Environmental Engineering Reference
In-Depth Information
3
Calculate intensity
I
at the scale of
P
by utilizing the set of coef-
ficients
w
k
determined at step (2) by means of Equation 10.2.
From one side the MS image is up-sampled to the Pan
scale (
MS
k
) with a quasi-ideal interpolator to contribute to the
estimation of parameters and to generate the synthetic intensities
I
k
. Fromthe other hand, eachMS band is filteredwith itsMTF and
up-sampled to the Pan scale to be fed to the parameter estimation
procedure together with the low-pass MTF filtered
P
image. The
estimation procedure is thus performed at reduced resolution
(MTF filteredMS and Pan) and the parameters obtained are then
utilized at the finest resolution. For each MTF a different
I
k
is
generated. Details
d
k
are obtained for each band as
P-I
k
.Fusion
is afterwards obtained by adding
d
k
, multiplied by the injection
gains
g
k
, to each expanded
MS
k
band. The fusion equation, which
accounts for
N
The implicit assumption that is done in step (3) is that the
regression coefficients that are computed at the resolution of
the MS image are practically the same as those that would be
computed at the resolution of the original Pan image, if MS
observations were available at full spatial resolution. This is
equivalent to assume that the spectral responses of the dataset
are practically unaffected by the change of the spatial resolution.
The fused image is then given by:
=
(
P
−
I
)
g
k
+
MS
k
MS
k
(10.3)
1) optimized parameters, i.e.,
N
gains
g
k
and
N
2
weights
w
k
,
j
canbewrittenasfollows:
×
(
N
+
Also in this case the injection model is accomplished by the
weight image
g
k
that can be constant or varying pixel by pixel,
the injection model being global or local, respectively. The choice
adopted in (Aiazzi,
et al
., 2009) is to take
g
k
as the regression
coefficients
β
(
I
,
MS
k
). The scheme is denoted with GSA when a
global
model is adopted, while GSA-CA denotes the scheme with
the
local
model.
⎛
⎝
P
⎞
⎠
,
N
MS
k
w
k
,
j
MS
j
MS
k
=
+
g
k
−
k
=
1,
...
,
N.
j
=
1
(10.4)
By separating MS and
P
factors, the equation may be reformu-
lated as:
N
10.4
Hybrid
MRA
-
component
substitutionmethod
MS
k
=
MS
k
γ
k
,
j
MS
k
+
γ
k
,
N
+
1
P
,
k
=
1,
...
,
N
(10.5)
+
j
=
1
Where the complete set of
N
×
(
N
+
1) parameters is repre-
sented by
A Pan-sharpening algorithm for very-high resolution MS images
has been proposed by (Garzelli, Nencini and Capobianco, 2008),
which is optimal in the minimum mean squared error sense
and computationally practical, even when local optimization is
performed. This solution adopts an injection model in which a
detail image extracted from the panchromatic band is calculated
for each MS band by evaluating a band-dependent generalized
intensity
I
k
from the
N
multispectral bands. The block diagram
of the GMMSE (Global MMSE) fusion method is reported in
Fig. 10.3.
g
k
j
=
N
+
1
γ
k
,
j
=
(10.6)
−
g
k
w
k
,
j
j
=
1,
...
,
N
and finally, in the compact form using lexicographically ordered
column-wise images,
MS
k
MS
k
=
+
Hγ
k
k
=
1, 2,
...
,
N
(10.7)
,
γ
k
,
N
+
1
]
T
k
Where
γ
k
=
=
[
MS
1
,
MS
2
,
...
,
MS
N
,
P
], represents the observation matrix of
[
γ
k
,1
,
γ
k
,2
,
···
=
1, 2,
...N
,and
H
Band-dependent
MTF filtering
Upsampling
to Pan scale
MS
Upsampling
to Pan scale
Estimation of
N gains and
N
2
weights
w
k,j
Computation of
band-dependent
intensity
g
k
I
k
Pan
MTF
MS
F
d
k
+
−
FIGURE 10.3
Hybrid multiresolution-component substitution fusion scheme: spatial details
d
k
are extracted from the Pan
image and injected in the expanded MS images. Weights
w
k
,
j
and gains
g
k
are jointly estimated at reduced resolution.
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