Performance Assessment of Fusion Techniques - Hyperspectral Image Fusion - page 126

( × × )

as a set of K
different bands, then we can form K subsets consisting of incrementally increasing
number of bands such that the k -th subset contains hyperspectral bands up to I k ,
i.e.,

If we consider a hyperspectral image of dimensions X Y K

{ I 1 , I 2 ,..., I k } . Each of the subsets can be fused using some fusion technique

F to produce what we refer to as the incrementally fused image. Let F k be the

incrementally fused image from the subset { I 1 , I 2 ,..., I k } as given by Eq. ( 9.1 ).

F k ( x , y ) = F ( I 1 , I 2 ,..., I k ) k = 1to K . (9.1)

This procedure generates a set of such K incrementally fused images { F k , k =

1to K . We want to analyze these incrementally fused images F k 's in terms of some

of the objective quality measures.
We define the consistency of a fusion technique as an asymptotic behavior while
dealing with an increasing number of images with respect to a given performance
measure. Formally we define the consistency of a fusion technique as follows:
[92]

}

Definition 9.1 Given a set of hyperspectral bands { I k ; k = 1 , 2 ,..., K } ,let

{ F k ; k = 1 , 2 ,..., K } represent a set of incrementally fused images given by

F k = F ( I 1 , I 2 ,..., I k )

F

using a fusion technique . Then for a given performance

( ) ≡ (

F k ),

=

measure g , the sequence g 1to K , represents an incremental

behavior of the given fusion technique. We define the technique to be consistent
with respect to the given measure g , if the sequence g

k g k

( k ) possesses the following

properties.

Property 9.1 The sequence is monotonic. An addition of

a new band during fusion must necessarily add more information to the resultant
image.

{ g ( k ) ; k = 1 , 2 ,..., K }

Property 9.2 The sequence { g ( k ) ; k = 1 , 2 ,..., K } is a bounded sequence. This

value must be finite as K . e.g., for entropy, the upper bound on the value is

defined by the bitdepth of an image band.

→∞

{ ( ) }

From Properties ( 9.1 ) and ( 9.2 ) , we can deduce that g k is a Cauchy sequence ,

ε>

i.e., given 0, however small, there are sufficiently large positive numbers m , n ,

such that . As one starts adding hyperspectral bands to the process

of fusion, the initial few ones contribute significantly and build up the process. How-
ever, as more bands get fused, their contribution gradually decreases, and the corre-
sponding incrementally fused images do not differ much. In other words, the property
of the sequence getting progressively closer indicates that the additive contribution
of individual image bands to the fusion process gradually becomes insignificant as
more image bands are being added.
As we have already explained, there exists a strong spatial correlation among
the consecutive bands of the hyperspectral image. This inter-band correlation leads
to a large amount of redundancy in the data. We have also explored a couple of
schemes for selection of a subset of few, but specific bands which capture most of the

| g ( m ) − g ( n ) | <ε