Digital Signal Processing Reference
In-Depth Information
Fig. 15.5
Second version of
color monogenic signal of
imageusedinFig.
15.4
.Here
again, the color axis and
phase data are not displayed
(black or gray) for
coefficients with low
amplitude
The color monogenic model is defined as follows:
cos
arg
N
u
2
A
2
s
=
s
+
N
s
+
j
(15.27)
'axis'
ϕ
2
where
u
indicates a direction in the 3D color space and
ϕ
2
is the usual 1D
phase. The gradient norm
=
s/
s
is obtained with Eq. (
15.25
). Finally, the amplitude
and phase can be retrieved with the sole Euclidean norm of
s
. The new color mono-
genic signal is built like a 4-vector whose spherical coordinates are the amplitude,
phase, and color axis:
N
s
color
M
s
R
s
G
s
B
T
,
=[
N
]
2
2
Amplitude:
A
=
s
+
N
∈[
0
;+∞[
,
arg
N
∈[
(15.28)
π
1D Phase:
ϕ
2
=
s
+
j
0
;
2
[
,
Color axis:
u
=
s/
s
.
Let us observe Fig.
15.5
illustrating this color monogenic signal. Like previously,
the analysis is done on a subband of the color image obtained with the same filter.
We can see that the amplitude is again coherent with geometrical structures (includ-
ing isoluminant ones) and highlights oriented elements equally regardless of their
orientation—due to Riesz transform isotropy. Its invariance to shift and rotation is
due to the sharing of geometric information with
ϕ
2
which forms a coherent coding.
This also allows coding a line by a 'simple line' in amplitude instead of a 'double
line' thanks to the encoding of the
kind of discontinuity
by
ϕ
2
. Orientation
θ
is
+
exactly that of Fig.
15.4
;
u
carries some information of the local color direction.
Based on this extension of the monogenic signal, we can derive the corresponding
color extension of the MWT presented previously.
15.4.3 Tensor Based Color Monogenic Wavelet Transform
The extension to the wavelet domain is direct since the above construction relies
on a marginal Riesz transform (non-marginality occurs when combining marginal
outputs into meaningful data). So we can again directly use polyharmonic spline
wavelets of Unser et al. but with a different combination of Cartesian coefficients.
This time components will be combined to carry out color AM/FM analysis.
Search WWH ::
Custom Search