The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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paper. The second section discusses the theoretical basis and 
transformation characteristics of multi-band and biorthgonal 
wavelet. Then the fusion method based on multi-band and 
biorthgonal wavelet is implemented to fuse 10-m SPOT 
panchromatic and 30-m multispectral TM images Next, the 
experimental result is compared with previous methods 
developed for image fusion, such as IHS ,PCA and two-band 
wavelet 
coefficients HL (variations along the columns), vertical 
coefficients LH (variations along the rows), diagonal 
coefficients LL (variations along the diagonals) (Gonzalez and 
Woods, 2001). The three high frequency image is called detail 
image, which contain information of local details. 
When a = rf and b= rfkdhe resulting equation is 
2. BIORTHOGONAL AND MULTI-BAND WAVELET 
2.1 MItiresolution Analysis 
Wavelets are functions in //^determined from a basic wavelet 
function by dilations and translations.They are used for 
representing the local frequency content of functions. The basic 
wavelet should be well localized in general, and the wavelet 
should have zero mean( Daubechics, I., 1992.). The basic 
method to construct a wavelet is Multiresolution Analysis.A 
Multiresolution Analysis (MRA) is defined by a sequence of 
closed subspaces (Vj ) j EZ ,which approximates L 2 R, and a 
function <£> EL]_ R s an orthonormal basis for V 0 . 
V'j.k(x) = 
ip (n j x - k) 
(4) 
WT,U,k) = {f(x),<P i M = X f f(x) V \ri"-k)dx (5) 
v« J R 
We denoted it as n-band wavelet. After n-band discrete wavelet 
transform, an image yields n2 images: one low-pass image and 
n2-l high-pass The high frequency image is called detail image, 
which contain information of local details and low frequency is 
approximate. image. 
{0}--- e V_ t e V 0 c F, c •• L 2 (R) 
{$ o.«;® <>,„(*) = - «)> n e z } 
Where O(x) is a scaling function 
2.2 Multi-band wavelet 
Generally, a wavelet family is described in terms of its mother 
wavelet, denoted as v|/(x). A daughter wavelet \|/ ab (x) is defined 
by the equation 
l / / aA x )= a) 
va 
Figure 1 two-band wavelet and three-band wavelet transform 
2.3 Biorthogonal wavelet 
Generally a function f(x) can be decomposed as a superposition 
of the orthogonal basisv|/j k (x), But to biorthogonal wavelet, a 
function f(x) can be written (7) 
where a, b e R and a^O; 
a is called the scaling or dilation factor and b is called the 
translation factor. 
A common choice is a=2 J and b=2 J k, where j and k are integers. 
The resulting equation is 
¥j,ki x ) = —(2 x - k) l (2) 
V2' 
WT f U,*) = (/№, k (tj)=-j= J^/(0i^*(2’ j x- k)dx (3) 
This equation(2) is two-band orthogonal wavelet. Accordingly 
the wavelet transform of function f(x) is (3) 
After 2-band discrete wavelet transform, an image yields four 
images: one low-pass image and three high-pass images. 
Namely, approximation coefficients (labeled LL), horizontal 
fix) = (WT f ,<p jtk (x)) = \(f(x),<p ik (x))<p jk (x)dx (6) 
fix) = (WT f , (pj, k (*)) = ¡(fix)jp* j,k {x))(p jk (x)dx (7) 
Where {(p jk O), (p* j,10)> - ô jr ô kl (8) 
When applying the biorthogonal wavelet to decompose function 
or images, the function (p j,k (x) is used, while the 
function (p . k (x) is applied to reconstrut function f(x) The 
biorthogonal wavelet systems generalize the classical 
orthogonal wavelet systems. They are more flexible and 
generally easy to design. One of the main reasons to choose 
biorthogonal wavelets over the orthogonal ones is symmetric. 
Symmetric wavelets and scaling functions are possible in the 
framework of biorthogonal wavelets. However, the 
orthogonality no longer holds in biorthogonal wavelet systems. 
However,that is the near orthogonal system.