Abstract

This paper proposes a recursive implementation of the bi- lateral filter. Unlike previous methods, this implementation yields an bilateral filter whose computational complexity is linear in both input size and dimensionality. The proposed implementation demonstrates that the bilateral filter can be as efficient as the recent edge-preserving filter- ing methods, especially for high-dimensional images. Let the number of pixels contained in the image be N , and the number of channels be D, the computational complexity of the proposed implementation will be O(N D). It is more efficient than the state-of-the-art bilateral filtering methods that have a computational complexity of O(N D2 ) [1] (linear in the image size but polynomial in dimensionality) or O(N log(N )D) [2] (linear in the dimensionality thus faster than [1] for high-dimensional filtering). Specifically, the proposed implementation takes about 43 ms to process a one megapixel color image (and about 14 ms to process a 1 megapixel grayscale image) which is about 18?faster than [1] and 86?faster than [2]. The experiments were conducted on a MacBook Air laptop computer with a 1.8 GHz Intel Core i7 CPU and 4 GB memory. The memory complexity of the proposed implementation is also low: as few as the image memory will be required (memory for the images be- fore and after filtering is excluded). This paper also derives a new filter named gradient domain bilateral filter from the proposed recursive im- plementation. Unlike the bilateral filter, it performs bilateral filtering on the gradient domain. It can be used for edge-preserving filtering but avoids sharp edges that are observed to cause visible artifacts in some computer graphics tasks. The proposed implementations were proved to be effective for a number of computer vision and computer graphics ap- plications, including stylization, tone mapping, detail enhancement and stereo matching.