Image Processing Reference
In-Depth Information
Fig. 10.28. Cartesian-separable Gabor decomposition where the × mark the filter centers and
the circles show the iso-amplitudes of the filters midway between the grid sites. The two axes
( dashed and solid ) illustrate, overlayed, the spectra of two linearly symmetric images
pair of Dirac pulses rotates on the same ring. Accordingly, it makes sense to attempt
to discretize the spectrum along a ring passing nearby the Dirac pulses. A log-polar
Gabor decomposition will do the sampling, but before sampling the spectrum along
the ring, the spectrum will be smeared. Thus, the Dirac pulses are smeared out in the
angular direction in an amount that corresponds to the size of the window for which
the Gabor filters are designed. In consequence, the more Gabor filters we use, the
more the structure tensor computed by Gabor decomposition will approach the one
discussed in Sect. 10.11.
In Fig. 10.30 we show on the left the quantity I 20 which is encoded by the hue,
and the brightness as obtained for the test image in Fig. 10.15. We used 18 directions,
i.e., the tune-on directions differed by increments of
π
18
. The band concentration of
the response is the same as before (cf. Fig. 10.25). This is to be expected because we
have not changed the radial widths of the log-polar frequency Gaussians. Instead we
see that the hue variation better follows the ideal hue variation shown in Fig. 10.15.
On the right in Fig. 10.30 we see I 11 modulating the brightness holding itself into
an annulus, and following the brightness of the image on the left,
, closely and
without oscillations. To assure ourselves, we also study the graph of these functions
along the same ring as before. This is shown in Fig. 10.31, where on the left we see
that the estimations follow the true directions nearly perfectly. On the right in the
first half of the graph, representing the clean part of the test image, we see that the
certainty,
|
I 20 |
, is nearly constant and follows I 11 apart from a minor bias. The small
bias is not surprising because even with 18 directions, the sampling density of the
spectrum afforded by the Gabor decomposition is inferior to what is available in the
original spectrum. The result is a smearing of the perfect energy concentration of
the Dirac pulse. The dilation of the concentration is noticed by the structure tensor
and is quantified as a small error in its attempt of fitting an ideal (infinitesimally
narrow) line. In the noisy part of the graph, the quality of the fit is lower as expected,
|
I 20 |
Search WWH ::




Custom Search