It was recently demonstrated in [4][arxiv:1105.4204] that the non-linear
bilateral filter \cite{Tomasi} can be efficiently implemented using an O(1) or
constant-time algorithm. At the heart of this algorithm was the idea of
approximating the Gaussian range kernel of the bilateral filter using
trigonometric functions. In this letter, we explain how the idea in [4] can be
extended to few other linear and non-linear filters [18,21,2]. While some of
these filters have received a lot of attention in recent years, they are known
to be computationally intensive.

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 [bilateralFilter]
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 Hilbert transform is essentially the only singular operator in dimension
1, which undoubtedly makes it the most important linear operator in harmonic
analysis. The Hilbert transform has had a profound bearing on several
theoretical and physical problems across a a wide range of disciplines; these
includes problems in Fourier convergence, complex analysis, potential theory,
modulation theory, wavelet theory, aerofoil design, dispersion relations and
high-energy physics to name a few.

We demonstrate that it is possible to filter an image with an elliptic window
of varying size, elongation and orientation with a fixed computational cost per
pixel. Our method involves the application of a suitable global pre-integrator
followed by a pointwise-adaptive localization mesh. We present the basic theory
for the 1D case using a B-spline formalism and then appropriately extend it to
2D using radially-uniform box splines. The size and ellipticity of these
radially-uniform box splines is adaptively controlled. Moreover, they converge
to Gaussians as the order increases.

We propose an amplitude-phase representation of the dual-tree complex wavelet
transform (DT-CWT) which provides an intuitive interpretation of the associated
complex wavelet coefficients.