Árbók VFÍ/TFÍ - 01.06.2002, Blaðsíða 244

for s>l. To implement this, the 6-tap Daubechies
DWT is used. It gives 8 subbands indexed by
j=0,l,...,7, where the index ;'=0 indexes the coarsest
level.
Curvelet subband s = 1 then corresponds to wavelet
subbands j = 0,1,2,3; curvelet subband s = 2 corre-
sponds to wavelet subbands j = 4,5; and curvelet
subband s = 3 corresponds to wavelet subbands j =
6,7. The coefficients in the curvelet subband s are
used as an input in the inverse DWT and give Ds,s =
1,2,3 as shown in Figure 3. Figure 4 shows the well
known image Lena and its subband decomposition.
The second step of the digital curvelet transforma-
tion is the tiling. The subband arrays Ds,s = 2,3 are
localized into squares. The subband array is left
alone. The subband array D2 is tiled into 8x8 over-
lapping blocks, each block being 64 x 64. The sub-
band array D3 is tiled into 16 x 16 overlapping
blocks each being 32 x32. The overlap between two
vertically adjacent blocks is a rectangular size b by
b/2 where b is the sidelength of the overlapping
blocks. The reason for this overlap is to eliminate the
artifacts associated with the boundaries of the
squares if no overlapping is used. Figure 5 shows
subbands 2 and 3 tiled. The last step of the digital
curvelet transformation is the digital ridgelet trans-
formation. For more details on thatrefer to [8] and [9].
Denoising Experiments
In this section denoising experiments are performed on both a synthetic image where the
distribution of the noise is known and a real SAR image where the distribution of the
noise is not known.
The synthetic image
The synthetic image used is an 256 x 256, 8-bit grayscale image of Lena. The pixel value
of the Lena image lies in the interval [0,1]. Gaussian white noise n with standard devia-
tion (7=0,1 is added to the image/. The image model is given with
y[k\>k2] = f[kiJc2]+ (7n 0 <Á:, /r 2 < 255. (H)
Figure 6 shows Lena without noise and with noise. It is clear that the noise is corrupting
the image and making further processing of it more difficult. The denoising algorithm
used is quite standard and is included here for completeness:
1. Calculate the curvelet transform of the noisy image by
y»=C(f + on\={Cf\+o{Cn\,
where /i is the curvelet index.