The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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In general, the n-D transform matrix can be written as T 
i 
Tn 
1 
yl(n-\)n 
Normalization 
factor in row 
l(n-i + l)(n-i+2) 
(5) 
A number of evaluations can be made on the above generalized 
IHS (GIHS) transform. First, a variation of the above transform 
may be derived based on a different calculation of the 
intensities. As shown in the equations, we have used the 
normalized vector as the intensity calculation. In fact, the 
average of the involved bands may also be used as intensity 
(Zhou et al.,1998; Nunez et al. 1999;Wang et al.,2005; 
Choi,2006; Gonzalez-Audicana et al.,2006), which would lead 
to different yet similar transform. Moreover, the order of the 
rows in the transform is not significant and the rows are 
interchangeable, however, it is recommended to keep the 
intensity as the first row. Finally, the generalized transform can 
be interpreted in terms of a wavelet transform in the spectral 
domain across different bands at one pixel location. It can be 
seen from Eq. 4 that the first row of the transformed image is 
the average of all the input bands (i.e., the intensity, up to a 
constant); this corresponds to the average spectral response at 
this pixel location and can be interpreted as the low frequency 
component in a wavelet transform. The second row is the 
difference between the average of the first three bands and the 
fourth band, which corresponds to a high frequency component 
among the bands. Similarly, the third row is the difference 
between the average of the first two bands and the third band, 
while the last row is the difference of the first two bands, all up 
to a normalization factor. Therefore, it is found that the 
generalized IHS (so is the classical IHS) transform is essentially 
equivalent to a wavelet transform in the spectral domain, where 
the first component is the intensity or band average, and the 
other components are band differences relative to band averages 
calculated in a sequential combination of the involved bands, all 
up to a constant. 
To apply the above transform to image fusion, the input 
multispectral bands will first be transformed to a transformed 
space (equivalent to IHS in 3-D) with Eq. 3, 4 or 5. The 
transformed intensities are then replaced by the gray values in 
the panchromatic image. As the last step, the fused image bands 
are obtained with a reverse transform T~ l . 
3. CRITERIA-BASED IMAGE FUSION 
The criteria-based image fusion method modifies the a-p 
method introduced by (Gungor and Shan,2005, 2006). The 
underlying principle is that the fused image should meet certain 
desired properties represented by a set of predefined criteria. 
The method forms the fused images as a linear combination of 
the input panchromatic and the upsampled multispectral images 
F k (m, n) = a k (m, n) • 7 0 (w, n) + b k (m, n) ■ I k (m, n) (6) 
where m and n are the row and column numbers, k - 1,2, ...., 
N (N = number of multispectral bands), F k is the fused image, 
7 0 is the input panchromatic image, I k is the k-th band of 
the resampled multispectral image, and a and b are the 
weighting factors for pixel location (m, n), which control the 
amount of contribution from the panchromatic image and 
multispectral bands, respectively. The fusion formulation needs 
to determine the a and b coefficients at every pixel location, 
for which rules or criteria must be set. The selected criteria will 
determine the properties of the fusion outcome. Considering 
that image fusion is to retain the high spatial information or 
details from the panchromatic image and the spectral 
information or color from the multispectral one, we introduce 
the following three criteria. 
Criterion 1: The variance of the fused image should be equal to 
the variance of the corresponding panchromatic image, such 
that its spatial details, described by the variance, can be retained 
in the fused image. Based on Eq. 6 this statement can be 
expressed as 
Cov(F k ,F k ) = a 2 k a 2 + 2 a k b k a ok + b 2 a 2 k = cr 2 (7) 
Criterion 2: The mean of the fused image should be equal to 
the corresponding mean of the multispectral image such that the 
color content, described by the mean, is retained in the fused 
image. Based on Eq. 6 the above statement can be expressed as 
mecm{F k ) = a k n 0 + b k n k = Mk (8) 
In Eq.7 and Eq.8, the notations for image location (m,n) are 
omitted for a clearer expression. a k and b k coefficients are used 
to construct the fused pixel at (m,n) . cr 2 , a k and cr ok are the