Bayesian Estimation - Hyperspectral Image Fusion - page 81

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
Chap. 2 . Accordingly, the array of spectral observations at

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