Image Processing Reference
In-Depth Information
band may not be well correlated with the last of the selected subset of bands, but
it may be well correlated with one of the bands selected earlier. In either case, the
newly available band should be considered as redundant for fusion purposes. The
band is selected for fusion if it has a conditional entropy higher than the threshold.
Thus, given a set of hyperspectral images
{
I
k
;
k
=
1
,
2
,...,
K
}
,the
p
-th image
band selected for fusion is given by Eq. (
4.2
).
min
k
k
I
r
|
I
k
)
≥
Θ,
p
=
argmin
r
H
(
r
>
,
(4.2)
I
k
represents the subset of selected bands up to the
k
-th available band in
I
,
where
I
r
|
I
k
)
represents the entropy of the image
I
r
conditioned on each of the element
of the set
I
.
The threshold
H
(
Θ
(
I
r
)
is empirically chosen as a suitable fraction of
H
, i.e.,
Θ
=
κ
H
(
I
r
),
0
<κ<
1
.
(4.3)
This procedure is continued until the entire dataset is exhausted. Any pixel based
fusion scheme can then operate over this selected subset
I
to generate an appropriately
fused image.
The aforementioned scheme exploits the statistical redundancy in the input data.
It discards bands that are very similar to the selected ones, and selects the ones that
are quite different in terms of additional information content. Therefore, although a
fewer images are selected, most of the information content in the data is captured
by the band selection process. The resultant fused image, thus, contains most of the
features of the entire dataset.
This scheme essentially selects the image bands by evaluating the additional
information content against all the previously selected bands. This scheme does not
take into consideration any possible pattern of the correlation among the bands.
Therefore, the band selection scheme is guaranteed to yield good results for any
organization of input data consisting of multiple images of the same scene to be fused.
However, a band selected once cannot be removed from the subset
I
throughout the
process; thus this scheme is based on the greedy selection of image bands.
4.3 Special Case: Ordered Data
Consider a typical problem of hyperspectral image data to be fused for efficient
visualization. The hyperspectral data consist of a set of narrow-width, but spectrally
contiguous bands acquired by an array of sensors. As the bands are ordered sequen-
tially with respect to the wavelength, the data content in the bands varies gradually
as the distance between two bands grows. Therefore, when the bands in the data
are ordered according to their wavelengths, the amount of correlation is usually a
decreasing function of the difference in their corresponding wavelength, and hence,
