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using a transform in order to get better insight into the properties of a signal, it should
be ensured that the signal can be perfectly reconstructed from its representation.
Otherwise the representation may be completely or partly meaningless.
Images are 2D functions of intensity. The analysis starts with defining a two-
dimensional scaling and wavelet functions:
s
//
ð
x
;
y
Þ
¼/
ð
x
Þ
/
ð
y
Þ
ð
6
:
3
Þ
s
ww
ð
x
;
y
Þ
¼w
ð
x
Þ
w
ð
y
Þ
ð
6
:
4
Þ
If f
0
ð
x
;
y
Þ
is the projection of f
ð
x
;
y
Þ
on the space V
0
generated by s
//
ð
x
;
y
Þ
,we
get:
f
o
ð
x
;
y
Þ
¼
X
1
X
1
a
o
ð
i
;
j
Þ
s
//
ð
x
i
;
y
j
Þ
ð
6
:
5
Þ
i¼
1
j¼
1
a
o
ð
i
;
j
Þ
¼\f
ð
x
;
y
Þ;
s
//
ð
x
i
;
y
j
Þ
[
The result is four sets of coefficients: approximation and horizontal, vertical and
diagonal direction details. During years, several transforms (curvelets, conturelets,
edgelets, etc.) emerged from wavelets [
25
], which define details in different
manners, but these are not concern of the chapter. These transforms can be used
when rotation is likely to happen. However, X-rays are always taken in fixed
position. Therefore, there is no reason to introduce other transforms. Anyway,
algorithm is in general enough to be modified for these transforms as well.
Image compression is usually obtained by thresholding wavelet coefficients.
Some efforts were made for compression by downsampling of coefficients vector.
In such case, interpolation is used in reconstruction.
6.3 Proposed Algorithm
Algorithm begins with the compression part. This part enables archiving of X-rays
in vector of wavelet coefficients. When doctor wants to see the image, recon-
struction part is used. After reconstruction, visualization is performed. Diagnostics
is facilitated by diagnostics visualization part of the algorithm by emphasizing
suspicious areas with different color. If color does not appear on the screen, this
means that patient is not candidate for asbestosis (there is no suspicious shadows in
the lung image).
Proposed algorithm can be explained in several steps (shown as algorithm in
Fig.
6.1
). Decomposition part has 6 steps as follows.
Step 1:
Wavelet decomposition of the pulmonary X-ray image.
2D
DWT
is
implemented
with
Matlab
command
wavedec2,
which
includes packaging the data into vector:
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