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R is the grey level of the filtered interest pixel,
Q VAR is the variance in filter window,
I is the mean grey level in the filter window,
U is the mean multiplicative noise and usually is 1,
CP is the central pixel in filter window,
Sigma is the multiplicative noise variance, it is estimated
based on a Rayleigh distribution and consistent with those
derived from actual data.
The Gamma MAP filter is based on a multiplicative noise
model with non-stationary mean and variance parameters.
Recent work has shown natural vegetated areas have been
shown to be more properly modeled as having a Gamma
distributed cross section. This algorithm incorporates this
assumption. The exact formula used is:
R = {
1
0BxI + ^[D)/(2a)
CP
C
I — U
c u < c, < c,
c, > c max
(7)
Where
B = a-NLOOK-1,
D = I 2 B 2 +4 a NLOOK I CP,
a = (l + C 2 )/(C 2 -C 2 ),
c u = \ Unlook ,
C, = JVARH,
c max =4i *c u ,
NLOOK is number of looks,
VAR is variance in filter window.
By experiments we find using both Gamma MAP and
Lee-Sigma filters to achieve better result than using Gamma
MAP or Lee-Sigma filter twice. So here the SAR image is first
filtered by Gamma MAP and then filtered by Lee-Sigma. The
proportions of original SAR image and denoised SAR image
are shown in figure 2. The speckle noise of denoised image has
been obviously removed and edge features have been
conserved.
Figure 2. Proportions of original SAR image (left) and
denoised SAR image (right)
registered multi-spectral image and SAR image are
decomposed by DT-CWT respectively, then the approximate
and detail parts of two images are fused according to some
rules at each level, finally the fused image is reconstructed.
This procedure is illustrated by figure 3. The fusion procedure
can be described in detail as following:
(1) Each band of the multi-spectral optical image and the SAR
image are geometrically registered to each other. After
geometrical rectification, their sizes are same.
(2) The gray level of SAR image is stretched tally with each
band of multi-spectral images respectively using histogram
equalization.
(3) Decompose the histogram-specified SAR and registered
multi-spectral optical images with DT-CWT to form their
multi-resolution and multi-directional descriptions. At the same
time, the moduli of their complex wavelet transform are
achieved.
(4) Since the aim of image fusion is to improve image
information quality, we should analyze characteristics of SAR
and optical images. Some objects, like lakes, roads or buildings,
are distinct in SAR image but more details are hard to
recognize. On the contrary, there are enough details and
spectral information in optical image. So we design different
fusion rules for low and high frequency parts fusion to integrate
the advantages of two images.
Image fusion begins with the coarsest level. The gray value of a
fused low frequency part pixel is determined by maximum gray
value rule. The bigger absolute gray value at cooresponding
pixel between SAR and optical images is selected. This rule
makes more approximate parts and spectral information in
optical image conserved.
The important information in SAR image is mostly in the high
frequency parts. But some important details in optical image
are also in the high frequency parts. So we decide to determine
the fused pixel by comparing energy values of corresponding
pixels in two images. The pixel with bigger energy value is the
fused pixel. The energy value of a pixel is calculated in its
centered neighbor window. Considering that DT-CWT of the
images can be interpreted as a complex including real part and
imaginary part, and the modulus can show clear directionality,
the energy values can be computed according to the moduli of
the high frequency parts. The procedure is illustrated in fig. 3.
The wavelet coefficients at point (/ j) of real and imaginary
parts in the SAR image are denoted as w R s (/, j) and W, s (i,j)
respectively. The wavelet coefficients at point (/ ; y) of real
and imaginary parts in the optical image are denoted as
W%(i,j) anc * W,°(i,j) respectively. The magnitudes at point
(z 5 j) in the SAR image and the optical image are achieved
respectively by
M s («, j) = V(^/(U)M*F/(U)) 2
M ° (I, J) = J{w R °(i,j)) 2 + {iv 1 0 «J)ï (8)
3.2 The fusion algorithm
We design an algorithm based on DT-CWT for fusing a
multi-spectral optical image and a SAR image. First the
The energy values at point ( /; y) in the SAR image and the
optical image are achieved respectively by
