Digital Signal Processing Reference
In-Depth Information
This reconstruction procedure takes account of the pyramidal sequence of im-
ages containing the multiresolution transform coefficients,
w
j
. The PMT has been
used for the compression of astronomical data (Starck et al. 1996; Louys et al. 1999)
and of optical readout biomolecular sensory data (Stoschek 2003).
4.4.3 Merging Wavelets and the Multiscale Median Transform
One of the advantages of the PWT over the MMT is the ability to have robust noise
estimation in the different scales, while the advantage of the MMT is a better sep-
aration of the structures in the scales. Using the MMT, a strong structure (like a
bright star, cosmic ray hit, etc.) will not be spread over all scales as when using a
wavelet transform. In fact, when there is no signal in a given region, a wavelet trans-
form would be better, and if a strong signal appears, it is the MMT that we would
like to use. So the idea naturally arises to try to merge both transforms and to adapt
the analysis at each position and at each scale, depending on the amplitude of the
coefficient we measure (Starck 2002).
A possible algorithm to perform this on a 2-D image
X
is the following:
Algorithm 13
2-D Median-Wavelet Transform Algorithm
Task:
Compute the 2-D median-wavelet transform of an
N
×
N
image
X
.
Initialization:
c
0
=
X
,
J
=
log
2
N
,
s
=
2, and
τ
=
5.
for
j
=
0
to
J
−
1
do
1. Compute
c
j
+
1
=
1)]. This step is similar to that used in the
MMT: it is a median smoothing with a window size depending on the scale
j
.
2. Compute
w
j
+
1
=
Med(
c
j
,
2
s
+
c
j
;
m
j
+
1
coefficients corresponding to median co-
efficients between two consecutive scales.
3. Detect in
w
j
+
1
the significant coefficients:
w
j
+
1
>τ
c
j
+
1
−
MAD(
w
j
+
1
)
/
0
.
6745
,
where MAD stands for the median absolute deviation used as an estima-
tor of the noise standard deviation (see equation (6.9)) and
τ
a threshold
chosen large enough to avoid false detections, for instance,
τ
=
5.
4. Set to zero all significant coefficients in
w
j
+
1
.
5. Compute
c
j
=
c
j
+
1
. Hence
c
j
w
j
+
1
+
is a version of
c
j
, but without the
detected significant structures.
6. Compute the 2-D starlet transform of
c
j
with
j
+
1 scales. We get
W =
1
c
j
+
1
}
{
w
,...,w
,
.
j
+
1
c
j
+
1
. Therefore
c
j
+
1
is smoothed with wavelets, but strong fea-
tures have been extracted with the median.
8. Compute the median-wavelet coefficients:
7. Set
c
j
+
1
=
w
j
+
1
=
c
j
−
c
j
+
1
.
9.
s
=
2
s
.
Output:
W ={
w
1
,...,w
J
,
c
J
}
, the median-wavelet transform of
X
.
