Environmental Engineering Reference
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these on the image signal depends on the decomposition level. In the first decom-
position level, the original froth image is used. When the wavelet signal is applied
to that image, it will cover 8 pixels. At the second decomposition level, the dyadic
sampling of DWT will make the wavelet cover a length of 16 pixels, and so on.
Therefore, as the number of decomposition level increases, the wavelet will capture
bubbles of larger sizes or, alternatively, lower frequency information. The correla-
tion between the wavelet and the image signal will be maximal when the size of the
bubble matches the width of the wavelet at a given decomposition level. Since the
wavelet function can be represented as a high-pass filter (see Figure 3.16), bubbles
with smaller diameter than the width of the wavelet will also be captured in the
detail images since these are higher frequency information. Larger bubbles, which
are lower frequency information, will remain in the approximation image. There-
fore, detail and approximation images will capture a
range
of bubble sizes at each
decomposition level and not a single size [37].
Wavelet size signatures can also be explained by analogy with experimental
determination of ore particle size distribution via sieving. The initial ore sample
is equivalent to the initial froth image (
i.e.
, before wavelet decomposition). Each
screen used in the sieving procedure can be viewed as a wavelet decomposition
level whereas the mesh size of each screen corresponds to the length covered by
the wavelet at each level. The material retained on a particular screen is equiva-
lent to the detail images captured by the wavelet function. The material passing the
screen (further sorted by screens of smaller mesh) is the approximation image at this
level, which is further decomposed. The main difference between these two analog
processes is the size selection process. The first screen retains the materials of the
largest sizes whereas the first decomposition level depletes the image with the finer
details (
i.e.
, high frequency information corresponding to smaller bubbles).
Computing the wavelet size signatures involves a series of steps described below
[37]:
1. First the MR-MIA II algorithm is applied to the set of froth images, involving
PCA decomposition of the set of images, and performing DWT on the
T
1
image
up to S levels of decomposition. This image is very close to the gray scale con-
version of the RGB image. The symlet order 4 wavelet was found to work well
on froth images [37].
2. The gray scale approximation images for each decomposition level,
A
0
,
A
1
,
A
2
,
,
A
S
, are used to obtain the size signatures. This is motivated by the fact
that bubbles in froth images have very little directionality. The detail images
D
s
,
s
...
S
, capturing strongly oriented features, will not be used directly in
this approach for computing size signatures. At each level of decomposition
s
,the
approximation images are depleted from a range of smaller bubbles (captured in
detail images). When applied at level
s
=
1
,
2
,...,
S
(
i.e.
, from high to low fre-
quencies) the approximation images
A
s
will contain information about bubbles
larger in size than the width of the wavelet at level
s
. The difference between
adjacent approximation images (
i.e.
,
A
s
and
A
s
=
1
,
2
,...,
1
) will eventually provide infor-
mation about the quantity of bubbles of size corresponding to level
s
that have
been removed or captured by the detail images (
i.e.
, associated with the wavelet
+
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