212 
> At level 1, there is one sample offset between the trees. 
(a\)„=(a°*h°) 2n (¿i)„=(« 0 *g\ 
( a a)» = (a° *h°) 2 „+i K)„=(«°*g°) 2 „ +1 
> Beyond level 1, there must be half a sample difference 
between the trees. 
(Jn.=(a J s*g°) 2 n+1 (2) 
The details d A an d d B can be interpreted as the real and 
imaginary parts of a complex process z = d A + id B ■ The 
essential property of this transform is that the magnitude of the 
step response is approximately invariant with the input shift, 
while only the phase varies rapidly. (A. Jalobeanu , 2000) 
Reconstruction is performed independently in each tree, and the 
results are averaged to obtain a ° at level 1, for symmetry 
between the two trees. 
• Level j (j> o): 
• Aty = 0 : 
i _ _ (4) 
a,° = 2«^ *h°) n +{d\ *g°\ +(3; •**). +(d' B *?*).) 
frequencies from negative ones vertically and horizontally. 
Figure 1 shows the transform of an isotropic synthetic image at 
level 3, which also contains details at different scales. 
Figure 1. Isotropic test image containing various scale 
information (left), magnitude of its complex wavelet transform 
at level 3 showing both directional and scaling properties 
(right) 
The DT-CWT is a good solution to image fusion because of its 
advantages. First, it is approximate shift invariant. If the input 
signal shifts a few samples, the fused image will be 
reconstructed without aliasing, which is useful to the not 
strictly registered images. Secondly, it can separate positive and 
negative frequencies and provide 6 subimages with different 
directions at each scale. So the details of DT-CWT can 
conserve more detail information than DWT. In addition, PR, 
limited redundancy and high computation efficiency make it 
suitable for image fusion execution. 
where _ jx p if n = 2p , ■_ jx p if n = 2p + \ 
{ 0 if rt = 2p + \ " [0 if n = 2p 
For 2-D signal, we can filter separately along columns and then 
rows by the way like 1-D. Kingsbury figured out in (Nick 
Kingsbury, 1998a) 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 CWT 
can provide six subimages in two adjacent spectral quadrants at 
each level, which are oriented at angles of ±15°, ± 45°, ± 
75°. The strong orientation occurs because the complex filters 
are asymmetry responses. They can separate positive 
3. SAR AND OPTICAL IMAGE FUSION BASED ON 
DT-CWT 
3.1 Speckle Denoising 
The SAR image is produced by coherently receiving echo. 
Echo overlapping inevitably produced speckle noise. Speckle is 
a serious obstacle of SAR image object recognition and makes 
some ground features disappear. (Xiao Guochao, 2001) So 
speckle has to be removed before image fusion. 
A few algorithms, such as Lee, Frost, Kuan, Gamma MAP, are 
successfully used to denoise speckle with an assumption that 
speckle is multiplicative noise. Here the Lee-Sigma (Lee S. J., 
1980) and Gamma MAP algorithms (Lopes A. et al. , 1993; 
Baraldi A. et al. , 1995) are chosen, because they can decrease 
the lost of edge features while removing speckle noise. 
The Lee-Sigma algorithm is described as following 
R = I + Kx(CP-UxI) 
K l (Sigma/U 2 ) (6) 
(!QVARI1 2 ) 
Where