Environmental Engineering Reference
In-Depth Information
subtracting a synthetic intensity image to the Pan image and
are thus produced without filtering the Pan image. Filtering can
occur for estimating weights
w
k
but not for producing details. In
the scheme, the original MS bands are preliminarily expanded to
the same spatial scale of the full-resolution Pan image
P
to obtain
the data set
MS
k
.
Weights
w
k
are fixed or can be computed as a function of
P
and MS
k
, as in Fig. 10.2, to generate the synthetic intensity
I
component:
applied adaptively as reported in (Aiazzi
et al
., 2009) originating
the context adaptive version (GSA-CA). As observed in (Otazu
et al
., 2005), the radiometric values of vegetated areas are likely
to be much smaller in the synthetic intensity
I
image than
in the true Pan. This effect causes the injection in the fused
MS bands of a radiance offset, which may give rise to color
distortion. In order to avoid this drawback, the idea reported in
(Alparone
et al
., 2004a) is to generate the synthetic intensity in
such a way that the spectral response of the sensor is considered,
by differently weighting the contributions coming from the
MS spectral channels. Conventional methods, however, only
consider the nominal spectral responses when available (Tu
et al
., 2004). Actually, the influence of other phenomena, such as
on-orbit working conditions, variability of the observed scene,
and post processing effects, can significantly modify the nominal
spectral response. In particular, atmospheric influence depends
on the viewing angle since the scattering effect is related to
the wavelength and on atmospheric conditions. The solution
proposed in (Aiazzi, Baronti and Selva 2007) is to perform a
linear regression between the low-pass (MTF) filtered Pan and
the MS bands by computing a synthetic intensity
I
that has a
minimum mean-square error (MMSE) with respect to the Pan.
This solution allows the histogram matching to be skipped since
the similarity of
I
and
P
is guaranteed by the MMSE criterion.
The steps of the procedure when working at the PAN scale are as
follows.
1
Obtain the
MS
k
images, by expanding the MS image to
the scale of
P
image and reduce the resolution of
P
image
at the same spatial scale by means of a low-pass filter that
should ideally represent an MTF capable to produce a
P
L
image with spatial frequency components similar to those of
the MS image.
2
Assume that the
I
component is given as in Equation 10.2
and estimate the set of coefficients
w
k
by means of a linear
regression algorithm (Ross 2004) in order to minimize the
MSE between
P
L
and
I
.
N
w
k
MS
k
,
I
=
(10.2)
k
=
1
Afterward,
I
is subtracted from
P
to produce a detail image
d
, i.e.,
d
I
. A second set of coefficients
g
k
is adopted to
modulate the detail image
d
before its addition to the expanded
data set
MS
k
in order to yield the final fused multispectral images
MS
k
. A histogram matching is usually performed on
P
to match
its mean and standard deviation to those of the synthesized
I
component. This procedure has the objective of reducing the
spectral distortions that are caused by the spectral mismatch
between
I
and
P
images. The scheme described in Fig. 10.2 is
general and agrees with the derivation presented by (Tu
et al
.,
2004) in which it is demonstrated that the substitution of
I
with
P
and the successive inverse transform is equivalent to
add
d
=
P
−
I
to the expanded
MS
k
data set. Depending on
the values of the two sets of weights
w
k
and
g
k
, the scheme is
suitable for describing any CS-based fusion algorithm (Aiazzi,
Baronti and Selva 2007). In particular, the GS algorithm for
which results are reported in the following is characterized
by input weights
w
k
all equal to 1/4 and output weights
g
k
given by the regression coefficients
β
(
I
,
MS
k
), expressed by the
covariance of
I
and
MS
k
, normalized by the variance of
I
.
The efficient CS algorithm we briefly review in this section is
the Gram-Schmidt (GS) adaptive (GSA) algorithm reported in
(Aiazzi, Baronti and Selva 2007). The algorithm can be also
=
−
P
MS
Upsampling
to Pan scale
w
k
Estimation of
N weights
Computation of
intensity
Pan
MTF
I
MS
F
d
+
−
g
k
Computation
of N gains
FIGURE 10.2
Component substitution fusion scheme: spatial details
d
are computed by subtracting the intensity component
I
from the Pan image and injected in the expanded MS images. Images
g
k
rule the injection gain. Weights
w
k
are estimated
in order to minimize the difference between Pan and
I
in order to improve the quality of the fused images.
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