IEEE Geoscience and Remote Sensing Magazine - September 2015 - 30
separate spatial and spectral information [6]. Subsequently, the transformed data are sharpened by substituting the
component that contains the spatial information with the
PAN image (or part of it). The greater the correlation between the PAN image and the replaced component, the
less spectral distortion will be introduced by the fusion
approach [6]. As a consequence, a histogram-matching
procedure is often performed before replacing the PAN image. Finally, the CS-based fusion process is completed by
applying the inverse spectral transformation to obtain the
fused image.
The main advantages of the CS-based fusion techniques
are the following: i) high fidelity in rendering the spatial
details in the final image [37], ii) fast and easy implementation [8], and iii) robustness to misregistration errors and
aliasing [38]. On the negative side, the main shortcoming
of this class of techniques is the generation of a significant
spectral distortion, cause by the spectral mismatch between
the PAN and the HS spectral ranges [6].
Following [4], [39], a formulation of the CS fusion
scheme is given by
k
k
W
X =L
Y H + g k ^P - O Lh,
(1)
X denotes the kth band of the esfor k = 1, f, m m, where W
timated full resolution target image, g = 6g 1, f, g mm@T is a
vector containing the injection gains, and O L is defined as
k
OL = / wi L
Y H,
mm
i =1
i
(2)
where the weights w = 6w 1, f, w i, f, w mm@T measure the
spectral overlap among the spectral bands and the PAN image [6], [40].
The CS family includes many popular pansharpening
approaches. In [26], three approaches based on principal
component analysis (PCA) [9] and Gram-Schmidt [13], [37]
transformations have been compared for sharpening HS
data. A brief description of these techniques follows.
1) Principal Component Analysis: PCA is a spectral transformation widely employed for pansharpening applications [9]. It is achieved through a rotation of the original
data (i.e., a linear transformation) that yields the so-called
principal components (PCs). The hypothesis underlying
its application to pansharpening is that the spatial information (shared by all the channels) is concentrated in the
first PC, while the spectral information (specific to each
single band) is accounted for the other PCs. The whole fusion process can be described by the general formulation
stated by Eqs. (1) and (2), where the vectors w and g of
coefficient vectors are derived by the PCA procedure applied to the HS image.
2) Gram-Schmidt: The Gram-Schmidt transformation, often exploited in pansharpening approaches, was initially
proposed in a patent by Kodak [13]. The fusion process
starts by using, as the component, a synthetic low resolution PAN image I L at the same spatial resolution as the HS
30
image2. A complete orthogonal decomposition is then performed, starting with that component. The pansharpening
procedure is completed by substituting that component
with the PAN image, and inverting the decomposition.
This process is expressed by (1) using the gains [37]
cov ^L
Y H, O Lh
var ^O Lh
k
gk =
(3)
for k = 1, f, m m, where cov ^$ , $ h and var ^ $ h denote the covariance and variance operations. Different algorithms are
obtained by changing the definition of the weights in (2).
The simplest way to obtain this low-resolution PAN image
simply consists of averaging the HS bands (i.e., by setting
w i = 1/m m, for i = 1, f, m m h. In [37], the authors proposed
an enhanced version, called GS Adaptive (GSA), in which
I L is generated by the linear model in (2) with weights estimated by the minimization of the mean square error between the estimated component and a filtered and downsampled version of the PAN image.
B. Multiresolution AnAlysis
Pansharpening methods based on MRA apply a spatial filter to the PAN image for generating details to be injected
into the HS data. The main advantages of the MRA-based
fusion techniques are the following: i) temporal coherence [5] (see Sect.27.4.4), ii) spectral consistency, and iii)
robustness to aliasing, under proper conditions [38]. On
the negative side, the main shortcomings are i) the implementation is more complicated due to the design of spatial
filters, ii) the computational burden is usually larger when
compared to CS approaches. The fusion step is summarized as [4], [39]
k
k
W
X =L
Y H + G k 7 ^P - PLh,
(4)
for k = 1, f, m m, where PL denotes a low-pass version of P,
and the symbol 7 denotes element-wise multiplication.
Furthermore, an equalization between the PAN image
and the HS spectral bands is often required. P - PL is often
called the details image, because it is a high-pass version of
P, and Eq. (4) can be seen as describing the way to inject
details into each of the bands of the HS image. According
to (4), the approaches belonging to this category can differ
in i) the type of PAN low pass image PL that is used, and
ii h the definition of the gain coefficients G k . Two common
options for defining the gains are:
1) G k = 1 for k = 1, f, m m, where 1 is an appropriately
sized matrix with all elements equal to 1. This choice
identifies the so-called additive injection scheme;
k
2) G k = L
Y H 8 PL for k = 1, f, m m, where the symbol 8 denotes element-wise division. In this case, the details are
weighted by the ratio between the upsampled HS image and the low-pass filtered PAN one, in order to reproduce the local intensity contrast of the PAN image
2GS is a more general method than PCA. PCA can be obtained, in GS, by
using the first PC as the low resolution panchromatic image [41].
ieee Geoscience and remote sensing magazine
september 2015
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