Image Processing Reference
In-Depth Information
=
k
I
l
11
z
kl
|
2
|
F
{k,l}
|
2
|
(10.89)
In fact, the quantities
I
l
20
and
I
l
11
are equal to
I
20
and
I
11
, respectively, if the
image is bandpass filtered in such a way that the absolute frequency contents dif-
fering from
are suppressed. Accordingly,
I
l
20
and
I
l
11
represent the direction
tensor of a specific (absolute) frequency (which is a ring with a “small” width in the
spectrum) of the image. Apart from requiring less computational resources, restrict-
ing the Gabor filter-based direction tensor to a specific absolute frequency is also
desirable in many situations where multi-scale analysis is needed, e.g., texture and
fingerprint image analysis, since this allows one to split the analysis over an array of
scales naturally. Originally suggested in [137], via a scheme that is a special case of
Eq. (10.84) applied to a frequency ring, such an orientation interpolation is sufficient
for the purpose of estimating the local direction for many applications. The original
scheme employed quadrature mirror filters with the angular bandwidth of
π
i.e., the
angular deviation from the tune-on direction was measured by the function cos
2
(
θ
)
as compared to the Gaussians discussed here. Regardless of the filters shape
5
|
z
kl
|
and
how they tessellate the spectrum, it should be emphasized that
I
l
20
must be com-
pleted with
I
l
11
to make it a direction tensor. Otherwise
I
20
alone cannot encode both
the minimum and the maximum error, apart from their difference.
Although the interpolated direction offers better direction accuracy, there is ob-
viously a limit on the minimum number of filters one can have in the decomposition.
To begin with, it is not possible to compute
I
l
20
, which consists of two real variables,
and
I
l
11
, which consists of one nonnegative real variable, from just one Gabor filter,
(actually two filters when mirrored). The
I
l
20
would point in the same direction, re-
gardless of the direction of the image. Even two Gabor filter directions differing with
π
2
(actually 4 filter sites when mirrored) are not enough because this would result in
an
I
l
20
that could not possibly point in other than the two directions: 2
ϕ
and 2
ϕ
+
π
where
ϕ
is the direction of one of the two filters. Using three filters with directions
separated by
π
3
is the minimum requirement on the Gabor decomposition to yield a
meaningful direction tensor.
In Fig. 10.24, this is illustrated by investigating the Gabor filter responses to
an image composed of a planar sinusoid. The intersection point of three circles is
marked with a small circle, and it represents the frequency coordinates of one of
the two Dirac impulses composing the sinusoid image. The three circles show the
frequency coordinates (the direction and absolute frequencies) at which the Dirac
impulse can be placed, judging from the individual Gabor filter magnitudes. Provided
that it is on such a ring, the magnitude response of the same Gabor filter is invariant
to the position changes of the Dirac pulse. Only by investigating no less than three
Gabor filter magnitudes, is it possible to uniquely determine the position of a Dirac
impulse. This result is nonetheless not surprising, because there are three freedoms
in a 2D direction tensor and these cannot be fixed by less than three independent
measuremen
ts.
5
Rotation invariant versions of quadrature filters have also been suggested [71].
