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
