Image Processing Reference
In-Depth Information
3
1
2
0.8
1
0.6
0
0.4
−1
−2
0.2
−3
0
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Fig. 10.18. The graphs in the
left image
represent the estimated (arg(
I
20
)) as well as the ideal
direction angle on the two radial lines shown in the FM test image. The graphs on the
right
represent
|I
20
|
and
I
11
on the same lines
filter with variance
σ
p
is applied. The result is squared pointwise and smoothed by
the Gaussian filter with the variance
σ
p
+
σ
w
. Despite the nonlinear squaring between
the two linear operators, the combined operator exhibits a bandpass character that is
tuned to a particular frequency. This is seen in the right graph of Fig. 10.18, where
the
tune-on frequency
, which the structure tensor is most sensitive to, has a well-
distinguished peak in the solid blue curve. The tune-on frequency of this particular
implementation at 0.62 radians, corresponding to a sinusoidal wave with a period
of 10 pixels, has been shown in dotted black in both graphs. For convenience, even
the points 1 through 4 defining the sampling ring above correspond to 0.62 radians.
The tune-on frequency can be changed in a variety of ways, a straightforward one
of which is by changing the derivative Gaussian variance
σ
p
. The arrows show the
limits of the frequency annulus to which the filters are most sensitive, as confirmed
by the graphs of
|
I
20
|
and
I
11
.
10.12 Application Examples
We present two applications to illustrate the use of the structure tensor.
Fingerprint Recognition
In Fig. 10.19, a
minutia
detection process is visualized. A minutia in a fingerprint
is a point that can be easily identified. Typically it is an end point of a ridge or a
bifurcation point. The first two images visualize the original fingerprint and its en-
hanced version, respectively. Image III represents
I
20
/I
11
, where the numerator and
denominator at each point are obtained by direct tensor discretization as discussed
above. Even here the minutiae are discernable as dark spots, indicating lack of linear
symmetry. The hue represents local direction. Image IV represents the presence of
linear symmety in parabolic coordinates, a subject that we will discuss in detail in
