Image Processing Reference
pixel in the image band has no correlation with the true scene, and hence, it can be
regarded as noise.
The problem of fusion is now equivalent to the estimation of parameters of the
image formation model. Ideally, one can compute these parameters from the pre-
cise information about the sensory system. However, this information is not always
available. Also, the method does not remain generic after incorporation of the device-
specific information. Therefore, let us assume no knowledge about the sensor device,
and address this problem blindly using only the available hyperspectral data. The
parameters of the image formation model should be computed from certain prop-
erties of the data. The model, however, being a first order approximation does not
reflect a perfect relationship between the input and the output images. The modeling
error may lead to an erroneous resultant image. Considering this as random perturba-
tions and to deal with these perturbations in an efficient and systematic manner, we
formulate the estimation problem in a Bayesian framework. The solution is obtained
through the maximum a posteriori estimate of the corresponding formulation.
In the next section we explain in brief the Bayesian framework and some of its
applications in vision and image processing. We describe the image formation model
developed by Sharma et al.  in Sect. 5.3 . The computation of the parameters of
this model is explained in Sect. 5.4 . The Bayesian solution and its implementation
details are provided in Sects. 5.5 and 5.6 , respectively. Some illustrative results are
provided in Sect. 5.7 , while Sect. 5.8 summarizes the chapter.
5.2 Bayesian Framework
We often come across with problems in various fields of engineering where the data
available to us are either incomplete or imperfect. The data, which might be available
in the formof a set of observations from some system, may contain additional compo-
nents, unwanted for the given application. When one wants to make some inferences
or calculate some parameters from such data, results are likely to be erroneous due to
the unwanted component, known as noise. Alternatively, one may not have an exact
model of the system under investigation, which can also lead to imprecise solutions.
Thus, the uncertainty or the incompleteness of the data or model are primary sources
of errors in the inference.
Bayesian estimation is a probabilistic framework that has been extensively used
for reasoning and decision making under uncertainty. It provides a systematic way of
approaching the solution when the knowledge about the system is incomplete. This
framework takes into account the posterior probability related to a specific event
pertaining to the quantity of interest. A posterior probability refers to the condi-
tional probability assigned after the relevant event has been taken into consideration.
Bayesian estimator provides a solution by maximizing the posterior probability. Due
to incomplete knowledge, one may not be able to obtain the exact solution, but can
obtain a solution that is close to the desired one. The process of Bayesian estimation
can be illustrated as follows. Let
be some underlying variable of interest while we