Image Processing Reference
In-Depth Information
been derived from certain measure related to the local contrast. A similar approach
has been employed for fusion of multi-exposure images to form a high dynamic
range (HDR)-like image in [145]. The basic formulation of the fusion problem using
a variational framework is provided in Eq. (
6.6
).
⎧
⎨
F
2
K
F
(
x
,
y
)
=
argmin
F
(
x
,
y
)
−
w
k
(
x
,
y
)
I
k
(
x
,
y
)
⎩
x
y
k
=
1
+
λ
var
∂
d
x
d
y
2
2
F
(
x
,
y
)
+
∂
F
(
x
,
y
)
,
(6.6)
∂
x
∂
y
where
λ
var
is the regularization parameter, and
F
is the fused image. This parameter
is set by the user to obtain a desired balance between the weightage assigned to the
matte (first term in Eq. (
6.6
)), and the weightage assigned to the smoothness of the
fused image. The second term in the above Equation is known as the regularization
term. In the present case of hyperspectral image fusion, we want the input bands
to merge smoothly without producing any visible artifacts in the fused image. As
most natural images have a moderate to high amount of spatial correlation, any
sharp and arbitrarily discontinuous region generated via fusion may turn out to be
a visible artifact. In order to avoid such artifacts, we would like the fused image
to have a certain degree of smoothness. One of the ways of accomplishing this
objective is to penalize the departure of the fused image from smoothness which
can be characterized by high values of its spatial gradient. We incorporate this in
the form of a regularizing term, and seek for the minimization of the corresponding
functional defined in Eq. (
6.3
). It may, however, be noted that several other types of
regularization or penalty functions have been proposed in the literature. The choice
of the regularization function is determined by nature of the data, and is of utmost
importance for obtaining the desired output.
We have the basic formulation of our fusion problemas given byEq. (
6.6
). Now, we
shall focus on designing an appropriate function for the calculation of fusion weights.
Pixels with higher local contrast bring out a high amount of visual information.
We quantify the local contrast of a given pixel by evaluating the variance in its
spatial neighborhood. Thus, fusion weights
w
k
(
,
)
for the
k
-th hyperspectral band
should be directly proportional to the local variance
x
y
2
σ
k
(
,
)
. However, in case
of homogeneous regions in hyperspectral bands, the variance becomes zero, and
such regions may not get an appropriate representation in the final fused image.
To alleviate this problem, we incorporate a positive valued additive constant
C
to
the weighting function. The constant
C
serves the similar purposes as the constant
from the contrast, we also want to balance the saturation level in the resultant fused
image. The fused image should have very little under- or over-saturated regions. It
is also desired that the fused image be quite close to the mean observation along
the band for radiometric consideration. Hence, we consider the weight
w
k
(
x
y
x
,
y
)
to
2
, with the proportionality constant being
be proportional to
(
F
(
x
,
y
)
−
I
k
(
x
,
y
))
