Variational Solution - Hyperspectral Image Fusion - page 89

Chapter 6
Variational Solution

6.1 Introduction

We have studied pixel-based image fusion as a linear combination of multiple input
images. The weights, or more precisely fusion weights, are the data-dependent terms
as they have been calculated from the set of input images (or input hyperspectral bands
in our case). For example, the bilateral filtering-based fusion technique calculates
the fusion weights ( w ) using a predefined function. The Bayesian fusion technique
is based on the computation of the sensor selectivity factor ( ) which indicates the

contribution of each pixel toward the fused image. Both of these fusion techniques
explicitly calculate the weights as a function of the input hyperspectral data. These
functions are also referred to as the weighting functions, while the weights are more
commonly known as the

β

matte in the graphics literature. The fusion weights act as
intermediate variables of the fusion process that define the relationship between the
fused image and the input hyperspectral bands. The purpose of the weighting func-
tion which generates the fusion weights, is to implicitly specify the model for the
fusion process. An explicit computation of fusion weights is, therefore, not required
so long as the underlying model is well specified. In other words, we do not nec-
essarily have to compute the fusion weights independently, if we can appropriately
model the weighting function as a data-dependent term to weigh the hyperspectral
bands. We now explore this possibility, and show how a fusion technique can be
developed without any explicit calculation of fusion mattes. In Chap. 5 , wehave
seen that imposing some constraints through a prior related to the smoothness of the
output fused image, gives it a natural and visually pleasing appearance. In order to
incorporate the smoothness constraint, we adopt an approach based on calculus of
variations. This chapter discusses how we can start with an initial estimate of the
fused image, and iteratively converge to obtain the desired resultant image based
on certain constraints as well as the predefined weighting function, without ever
explicitly computing the weights.
In case some of the readers are not familiar, we discuss in very brief about the
calculus of variations in Sect. 6.2 , which will be used as a tool to address the fusion

α