Environmental Engineering Reference
In-Depth Information
To illustrate how to interpret the score and loading vectors in the context of digital
image analysis, consider the example presented in Figure 3.6 where a RGB froth
image taken from a zinc (
i.e.
sphalerite) flotation plant is decomposed using PCA.
Only the first two components are considered in this example. This figure shows
the
T
1
and
T
2
score images obtained after scaling score values between 0 and 255
according to the following equation:
Round
255
T
a
(
x
,
y
)−
min
(
T
a
(
x
,
y
))
T
a
(
x
,
y
)=
))
×
,
(3.6)
max
(
T
a
(
x
,
y
))−
min
(
T
a
(
x
,
y
where
x
and
y
correspond to the pixel location in both spatial directions. The cor-
responding loading vectors (
p
1
and
p
2
) containing the weights of the linear com-
binations of the three color channels explaining most of the color variations across
the RGB image
X
are also shown in this figure. The first PC clearly capture varia-
tions in gray colors across the froth (
i.e.
, the dominant color) since all
p
1
weights
have a similar magnitude and the same sign. The darker regions in the original froth
image correspond to pixels having low RGB intensities. Taking a linear combina-
tion of these low RGB intensities according to the
p
1
weights will result in small
negative
t
1
values (
i.e.
, close to zero). On the other hand, light gray regions corre-
spond to high RGB values and, therefore, to strong negative
t
1
values. Once scaled
between 0 (min(
t
1
)) and 255 (max(
t
1
)) and displayed as a
T
1
image, darker regions
in the froth image will appear as light gray to white, whereas clearer regions will
appear as dark gray to black in the
T
1
image. Hence, the
T
1
image captures the gray
color contrasts within the froth image accounting for 99
.
88% of the color variations
(
i.e.
, RGB intensities) within the image. On the other hand, the second PC explains
only an additional 0.085% of the variance of
X
, but captures a contrast between the
red and the blue channels (see
p
2
weights) revealing some interesting features. The
opposite signs of the R and B weights indicate that these two intensities vary in a
negatively correlated fashion (
i.e.
, when R increases B decreases and
vice versa
).
This allows capture of the brownish areas on the froth, corresponding to sphalerite
particle agglomerates, which could be used to estimate froth grade. This PC also
captures the blueish colors, but these are most probably caused by light reflections
from bubbles. The unexplained variance of
X
after two PCs (
i.e.
,
T
3
) essentially
corresponds to noise and are left as residuals
E
.
However, the most useful approach for exploring image features involves inves-
tigating the clustering patterns of the pixels obtained by plotting the score vectors
t
a
against each other (see the
t
1
-
t
2
score scatter plot in Figure 3.6), as typically done
when analyzing large historical process databases. The only difference between tra-
ditional data and image analysis lies in the fact that, in the latter case, the score
scatter plots are visually enhanced by displaying them as 2-D density histograms
due to the very large number of observations projected into this plot. Indeed, there
are as many observations (
i.e.
, points) in the score plots as the number of pixels
forming the image, that is 516
178536 observations in the
t
1
-
t
2
plot for the
image shown in Figure 3.5. It would therefore be very difficult to see any cluster
in such a plot, especially considering that 516
×
346
=
×
346 pixels is a very low resolution
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