Abstract
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 (\(\beta \)) 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 \(\alpha \) 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 function 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 necessarily 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, we have 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.