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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
3. THE 2-D DUAL-TREE COMPLEX WAVELET 
TRANSFORM 
For 2-D signals, we can filter separately along columns and then 
rows by the way like 1-D. Kingsbury figured out in (Nick 
Kingsbury: 11998a) that, to represent fully a real 2-D signal, we 
must filter with complex conjugates of the column and row 
filters. So it gives 4:1 redundancy in the transform. Furthermore, 
it remains computationally efficient, since actually it is close to 
a classical real 2-D wavelet transform at each scale in one tree, 
and the discrete transform can be implemented by a ladder filter 
structure. 
The quad-tree transform is designed to be, as much as possible, 
translation invariant. It means that if we decide to keep only the 
details or the approximation of a given scale, removing all other 
scales, shifting the input image only produces a shift of the 
reconstructed filtered image, without aliasing. ( A. Jalobeanu , 
2000) 
The most important property of CWT is that it can separate 
more directions than the real wavelet transform. The 2-D DWT 
produces three bandpass subimages at each level, which are 
corresponding to LH, HH, HL, and oriented at angles of 0? , + 
45°, 90°. The 2-D CWT can provide six subimages in two 
adjacent spectral quadrants at each level, which are oriented at 
angles of +15°, +45°, +75°. This is shown in fig 3. The strong 
orientation occurs because the complex filters are asymmetry 
responses. They can separate positive frequencies from negative 
ones vertically and horizontally. So positive and negative 
frequencies won't be aliasing. The orientations of details is 
shown in fig 4. Fig 5 shows the transform of an isotropic 
synthetic image at level 3, which also contains details at 
different scales. The orientation selectivity is more clear under 
each scale in comparison with the classical wavelet transform. 
Since CWT has so many advantages, we consider use CWT to 
carry out image fusion instead of DWT. Then we design an 
image fusion method based on CWT in next section. 
4. ANIMAGE FUSION APPRAOCH BASED ON CWT 
The DWT has already been used for image fusion ten years ago. 
Though image fusion approaches by wavelet transform have 
been improved to be adaptive to process varied images, two 
disadvantages (lack of shift invariance and poor direction 
selectivity) still exist. They have hampered the further 
application of wavelet transform in image fusion. 
The CWT is a good solution to this problem. It is approximate 
shift invariant. If the input signal shift a few samples, the fused 
image will be reconstructed without aliasing, which is useful to 
the not strictly registered images. Morover it can separate 
positive and negative frequencies and provide 6 subimages 
with different directions at each scale. So the details of CWT 
can conserve more spatial information than DWT. The spatial 
531 
  
    
REAL DW 
Figure 3. 2-D impulse responses of the complex wavelets at 
level 4 (6 bands at angles from -75° to +75°) and equivalent 
responses for a real wavelet transform (3 bands) 
-N2 
  
  
  
X 
IN. 
/ 
  
Figure 4. Directional selectivity of the frequency space 
corresponding to the complex wavelet transform 
  
Figure 5. Left : isotropic test image containing various scale 
information, right: magnitude ofits complex wavelet transform 
at level 3 showing both directional and scaling properties 
can conserve more spatial information than DWT. The spatial 
resolution of the fused image is more closer to the 
high-resolution image. PR, limited redundancy and high 
computation efficiency make it suitable for image fusion. 
We design an approach based on the quad-tree complex wavelet 
transform for fusing a low resolution multi-spectral image and a 
high resolution panchromatic image. First the registered 
multi-spectral image and panchromatic image are decomposed 
by complex wavelet respectively , then the approximate and 
detail parts of two images are fused according to some rule at 
each level, finally the fused image is reconstructed. This is 
illustrated by fig 6. The fusion procedure can be described 
detailedly as following: 
(1)Each band of the low resolution multi-spectral image and the 
high resolution panchromatic image are geometrically 
registered to each other. After geometrical rectification , ihe 
images have the same size. 
(2)The panchromatic image is stretched tally with each band 
of multi-spectral images respectively according to the 
histogram. 
(3)Decomposed the histogram-specified panchromatic image