Research Abstracts - 2007

A Signal-Processing Approach

The bilateral filter is a nonlinear filter that smoothes a signal while preserving strong edges. It has demonstrated great effectiveness for a variety of problems in computer vision and computer graphics. Frédo Durand and Julie Dorsey have described a fast version at SIGGRAPH 2002 [1]. Unfortunately, little was known about the accuracy of this acceleration. We propose a new signal processing analysis of the bilateral filter, which complements the recent studies that analyzed it as a PDE or as a robust statistics estimator. Importantly, this signal-processing perspective allows us to develop a novel bilateral filtering acceleration using a downsampling in space and intensity. This affords a principled expression of the accuracy in terms of bandwidth and sampling. The key to our analysis is to express the filter in a higher-dimensional space where the signal intensity is added to the original domain dimensions. The bilateral filter can then be expressed as simple linear convolutions in this augmented space followed by two simple nonlinearities.

A Fast Approximation

This new formulation allows us to derive simple criteria for downsampling the key operations and to achieve important acceleration of the bilateral filter. We have shown that, for the same running time, our method is significantly more accurate than previous acceleration techniques. From a practical point of view, we are able to process large images in very short time. For instance, a 2-megapixel picture is filtered in less than a second and the result is visually identical to the exact computation that requires several minutes, even with optimized code. In comparison, the previous technique shows important visual differences (see the figure below).

Conclusion

This work introduces a novel interpretation of the bilateral filter. It contributes to a better understanding of the nonlinear filters by unveiling a strong link between linear and nonlinear techniques. We believe this work can yield further theoretical progress in this domain.

From a practical point of view, the achieved speed-up and faithfulness pave the way for high-quality and interactive image manipulations. Being able to process in a second picture at screen resolution enables a direct user interaction where only an off-line computation were available before. Such a tool opens new possibilities in the field of computational photography and video processing.

This work has been published at the European Conference on Computer Vision 2006 [2]. Code is available on our website.

References:

[1] Frédo Durand and Julie Dorsey. Fast Bilateral Filtering for the Display of High-Dynamic-Range Images. In Proceedings of SIGGRAPH 2002.

[2] A Fast Approximation of the Bilateral Filter using a Signal Processing Approach. Sylvain Paris and Frédo Durand. In Proceedings of the European Conference on Computer Vision. 2006.