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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
designate the approximations at the resolution of 2" and the
coarser resolution 2"*!. Cm,n also denotes the difference between
one approximation and the other. To calculate Gun: and Can
coefficients, a scaling function is necessary. Then, the
convolution of scaling function and the signal is implemented
at every scale using a low pass FIR (Finite Impulse Response)
filter 4, to calculate a,,, coefficients (Nikolov et al, 2001). This
process can be designated with the following equation (Nikolov
et al, 2001).
Umn 7 2 Pay (3)
Similarly, by using a related high pass FIR filter gy ihe c,
coefficients are calculated using the following equation
(Nikolov et al, 2001).
C n.n = 2 E2n-k alk (4)
t
For 2-D DWT, it is just necessary to separately filter and
downsample the image in the horizontal and vertical directions
(Nikolov et al, 2001). By doing this, the spatial resolution is
halved at each level by subsampling the image by a factor two.
Each image provides four sub-images at each resolution level
corresponding to one approximation image (low spatial
resolution) and three detail (horizontal, vertical and diagonal)
images (Chibani and Houacine, 2002). The same input image
can be obtained by inverse DWT using calculated wavelet
coefficients.
3. IMAGE FUSION ALGORITHM
3.1. Preprocessing of input images
In image fusion, the first step is to prepare the input images for
the fusion process. This includes registration and resampling of
the input images (Zhou, 1998). Registration is to align
corresponding pixels in the input images. This is usually done
by geo-referencing the images to a map projection such as
UTM (Universal Transverse Mercator). If the images are from
the same sensors and taken at the same time, they are usually
already co-registered and can be directly used for fusion
processing. However, if the images are from different sensors,
and even if they are georeferenced by the image vendors, a
registration process is likely still necessary to ensure that pixels
in the input images exactly represent the same location on the
ground.
Image registration can be performed with or without ground
control. The most accurate way is to rectify the images using
ground control points. However, in most cases, it is not
possible to find ground control points in the input images. In
such situations, taking the panchromatic image, which has a
better spatial resolution, as the reference image and registering
the multispectral images with respect to the panchromatic one
can be a good solution to refine the rectified multispectral
images.
Image fusion essentially occurs when the involved images or
their transformation have the same spatial resolution. In the
selected wavelet decomposition, the dimension of the newly
decomposed image becomes half the size of the image at the
previous level (Chibani and Houacine, 2002). Therefore,
another important task in the preparation phase is to make the
proportion between the pixel spacing of the panchromatic and
multispectral images to be a power of two. The panchromatic
and multispectral images of the same sensors (i.e. QuickBird,
1245
SPOT and Ikonos panchromatic and multispectral images
respectively) may inherently meet this requirement. For
example, the proportion between the pixel sizes of the
panchromatic and multispectral images is 2? for QuickBird
images (0.7m. versus 2.8m. for panchromatic and multispectral
bands respectively). For this reason, no resampling is needed
for these images. Their pixel sizes will be the same if one-level
and three-level discrete wavelet decomposition are performed
to these images respectively. If the pixel size of the input
images does not have the 2" multiplier relationship, resampling
is needed.
However, resampling will deteriorates the quality and structure
of the image involved. For this reason, it is expected that the
resampling should be performed at minimum extent. (Du et al,
2003) propose an algorithm to find the minimum resampling
needed. According to this algorithm, a coefficient S that makes
the pixel sizes (P,, Pg) of two images (A and B) equal is
determined using the equation P4 — SPg. Then, another number
Sy, which is the nearest number to S that is the power of two, is
found. Finally, the image A, which has a larger pixel size, is
resampled to have a pixel size of S *Pg. This approach ensures
that the resampled image now has a pixel spacing that the
proportion between the Py and resampled image is the power of
two. It also ensures that the (S-S,) is the minimum number that
meets. this requirement. This resampling approach is used in
our study.
3.2 Implementing wavelet transform
Wavelet transform based image fusion involves three steps;
forward transform, coefficient combination and backward
transform. In the forward transform, two or more registered
input images are calculated to get their wavelet coefficients.
These coefficients respectively represent the approximation,
horizontal, vertical and diagonal components of the input
images (Hill et al, 2002). Figure | below illustrates a 2-D
forward DWT process (Misiti, 2002).
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ye iue 2
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ass filter £T
Input image colum vertical
Aj h, low 7 p^
row ne 142 Ds,
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low pass 241 colum
filter 2, low | i 5 cD d 1
pass filter J
diagonal
Figure 1. 2-D forward DWT to get approximation, vertical,
horizontal and diagonal wavelet coefficients
The same process needs to be applied to all input images one
by one. Then, these wavelet coefficients from the different
input images are combined according to certain fusion rules to
get fused wavelet coefficients.
3.3 Fusion Rules
This is where the fusion essentially occurs. The wavelet
transform coefficients obtained from the input images need to
be combined to form a new set of coefficients to be used for
backward transform. There are various fusion rules to form the
fused wavelet coefficients matrix using the coefficients of the
input images. In this study, taking the largest absolute values of
