Image Processing Reference
InDepth Information
(
I
r

I
k
)
Theorem 4.1.
If the average conditional information H
follows an exponen

−

λ
R
,
tial function with respect to the spectral distance
r
k
with the rate parameter
1
then the average saving in the computation is given by
S
=
.
λ
R
1
−
ln
(
1
−
κ)
Proof:
Let the conditional information be modeled as,
1
e
−
λ
R
(
r
−
k
)
H
(
I
r

I
k
)
=
H
(
I
r
)
−
,
r
=
k
,
k
+
1
,
k
+
2
,...
(4.5)
To obtain the computational savings, we use the expressions from Eqs. (
4.3
)(
4.4
)
of the band selection process.
e
−
λ
R
(
r
−
k
)
)
≥
κ
(
I
r

I
k
)
=
(
I
r
)(
−
(
I
r
),
H
H
1
H
e
−
λ
R
(
r
−
k
)
≥
κ
or,
1
−
1
λ
R
1
or,
(
r
−
k
)
≥
ln
−
κ
.
(4.6)
1
Thus, a band is selected if its spectral distance from the reference band exceeds the
RHS of Eq. (
4.6
). As the spectral distance increases, one selects lesser number of
bands, resulting in higher savings in computation. When a new band is selected after
discarding (
r
−
S
k
) bands, the fractional savings (
) in computation can be calculated
(
r
−
k
)
S
=
κ
as
. In terms of the threshold
, we can write,
(
r
−
k
+
1
)
1
λ
R
1
ln
−
κ
1
S
=
1
λ
1
,
1
ln
+
1
−
κ
R
1
S
=
.
(4.7)
λ
R
1
−
ln
(
1
−
κ)
The value of
λ
R
is dependent on the statistics of the spectrally ordered hyperspectral
data. A higher value of
λ
R
implies a highly decreasing nature of the conditional
information. Thus, the computational saving is directly proportional to the rate para
meter. Also, for very small values of
κ
, there is practically no saving in computation,
while higher values of
lead to high values of the denominator of the fractional term
in the expression, indicating a high amount of saving at the cost of selecting a very
few image bands, thus sacrificing in the quality of the fusion results. As the extreme
case, when
κ
κ
is set to zero, the amount of savings equals zero as the system always
selects every possible band in the data. On the other hand, the savings turn out to be
1 (i.e., 100%) when
equals one when no other band gets selected.
The expression in Eq. (
4.7
) gives the theoretical upper bound on the computational
savings. This expression, however, does not consider the processing overhead of the
calculation of the conditional information. For most of the fusion techniques that
operate on a per pixel basis, the amount of processing for the computation of entropy
κ