Bilateral Filtering for Gray and Color Images

1. Introduction
2. The Idea
3. The Gaussian Case
4. Experiments with Black-and-White Images
5. Experiments with Color Images
6. References

Introduction

Filtering is perhaps the most fundamental operation of image processing and computer vision. In the broadest sense of the term "filtering", the value of the filtered image at a given location is a function of the values of the input image in a small neighborhood of the same location. For example, Gaussian low-pass filtering computes a weighted average of pixel values in the neighborhood, in which the weights decrease with distance from the neighborhood center. Although formal and quantitative explanations of this weight fall-off can be given, the intuition is that images typically vary slowly over space, so near pixels are likely to have similar values, and it is therefore appropriate to average them together. The noise values that corrupt these nearby pixels are mutually less correlated than the signal values, so noise is averaged away while signal is preserved.
The assumption of slow spatial variations fails at edges, which are consequently blurred by linear low-pass filtering. How can we prevent averaging across edges, while still averaging within smooth regions? Many efforts have been devoted to reducing this undesired effect. Bilateral filtering is a simple, non-iterative scheme for edge-preserving smoothing.

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The Idea

The basic idea underlying bilateral filtering is to do in the range of an image what traditional filters do in its domain. Two pixels can be close to one another, that is, occupy nearby spatial location, or they can be similar to one another, that is, have nearby values, possibly in a perceptually meaningful fashion.
Consider a shift-invariant low-pass domain filter applied to an image:

The bold font for f and h emphasizes the fact that both input and output images may be multi-band. In order to preserve the DC component, it must be