Image Processing Reference
In-Depth Information
high amount of additional information than the presently available fused image. The
threshold, as before, is set to an appropriate fraction of the entropy of the band under
consideration. Once the band is selected, a new fused image is formed by combining
all the selected bands using the fusion technique
. Subsequently, for every addi-
tional image band, the conditional entropy with respect to the corresponding fused
image obtained by combining all of the previously selected bands is calculated.
Thus, given a set of hyperspectral images
F
{
I
k
;
k
=
1
,
2
,...,
K
}
,the
p
-th image
band selected for fusion is given by Eq. (
8.1
).
H
F
r
−
1
)
≥
θ
,
p
=
arg min
r
(
I
r
|
(8.1)
where
H
represents the entropy of the image
I
r
conditioned on the fused
image
F
r
−
1
, obtained from all of the already selected bands up to
I
r
−
1
using the same
fusion technique
(
I
r
|
F
r
−
1
)
. It should be noted that the actual number of bands undergoing
fusion are much less than
F
θ
(
r
−
1
)
. The threshold
is chosen again as a suitable
fraction
κ
of
H
(
I
r
)
,
θ
=
κ
H
(
I
r
),
0
<κ<
1
.
(8.2)
This procedure is continued until the entire dataset is exhausted. This scheme exploits
the redundancy in the input data with respect to the intermediate output, as opposed
to purely input-based band selection. Hence this is an example of
fusion process-in-
the-loop
technique for band selection. The resultant fused image contains most of
the features of the entire data as it rejects an image band when it is highly similar to
the existing output at the corresponding fusion stage.
The methodology of the selection of the bands based on the output is thus directly
dependent upon the chosen fusion technique. Several pixel-based techniques for
fusion of hyperspectral images have been proposed [71, 79, 88, 189]. Different fusion
techniques measure the saliency of the pixels in different ways, and assign suitable
weights to them. The resultant fused images are generated by a linear combination
of the pixels across all the bands. During the process of fusion, different fusion rules
introduce different amounts of loss and noise, and thus, an
apriori
estimation of the
entropy of the output is a difficult task. Further, due to the dependencies on the fusion
technique, it is difficult to device a generalized model for
H
and hence to
estimate the savings in computation as was done for the input-based band selection.
Similar to the input-based selection of bands, this type of output-based selection is
also based on the greedy technique as one cannot undo the selection of any band.
At this point one may point out that ideally one should fuse a candidate band
first and then we should compute the conditional entropy of the newly fused image
with respect to the previous fusion result, and if this measure exceeds a threshold
then the candidate band should be selected. While this sounds logical, this is of no
practical value as the majority of the computational requirement is consumed by the
fusion process and not the computation of conditional entropy. Thus, if the fusion
(
I
r
|
F
r
−
1
)
