The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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high-resolution spatial image through adaptive wavelet package 
fusion algorithm to obtain high-resolution colour image. This 
method can avoid the limits brought by using single feature for 
fusion and make fusion information more abundant. 
2.1 Band selection model by the optimal index principle 
The greater the image’s standard deviation the more 
information it will have, while the smaller the correlative 
coefficient between image bands, the more independent they 
will be and the less redundant information they will have. So 
here the optimal index model (OIF) can be adopted. 
£№ = ¿$,/¿1/1,1 (1) 
1=1 1=1 
Where S,• =standard deviation of the i image band 
Ry ^correlative coefficient between i and j image 
band. 
The numerator is the sum of the mean square error of three 
image bands, and the denominator is the sum of the correlative 
coefficient between them. It is clear that the smaller the 
correlation, the higher sparsely (information) these bands will 
have. The 3 image bands of the whole image data that give the 
highest OIF value are selected as the optimal image bands. If 
we are interested in some special regions, we can define them 
and get the optimal bands for them correspondingly. 
2.2 Wavelet Package Algorithm 
Wavelet package is composed of a few wavelet functions. 
Assume that there is a wavelet function cluster [w n {x),n e N] 
and these functions have the following relationship (Zhu,2000). 
w 2n( x )=T, h k w n ( 2x ~ k ) (2) 
k 
W 2„ + lM =Yj^k W n{^ X ~ k ) (3) 
k 
When n=0, > v 0 (jc)=<p(jc)and W| (x) = v'W’ where <p( x ) is scale 
function, y/{x) is wavelet function, [h k } and are 
coefficients of low-pass and high-pass QMF filter and 
correspond to scale function and wavelet function respectively. 
Then |w H (x),x e N) is called wavelet package determined by 
w 0 (.x)=tp(x). According to equation (2) and (3), w n (x)can be 
reconstructed by w 2 „(x) and w n (x): 
w ni 2x - 0= Y,{Pl-2k W 2n(* - *)+ ?/-2* W 2„ + l(* “ *)} (4) 
k 
Where ({/?*}, }) is QMF filter corresponding to {{h k }, {g*}). 
So the decomposing and reconstructing algorithm can be 
described as follows: 
Assume that A * f{x) is approximation of wavelet package 
w„ on scale 2 1 , which is 
A n A x ) = X S A -Wn&x-k) (5) 
k 
Where S J nk = 2 2 f f(x)w n il J x-k)dx 
’ J-oo 
Then according to equation (1),(2), the decomposing algorithm 
is 
=2X-A 
m 
^2n.+l,I ~ XI ^m-21^n,m 
And according to equation (3), the reconstructing algorithm is 
Sn,k ~ y ^ J Pk-2l^2n,l + ' S y. C lk-2l^ J 2n+U ( 7 ) 
/ / 
2.3 Wavelet package fusion algorithm for hyperspectral 
data based on optimal index principle 
To process hyperspectral data with wavelet package fusion 
algorithm, optimal bands have to be selected first by optimal 
index algorithm to obtain synthesized low-resolution color 
image, and based on which, registered panchromatic image is 
obtained. Then the color image and registered image are 
decomposed by wavelet package algorithm. On each 
decomposing layer, every image part, no matter high frequency 
part or low frequency part, of the previous layer is analyzed 
iteratively and comprehensively. After analyzing, sub-images 
that are decomposed from the original two images are fused 
through related fusing strategy, and the final fusion image is 
reconstructed through inverse transformation. 
Usually, the more the wavelet package decomposing layer, the 
more details the fusion result will have. However, this is 
achieved at the expense of increasing computing amount. With 
the increasing of analyzing layer, the computing amount 
increases quite rapidly and information will be more and more 
heavily lost at the top layer. Thus wavelet package 
decomposing layers are sometimes chosen between 3 and 5 and 
more layers are not advisable. Here, wavelet package fusion 
algorithm for hyperspectral data based on optimal index 
principle can be summarized as follows: 
(1) Select optimal bands of hyperspectral data based on 
optimal index and construct synthesized low-resolution 
color image. 
(2) Initialize the wavelet package decomposing layer, the 
decomposing coefficients and so on.