It is well-known that spatial averaging can be realized (in space or frequency domain) using algorithms whose complexity does not depend on the size or shape of the filter. These fast algorithms are generally referred to as constant-time or O(1) algorithms in the image processing literature. Along with the spatial filter, the edge-preserving bilateral filter [Tomasi1998] involves an additional range kernel. This is used to restrict the averaging to those neighborhood pixels whose intensity are similar or close to that of the pixel of interest. The range kernel operates by acting on the pixel intensities. This makes the averaging process non-linear and computationally intensive, especially when the spatial filter is large. In this paper, we show how the O(1) averaging algorithms can be leveraged for realizing the bilateral filter in constant-time, by using trigonometric range kernels. This is done by generalizing the idea in [Porikli2008] of using polynomial range kernels. The class of trigonometric kernels turns out to be sufficiently rich, allowing for the approximation of the standard Gaussian bilateral filter. The attractive feature of our approach is that, for a fixed number of terms, the quality of approximation achieved using trigonometric kernels is much superior to that obtained in [Porikli2008] using polynomials.
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