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Wavelet transform is a mathematical tool that can detect local 
features in a signal process. It can also be employed to 
decompose two-dimensional signals - a digital image - into 
different levels of resolution for a multi-resolution analysis. 
This multi-resolution characteristic is utilized for fusing images 
with different resolutions. 
2.1 Wavelet Theory 
2.1.1 Wavelet Transform 
To images or signals, the wavelet transform is to decompose or 
analyze them. The process is called decomposition or analysis. 
From the signal analyst's point of view, wavelet analysis is a 
decomposition of the signal on a family of analyzing signals, 
which is usually an orthogonal function method. From an 
algorithmic point of view, wavelet analysis offers a harmonious 
compromise between decomposition and smoothing techniques. 
Namely a signal or an image will be decomposed into a 
hierarchical set of approximation and details after wavelet 
transform. The basic wavelet equations are as follows: 
1 t-b 
WaA 0= -¡=¥i )a>0,b<BR (1) 
da a 
WTj (a, ti) = (f(t), Y gJ , (i)} = -7= f )dl<2) 
v a r a 
V,,tU)= - k) j,kez (3) 
WT f (J,k) = {/(1X^(1)) = 4= J/(0)/(T‘ - k )dt (4) 
where If/ a h (0 continuous wavelet 
WTj- (¿7, b) continuous wavelet coefficient 
WjA 0 discrete wavelet 
WT f (j,k) discrete wavelet coefficient 
f(t) = signal or image 
a = scale of wavelet 
b = shift factor of wavelet 
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 
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. 
2.1.2 Wavelet Inverse Transform 
A virtue of wavelet transform is that the components can be 
assembled back into the original without loss of information. 
Where wavelet analysis involves filtering and downsampling, 
the wavelet reconstruction process consists of upsampling and 
filtering.Upsampling is the process of lengthening a signal 
component by inserting zeros between samples. The 
reconstruction function or inverse transform eqation is as 
follows: 
v Ip r dadb 
/(0-— J + J WT f (a,b)y/ b {t)——(5) 
°r °r a 
/(0= 2 YiWT f {j,k)y/ jk {t) (6) 
j^Zk^Z 
where is a constant depending on (// 
2.2 Wavelet-based Image Fusion Method 
Figure 1 schematic of conventional wavelet fusion method 
Wavelet-based fusion technique integrates the high-frequency 
components of the higher resolution data with the low- 
frequency components of the lower resolution data (or the lower 
resolution data) in a Multi-Resolution Analysis (MRA). 
Currently wavelet-based image fusion methods used are mostly 
based on two computation algorithms: the Mallat algorithm 
(Mallat, 1989; Ranchin et al, 2000) and the a trous algorithm 
(Aiazzi et al, 2002; Shensa, 1992).In this paper the Mallat 
algorithm is used.The fusion process has been well described 
(Yocky, 1996; Aiazzi et al, 2002).The first step is to co-register 
two images precisely. It is generally preferable to register the 
lower resolution image to the higher resolution image. In other 
words the high resolution image is used as reference image. 
However if the lower resolution image has georeference that is 
to be retained, it may be desirable to use it as the reference 
image. Secondly the histograms between images are matched. 
The next step is to decompose the low resolution image and the 
high resolution image based on wavelet transform. Sometimes