jonathan corriveau thesis advisor: dr. shreekanth mandayam
DESCRIPTION
Three-dimensional shape characterization for particle aggregates using multiple projective representations. Jonathan Corriveau Thesis Advisor: Dr. Shreekanth Mandayam Committee: Dr. Beena Sukumaran and Dr. Robi Polikar. Rowan University College of Engineering 201 Mullica Hill Road - PowerPoint PPT PresentationTRANSCRIPT
Three-dimensional shape characterization for particle aggregates using multiple projective representations
Jonathan CorriveauThesis Advisor: Dr. Shreekanth Mandayam
Committee: Dr. Beena Sukumaran and Dr. Robi Polikar
Rowan UniversityCollege of Engineering201 Mullica Hill RoadGlassboro, NJ 08028
(856) 256-5330http://engineering.rowan.edu/
Thursday, April 20, 2023
Outline
Introduction Objectives of Thesis Previous Work Approach Results Conclusions
Characterizing Shapes Shapes are described by names
Circle, Triangle, Rectangle, etc. Not possible for complicated shapes
Shapes need to be described by numbers Most shapes can be described by a set of
numbers Computers need numbers Similar shapes must have similar values Few as possible is desirable
Shapes
Rectangle Circle Triangle
Arbitrary Shape
Application
Computer Vision Face Recognition Fingerprint matching
Image 1 Image 2 Image 1 Image 2
Images Match Images Do Not Match
Phi 1: 1.058 1.2377Phi 2: 2.664 3.403Phi 3: 9.4284 7.8057Phi 4: 14.2453 13.702Phi 5: 29.8432 27.6783Phi 6: 16.0222 16.1285Phi 7: 29.4245 28.2324
Application
Character Recognition
Phi 1: 1.0292Phi 2: 2.5359Phi 3: 8.917Phi 4: 14.1381Phi 5: 29.2098Phi 6: 15.4456Phi 7: 29.1866
Descriptor DatabaseCharacter Descriptors a b
Motivation Soil Behavior
Strong relationship between stress-strain behavior of soils and the inherent characteristics of its individual particles
Inherent Particle Characteristics
Hardness, Specific Gravity Distribution
Shape and Angularity
Particle Size and Size Distribution
SEM Picture of Dry Sand
Aggregate Mixtures
Michigan Dune Sand#1 Dry Sand
Daytona Beach Sand Glass Beads
Motivation
Currently 2-D methods are not enough to characterize a soil mixture for discrete element model
Only behavior trends can be captured using 2-D models
3-D information allows a much more accurate model
3-D Shapes 3-D shapes are difficult to characterize as a set of numbers
Require sophisticated equipment Large databases of numbers to record the position of each
coordinate Aggregates of 3-D objects
A collection of 3-D particles must be characterized by a set of numbers
2-D Shapes
Computationally inexpensive Many methods already exist for characterizing
2-D shapes Can easily be implemented on a computer
with only digital images Question: How can 2-D methods help with
finding a 3-D solution?
Objectives of Thesis
Design automated algorithms that can estimate 3-D shape descriptors for particle aggregates using a statistical combination of 2-D shape descriptors from multiple 2-D projections.
Demonstrate consistency, separability and uniqueness of the 3-D shape-descriptor algorithm by exercising the method on a set of sand particle mixes.
Preliminary efforts towards the demonstration of the algorithm’s ability to accurately and repeatably construct composite 3-D shapes from multiple 2-D shape-descriptors.
Desirable Descriptor Qualities
Fundamental Qualities Uniqueness Parsimony Independent Invariance
Rotation Scale Translation
Original
Rotation Scale Translation
Additional Qualities
Reconstruction Allow for a shape to be constructed from
the descriptors Interpretation
Relate to some physical property Automatic Collection
Collection and evaluation automation Removes human error
Previous Work
Proponents Method Explanation
Sebestyn andBenson
“unrolling” a closed outline
The concept of creating a 1-D function from a 2-D boundary. Introduced by
Benson into the field of geology.
Hu2-D Invariant
Moments2-D moments that invariant to translation,
rotation, scale and reflection.
Ehrlich and Weinberg
Radius ExpansionIntroduced Fourier analysis for radius
expansion into sedimentology.
Medalia Equivalent EllipsesFits an ellipse to have similar properties
to the actual shape. Does not need outline.
Davis and Dexter Chord to PerimeterMeasures chord lengths between various
points along an outline.
Previous WorkProponents Method Explanation
Zahn and Roskies Angular BendIntroduced by Sebestyn, but made widely
known by Zahn and Roskies. Discretize an outline into a series of straight lines and angles
GranlundFourier
DescriptorsUses x+jy from the coordinates of an outline
to be analyzed by Fourier analysis.
Sadjadi and Hall3-D Invariant
Moments3-D moments that are invariant to translation,
rotation, and scale.
Garboczi, Martys, Saleh, and Livingston
Spherical Harmonics
A process similar to 3-D Fourier analysis, and requires 3-D information.
Sukumaran and Ashmawy
Shape and Angularity
Factor
Compares shapes to circles and measures their deviation. Uses a mean and standard deviation
of many particles to compare a mixes.
