Environmental Engineering Reference
In-Depth Information
function) at that level. This will allow estimation of bubble size distribution as
described in the following steps.
3. A threshold is applied to each
A
s
image to remove dark pixels and valleys from
the image, thus enhancing the areas covered by the bubbles observed at this level,
which size is greater than the wavelet width at level
s
.Let
A
s
be the total area
covered by the bubbles in approximation image
A
s
. This correspond to the total
number of pixels within the image remaining after the threshold was applied
(
i.e.
, pixels belonging to bubbles). The fractional area covered by these bubbles
is computed as follows based on the total area seen by the camera (
i.e.
, field of
view) or, alternatively, the total number of pixels of the image
A
T
:
A
s
A
T
.
F
s
=
(3.14)
4. To obtain the range of bubbles sizes extracted at each level, adjacent fractional
areas
F
S
are subtracted, that is, for six decomposition levels (
i.e.
,
S
=
6),
F
1
=
(
A
0
−
A
1
)
A
T
=
(
A
1
−
A
2
)
A
T
=
(
A
4
−
A
5
)
A
T
. The last approximation image is usually
left out since it often contains very low frequency information such as variations
in lighting conditions [51].
5. Finally, the number of bubbles having a size within the range captured at each
level
N
s
is computed as shown below, assuming circular bubbles. Indeed, hori-
zontal and vertical diameters
D
H
,
s
and
D
V
,
s
are assumed equal, and correspond
to the width of the wavelet at level
s
, that is the largest bubble size captured by
the wavelet function at level
s
.
,
F
2
,
...
,
F
5
A
T
F
s
N
s
=
4
D
H
,
s
D
V
,
s
.
(3.15)
π
This procedure leads to very robust estimates of bubble sizes. It is not corrupted
by noise, which is captured by the detail images at the first level, and not influenced
by lighting variations, which for the resolution of images considered (
i.e.
, 720
×
480), appear in the sixth decomposition level. When
A
s
and
A
T
are computed in
terms of number of pixels, the bubble sizes are relative measures or scaled bubble
diameters. These could be calibrated to provide bubble diameters in engineering
units (
i.e.
, cm) by measuring the actual area covered by the field of view
A
T
,and
computing
A
s
as a proportion of the field of view.
To illustrate estimation of bubble size distribution using the wavelet size signa-
tures, a histogram of the number of bubbles of different sizes (
i.e.
,
N
s
,
s
5)
for three different froth images are shown in Figure 3.17, where the scaled bubble
diameter (
D
H
,
s
or
D
V
,
s
) related with wavelet width at the various decomposition
levels
s
. The difference in bubble size distribution observed in the images is clearly
distinguished in the size histogram.
=
1
,
2
,...,
Search WWH ::
Custom Search