Environmental Engineering Reference
In-Depth Information
using today's imaging technology, which currently provides spatial resolutions as
high as 14 megapixels. Constructing a 2-D density histograms from a
t
i
-
t
j
scatter
plot (
i
j
) involves: (1) discretizing the score plot into a certain number of bins by
applying a grid of a user-defined mesh size; (2) counting the number of occurrences
(
i.e.
number of pixels) falling into each bin; and (3) applying a color map showing
pixel density variations. A grid of 256
=
256 typically provides sufficient resolution
for exploring the information contained within a single image [25, 29]. A lower res-
olution is often used after the initial exploration stage, when the features of interest
are known and are used to develop classification of regression models, for example.
The feature exploration stage of MIA is illustrated in Figure 3.7 using another
froth image, again collected from a zinc flotation plant. After decomposing the froth
image using PCA, a 256
×
256
t
1
-
t
2
score density histogram is computed and dis-
played using a hot color map. The regions in black indicate that no pixels have that
particular combination of
t
1
and
t
2
values, whereas colors of increasing brightness,
from dark red to yellow and, eventually, to white indicate an increasing density of
pixels falling into specific areas of that plot. Then, the feature exploration proceeds
iteratively with (1) drawing of a mask (
i.e.
, ROIs) on the score density histogram,
followed by (2) identification of the pixels falling under that mask (
i.e.
, pixels having
t
1
and
t
2
values falling within the region segmented by the mask) and, (3) display-
ing these selected pixels on the original image using an easily distinguishable color.
Pixels falling within a certain neighborhood have similar features or spectral signa-
tures whereas those pixels falling in distinct regions of the density histogram will
capture different features.
This procedure is illustrated in Figure 3.7 with the blue and green masks. These
masks are initially of arbitrary size and shape, but are refined in later stages using
either automatic masking techniques [40] or using regression [32, 38]. After identi-
fying the pixels falling underneath each mask and overlaying them onto the original
image, one can visually interpret the color features extracted by each mask; the blue
and green masks, respectively, capture the sphalerite agglomerates (
i.e.
brown areas)
and the clear (black) windows on the bubbles associated with poor bubble loading.
Since MIA is performed on a pixel-by-pixel basis, these features are easily extracted
regardless of their size, shape, and location within the image. This is an important
advantage of the method since these features are stochastically distributed from one
image to the other.
Once the initial feature exploration stage of individual images is completed, the
subsequent steps usually involve collecting images under different operating con-
ditions according to some designed experiments (or not) for the purpose of on-line
monitoring or classification of features of interest, or for predicting some key pro-
cess variables. Comparing features of a set of
J
images requires building a common
PCA model for all the images within the set. This is accomplished by applying SVD
to the global kernel matrix
Z
×
J
∑
X
j
=
X
j
[29], which will result in a common set
j
=
1
of loading vectors
p
a
for all
J
images. The score vectors of the
j
th image
t
a
are then
computed as
t
a
X
j
p
a
. A common scaling range is also required for the scores
prior to displaying them as density histograms. This scaling range corresponds to
=
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