where is the orientation,
s
controls the spread about that orientation and
l
is the angle is
local orientation focus. The other spreading function is a band-pass filter, here a log-Gabor filter
lg with M different scales.
lg
m
=
⎧
⎪
⎨
⎪
⎩
0 =0
1
√
2
e
−
log
m
2
2
log
2
=0
(4.37)
where is the scale, controls bandwidth at that scale and
m
is the centre frequency at that
scale. The combination of these functions provides a 2D filter l2Dg which can act at different
scales and orientations.
l2Dg
m
l
=g
l
×lg
m
(4.38)
One measure of phase congruency based on the convolution of this filter with the image P
is derived by inverse Fourier transformation
−1
of the filter l2Dg (to yield a spatial domain
operator) which is convolved as
S
m
xy
=
−1
l2Dg
m
l
xy
∗P
xy
(4.39)
to deliver the convolution result S at the mth scale. The measure of phase congruency over the
M scales is then
PC
xy
=
M
m=1
S
m
xy
M
m=1
S
m
xy
+
(4.40)
where the addition of a small factor numerator again avoids division by zero and any potential
result when values of S are very small. This gives a measure of phase congruency, but is
certainly a bit of an ouch, especially as it still needs refinement.
Note that keywords recur within phase congruency: frequency domain, wavelets and con-
volution. By its nature, we are operating in the frequency domain and there is not enough
room in this text, and it is inappropriate to the scope here, to expand further. Despite this,
the performance of phase congruency certainly encourages its consideration, especially if local
illumination is likely to vary and if a range of features is to be considered. It is derived by
an alternative conceptual basis, and this gives different insight, as well as performance. Even
better, there is a Matlab implementation available, for application to images, allowing you to
replicate its excellent results. There has been further research, noting especially its extension in
ultrasound image analysis (Mulet-Parada and Noble, 2000) and its extension to spatiotemporal
form (Myerscough and Nixon, 2004).
4.8 Localized feature extraction
Two main areas are covered here. The traditional approaches aim to derive local features by
measuring specific image properties. The main target has been to estimate curvature: peaks
of local curvature are corners, and analysing an image by its corners is especially suited to
images of artificial objects. The second area includes more modern approaches that improve
performance by using region or patch-based analysis. We shall start with the more established
curvature-based operators, before moving to the patch or region-based analysis.
152 Feature Extraction and Image Processing
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