Image Processing Reference

In-Depth Information

x
- and
y
-directions, respectively. The solution of such problem is typically given by

the Euler-Lagrange equation. We shall provide more details on the Euler-Lagrange

equation with reference to the calculus of variations in the next chapter. (Refer to

β
k
(

x

,

y

)

as given in Eq. (
5.9
).

1

β
(
m
+
1
)

k

)
=
β
(
m
)

k

β
(
m
)

k

(

x

,

y

(

x

,

y

)
−

(

x

,

y

)
−
(

I
k
(

x

,

y

))

,

(5.9)

2

λ
β

where

where the product of the

corresponding quality factor terms is too close to zero or one, may lie outside the

desired range of (0,1] when solved using Eq. (
5.9
). Ideally, this can be prohibited by

introducing a set of some explicit constraints to Eq. (
5.8
). However, this process con-

verts the problem of

(

m

)

is the iteration number. The values of

β
k
(

x

,

y

)

computation into a constrained optimization problemwhich is

computationally demanding. To speed up the computation process we refrain from

the use of explicit constraints. Experimentally we found that very few values of

the selectivity factors lie outside the range, and they have been clipped to the corre-

sponding extrema, i.e., 0 or 1. This enables us to solve the unconstrained optimization

problem without much degradation in the quality of the results.

The hyperspectral data also exhibit a moderate-to-high correlation along the spec-

tral array. Therefore, one may want to impose the smoothness constraint over the

spectral dimension (
k
) as well, and obtain the sensor selectivity factor

β

which

is globally smooth. This requires incorporation of additional terms to Eq. (
5.8
), which

are related to the derivative of

β
k
(

x

,

y

)

across the spectral dimension
k
. However, such a

process for computation of the sensor selectivity factor

β

will be very slow as com-

pared to the existing one, and thus, we refrain from the use of this additional derivative

term. It should, however, be noted that the penalty for departure from the smooth-

ness is chosen to be quite weak. The purpose of introducing this penalty is to address

the spatial relationship among the pixels, and prohibit arbitrary spatial variations

in

β

. However, one should be careful in selecting the penalty factor, and avoid a

strong penalty which can oversmooth the

β

β

-surface. A strong penalty factor is not

desired.

The aforementioned procedure generates a

-surface for every band in the hyper-

spectral data. Let us consider an original band of hyperspectral image as shown in

Fig.
5.1
a. This band depicts a small region of Palo Alto (CA, USA), constituting

the urban area. This is the 50-th band of the urban data by the Hyperion sensor.

Figure
5.1
b shows values of

β

computed using Eq. (
5.8
). These values, which are

formed by the product of two measures associated with the visual quality of the cor-

responding scene pixels, attempt to quantify the usefulness of the underlying pixel

towards the final result of fusion. It may be observed that higher values of

β

cor-

respond to the areas that are sharp and clearly visible in the image, while smaller

values of

β

β

have been obtained for dull, less informative areas.