  The bilateral filter is a versatile non-linear filter that has found diverse
applications in image processing, computer vision, computer graphics, and
computational photography. A widely-used form of the filter is the Gaussian
bilateral filter in which both the spatial and range kernels are Gaussian. A
direct implementation of this filter requires $O(\sigma^2)$ operations per
pixel, where $\sigma$ is the standard deviation of the spatial Gaussian. In
this paper, we propose an accurate approximation algorithm that can cut down
the computational complexity to $O(1)$ per pixel for any arbitrary $\sigma$
(constant-time implementation). This is based on the observation that the range
kernel operates via the translations of a fixed Gaussian over the range space,
and that these translated Gaussians can be accurately approximated using the
so-called Gauss-polynomials. The overall algorithm emerging from this
approximation involves a series of spatial Gaussian filtering, which can be
implemented in constant-time using separability and recursion. We present some
preliminary results to demonstrate that the proposed algorithm compares
favorably with some of the existing fast algorithms in terms of speed and
accuracy.
