Conceptually, each point in the FF plane corresponds to a chirp component in the original signal, which also corresponds to a linear portion of the time-frequency (TF) plane. The Radon transform (also known as the Hough transform) is formulated as a family of line integrals through a two-dimensional function. It is known for its ability to extract straight lines from images. For a good survey paper on the Radon transform, see Illingworth and Kittlerillingworth. This property allows us to use it as an alternate means of computing the FF plane of the chirplet transform, by using the TF plane as our input image.
The Radon transform provides us with a simple means of computing the FF plane of the autochirplet transform, by using the Wigner distribution, and arriving at a transform space that tells us basically the same information as the chirplet FF-plane, except that we benefit from the greater resolution of the Wigner distribution. It is well known that the cross components of the Wigner distribution are of an oscillatory nature, while the autocomponents give a net positive contribution. Therefore, since the Radon transform is integrating along lines, the cross terms of the Wigner distribution are cancelled out along each line, so that the points in the Radon transform of the Wigner distribution only ``see'' the autocomponents of the Wigner distribution (Fig. 11(a)).
Figure: FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY
The Radon transform is usually computed from the normal equation of a line:
x (
) + y () =
as an integral along each of these lines in the original space.
The parameter space is sampled uniformly, in the 
coordinates.
It is easier to compare
the Radon transform of the Wigner distribution
with the chirplet CF plane (Fig. 10)
if, by first, without loss
of generality, we normalize 
and 
to be
on the interval from -1/2 to 1/2 and
the TF distribution to have time and frequency coordinates
on the same interval from -1/2 to 1/2.
Then we make the substitution:
() = /f_avg
and
() = 1/f_dif\!f
where 
and 
.
A simpler (perhaps equally well known) form of the Radon transform
parameterizes the lines in
terms of their slopes and intercepts. This parameterization
has the advantage that it maps lines
to points, and points to lines, while it has the disadvantage
that there is a singularity when lines of infinite slope (vertical lines)
are encountered. Because of the Nyquist limit, however, we do not have
this problem when the input to the Radon transform is a time-frequency
distribution.
Thus we may be tempted
to use the slope-intercept form of the Radon transform, except that
we would prefer to have a parameterization in that matches the
FF plane rather than the CF plane, for reasons previously discussed.
The ``Nyquist boundaries'' we referred to earlier are most evident
if we simply consider the discrete Radon transform of a matrix
of identically non-zero values (Fig. 12),
where we can observe the same diamond shape which initially
prompted us to use and rather
than 
and 
.
Figure: FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY
We may overcome the problems associated with boundaries by defining a new version of the Radon transform, where we use the following pair of parameters:
: the leftmost ordinate
on the line (the ordinate for an abscissa of -1/2).
: the rightmost ordinate on
the line (the ordinate for an abscissa of +1/2).