Digital Signal Processing Reference
In-Depth Information
(a)
(b)
Log Counter−Harmonic Mean P=5,10,20
(sup in red, in green P=0)
Set of matrices (in blue)
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−2.5
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
Fig. 1.5
a
Set
A
of
N
=
3PDS
(
2
)
matrices.
b
Counter-harmonic matrix Log-Euclidean mean for
various positive values of
P
(a)
Counter−Harmonic Mean P=2,5,10
(sup in red, inf in mag)
(b)
Log Counter−Harmonic Mean P=2,5,10
(sup in red, in green P=0)
3
2.5
2
2
1.5
1
1
0.5
0
0
−0.5
−1
−1
−1.5
−2
−2
−2.5
−3
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
Fig. 1.6
Given a set
A
of
N
=
10 PDS
(
2
)
matrices:
a
counter-harmonic matrix mean for various
values of
P
(red color for
positive value
of
P
, magenta color for
negative values
of
P
);
b
counter-
harmonic matrix Log-Euclidean mean for various positive values of
P
1.5 Application to Nonlinear Filtering of Matrix-Valued
Images
N
i
The different strategies of supremum and infimum of a set
1
discussed in
this study can straightforward be used to compute morphological operators on matrix-
valued images. Hence, let
f
A ={
A
i
}
=
be a matrix-valued image to be
processed. Figure
1.7
a gives an example of such image for
n
(
x
)
∈
F(
E
,
PDS
(
n
))
=
2. This visualization
of PDS
images uses the functions developed by G. Peyré [
32
]. We notice that,
in order to make easier the representation of their “shape”, all the ellipses have a
normalized “size”; in fact, the original size given roughly by
λ
1
+
λ
2
is coded by
their color using the cooper color map (which varies smoothly from black to bright
copper). Figure
1.7
b, c depicts precisely the images of
S
1
=
λ
1
+
λ
2
and
λ
1
/λ
2
.This
(
2
)
