilateral Filtering

“…This makes the real-time implementation challenging for practical values of θ. Several fast approximations [3]- [18] have been proposed to accelerate the brute-force computation of (1). The present focus is on approximations that use trigonometric functions [8], [10], [14], [18].…”

“…Several fast approximations [3]- [18] have been proposed to accelerate the brute-force computation of (1). The present focus is on approximations that use trigonometric functions [8], [10], [14], [18]. We briefly describe their relation to the present work and summarize our contributions.…”

“…It was observed that, by approximating ϕ using trigonometric polynomials, we can compute (1) using fast convolutions. This was later refined and extended in [10]- [18]. In particular, state-ofthe-art results were obtained using Fourier series approximation of a truncated ϕ in [10], where the truncation is simply the intensity range of the input image.…”