Image Processing Reference

In-Depth Information

Fig. 5.1
Illustration of the

computation of sensor selec-

tivity factors (
β
)fortheurban

image from the Hyperion.
a

Original image is the 50-th

band in the urban data, and

b
computed

(a)

(b)

β

image for the

same band

5.5 Bayesian Solution

We would like to estimate the true scene, i.e., the fused image
F
from the noisy

observations (bands). These observations
I
k
, are related to the fused image
F
through

a first order model. A pixel in the fused image needs to be estimated from the

array of observations across all hyperspectral bands at the same spatial location,

(

. Therefore, during this process, we mostly deal with the 1-D spectral data at a

given location in a 2-D spatial plane. Therefore, instead of working with 2-D bands

I
k
,

x

,

y

)

K
, we work with spectral array notation as explained earlier in

k

=

1

,

2

,
···
,

(

x

,

y

)

shall be referred to

K
. The observation in the
k
-th band, i.e., the
k
-th element

of this vector is denoted by
s
k
(

as
s

(

x

,

y

)

such that
s

∈ R

. It should, however, be noted that the change

of notation is purely meant for ease of understanding, and to avoid any possible

confusion when the same data are being referred from different dimensions.

The maximum likelihood (ML) solution is a commonly preferred approach for

estimation from a noisy data as discussed earlier. The ML solution produces an

output by minimizing the quadratic norm of the error between the estimated fused

pixel and the corresponding quantity related to input pixel scaled by

x

,

y

)

when the

noise is assumed to be Gaussian. The ML solution estimates each pixel of the fused

image
F
independent of the neighborhood pixels, and thus, the spatial correlation

within pixels of the output image has not been taken into consideration. Most of

the natural images are spatially smooth, except at the edges. This moderate smooth-

ness is one of the important characteristics of natural images. As the ML solution

operates on a per pixel basis, it may produce an image that lacks the properties

β