It may seem that we can also see the resolution length distribution in the
Fourier spatial frequency spectrum of an image. But even a seemingly
simple image
like our first example, the checkers image, which has only a few components,
the checkerboard, the checker pieces, and a few lines on the floor,
has a complicated spectrum, as shown in Fig. 9 (left).
Automatically
finding the relevant scales is not easy. Although our eye may
(given we know what to look for) catch the basic frequency and the higher
harmonics of the tiles by looking at the full spectrum, once we sum
up all the contributions of a particular frequency
(Fig. 9 right)
we lose them
. So even in this extremely simple image, using the
spectrum, a human can just barely find the characteristic frequencies.
We claim that an algorithm to find relevant frequencies operating on either
the 2D or the radially summed up spectrum would have to be significantly
more complex than the one we use to find the information measure presented
in this paper. We tried many other ways of using the Fourier spectrum
in an information decomposition, but we found none nearly as good
as the Kullback contrast.
Figure 9: Spatial frequency spectrum plot of the checkerboard image
(left) and its radial sum (right).
The computational costs of the power spectrum method and the algorithm
for our information
measure are comparable, both being
algorithms. In a
non-optimized computer
implementation on a one-processor Sparc station, both take around 20 seconds
to process a size
image. Implementation on image processing
hardware should allow real-time performance.