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A stochastic approach for measuring bubble
size distribution via image analysis
A solu t ion to the bubble c lusters p rob lem
W. Kracht, X. Emery, A. Egaa
ALGES laboratory, Mining Engineering Department
Universidad de Chile
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Introduction
Bubble size distribution measurement
Classical image analysis
Stochastic approach
Geometric covariogram
Results on simulated images
Conclusions
Outline
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Bubble size estimation, e.g., drift flux analysis.
It is not clear what is the role of chemistry on the model
UCT: bubbles sampled with a capillary.
Underestimation of BSD
Bubble Size Distribution (BSD) measurement
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Sampling for image technique.
Bubbles sampled from the pulp and imaged in a viewing
chamber (bubble viewer)
Bubble Size Distribution (BSD) measurement
CameraViewing chamber
Lamp
Sampling tube
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Image converted from 24-bit RGB to 8-bit gray scale and
later to a binary image.
Classical Image analysis: pre-process
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Object recognition: each set of adjacent black pixels is
recognized as an object and four cases are defined:
Classical Image analysis: segmentation
Bubble Clusters
Bubble Clusters(edge)
Single Bubbles
Single Bubbles
(edge)
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Three alternatives:
Watershed
Manual processing
Assume they are representative and neglect them
Cluster processing
Watershed Manual processing Neglect them
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The images can be modelled as a stochastic process:
A set of circular objects, with varying diameters, disposed
randomly over the area of the picture (Boolean model)
Stochastic approach
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The geometric covariogram of an object has information
of its geometric properties
Geometric covariogram of a disc (2D-bubble)
Geometric covariogram of a disc (800 pixels)
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The geometric covariogram of a distribution of discs or
bubbles is represented by the average of single
covariograms
Geometric covariogram of a distribution
BSD (5% 200p, 20% 400p, 5% 800p)BSD (100% 800p)
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The geometric covariogram of an image can be
estimated directly from the image before segmentation.
Instead of calculating the covariogram of single objects,
one calculates the probability that two points distant by a
distance h, simultaneously belong to the background of
the image.
This approach takes much shorter than classical image
analysis (1 to 10% of the time for segmentation-
watershed).
Extracting geometric properties of an image
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100 bubble images were simulated as a Boolean model
with sizes following a log-normal distribution with d10 =
1.0 mm and d32 = 1.4 mm.
Simulated images
0
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0 1 2 3 4
Diameter, mm
N
umberfrequency,
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The 100 images are processed by both methods:
classical image analysis, neglecting bubble clusters
(left); and the stochastic approach (right).
Results
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12
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0 1 2 3 4
Diameter, mm
Numberfrequency,%
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0 1 2 3
Diameter, mm
Numberfrequency,%
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Neglecting bubble clusters makes large bubbles to be
underestimated.
Indeed, in the simulation, d10 and d32 are underestimated
by 10 and 14% respectively.
The bubble surface area flux is then overestimated by
16-17%.
Results
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Both, bubble clusters and bubbles truncated by the edge
of the image should be considered.
Not considering them leads to bubble size
underestimation.
It is possible to determine the bubble size distribution
without applying classical image analysis.
If bubble clusters are not a problem, the sensor (bubble
viewer) geometry can be modified: there is no need for
the sloped window.
Conclusions
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ALGES laboratory, where the technique is being
developed.
AMTC (Advanced Mining Technology Center) for
supporting this investigation.
Acknowledgements
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A stochastic approach for measuring bubble
size distribution via image analysis
A solu t ion to the bubble c lusters p rob lem
W. Kracht, X. Emery, A. Egaa
ALGES laboratory, Mining Engineering Department
Universidad de Chile