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Mapping without the sun
Zhang, Jixian

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
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
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)
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