Radius Expansion
R1
R2R3
R4
Radius Expansion
x
y
R1()
R2()
Angular Bend
1 2
L1L2
L3
Complex Coordinates
y
x
(x1, y1)
Chord to Perimeter The covered perimeter length divided by total
perimeter determines the amount of irregularity Small ratio measures small irregularities Approaching one measures large irregularities
Chord Length
Perimeter Length
Equivalent Ellipses Two factors are calculated from ellipses
Anisometry – ratio of long to short axis of ellipse
Bulkiness – ratio of areas of figure and ellipse
Approach: Premise
2-D images of 3-D particles in an aggregate mix can be used to denote 2-D projections of a composite 3-D particle that represent the entire mixture
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2-D facets of 3-D particles in mix
3-D Descriptors2-D Descriptors from Mix 2-D Descriptors from Particle
2-D facets of Composite Particle
Composite 3-D “Reconstruction”
Overview of Approach
Particles
Orientation Every particle observed offers a different
angle of a composite particle Many different facets should be represented
by the images Regularity
Similar particles should have similar shapes
Aggregate Mixtures
Michigan Dune Sand#1 Dry Sand
Daytona Beach Sand Glass Beads
Similar shapes should have similar descriptors Find a distribution for each descriptor from all
particle images Calculate both the mean and variance that
characterize the distribution Allows a set of 2-D projections to represent a
composite 3-D object using a small set of numbers
Statistics
[S1, S2, S3, S4,…… SN]
[S1, S2, S3, S4,…….SN]s3
f(s3)
m3
2
1
3-D aggregate mixes can be characterized by a set of numbers
Multiple 2-D images can be used to construct a single composite 3-D object
Very little equipment required Microscope and Camera (data collection) Computer (analysis)
From 2-D to 3-D
Shape Characterization Methods
Complex Coordinate Fourier Analysis Allows random generation of projections
from 3-D descriptors Invariant Moments
Requires less computation, less preprocessing, and is more parsimonious, but does not allow projection generation
Fourier Analysis
Object must be described as a function Function should be periodic
Fourier Transform can be applied to analyze the frequencies Low Frequencies hold general shape
information, while high frequencies carry more detail
Effective for compression since reconstruction is possible with fewer values than the original
Fourier Descriptors
0 500 1000 1500 200050
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Fourier Descriptors
Descriptors
Near Zero Values
Moments
Statistical moments Normalized combinations of mean,
variance, and higher order moments Moments of similar objects should share
similar moment calculations 2-D moments evaluate the images
without having to extract the boundary Parsimonious (only 7 moments)
2-D Central Moments
Equation of 2-D moment is given as:
Central moments:
dxdyyxfyxm qppq ,
dxdyyxfyyxxqp
pq ,
For a digital image the discrete equation becomes:
Normalized Central Moments are defined as:
Moments
yxfyyxxqp
yxpq ,
00
pqpq where,where, 1
2
qp
Invariant Moments
02201
211
202202 4
20321
212303 33
20321
212304
20321
21230210303217 33
20321
21230123012305 33
20321
2123003210321 33
20321
2123003213012 33
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2-D facets of 3-D particles in mix
3-D Descriptors2-D Descriptors from Mix 2-D Descriptors from Particle
2-D facets of Composite Particle
Composite 3-D “Reconstruction”
Overview of Approach
Creation of Composite Particle
“Reconstruction” of 3-D Composite Particle
Three techniques were tested for constructing a 3-D composite particle using 2-D projections Extrusion Rotation into 3-D Tomographic
Extrusion Method
Rotation into 3-D Method
Tomographic Method
Implementation and Results
Experimental Setup Normalization and Results of Complex
Coordinate Fourier Analysis Invariant Moment Results Preliminary “reconstruction” results of
the different methods introduced
Experimental Setup
#1 Dry Sand
Daytona Beach Sand
Glass Bead
Optical Microscope, Digital Camera, and Computer
Data SamplesEquipment
Preprocessing of Images
Final Image Cleaned
InvertedBlack and WhiteOriginal Image
Obtaining Fourier Descriptors
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x Coordinates
y C
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Number of Points
|x+
jy|
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Coefficient Number
Am
plit
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Edge detection of the image Plot of coordinates extracted from image
Plotted as a 1-D Function FFT of 1-D Signal
100 150 200 250 300-200
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x Coordinates
y C
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rdin
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Reconstruction using all descriptors Reconstruction using 20 descriptors
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Coefficient Number
Am
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Coefficient Number
Am
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ud
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x Coordinates
y C
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Frequency Normalization Process
Original Image Half-Sized Image
Original Functions and FFTs
100 200 300 400
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Number of Points
X C
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Number of Points
X C
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Fourier Coefficient
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itud
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Fourier Coefficient
Ma
gn
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Original Image Half-Sized Image
After Normalization
0 50 100 150 200 250-1
-0.5
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1
Number of Points
No
rma
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d X
Co
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ina
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0 50 100 150 200 250-1
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Number of Points
No
rma
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d X
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Fourier Coefficient
Ma
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Fourier Coefficient
Ma
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Original Image Half-Sized Image
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2-D facets of 3-D particles in mix
3-D Descriptors2-D Descriptors from Mix 2-D Descriptors from Particle
2-D facets of Composite Particle
Composite 3-D “Reconstruction”
Overview of Approach
Statistics of Fourier Descriptors
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Descriptor Value
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#1 Dry Sand Standard Melt Sand
Daytona Beach Sand Michigan Dune Sand
Ellipsoid Model for 3-D Shape Characterization
Radius in X – Variance of First Descriptor
Radius in Y - Variance of Second Descriptor
Radius in Z – Variance of Third Descriptor
Center of Ellipsoid – 3-D Coordinate of Descriptor Means
x
z
y
Separability of Soil Mixes using Fourier Descriptors
Glass Bead
#1 Dry
Melt
Michigan Dune
Daytona Beach
Classification Effectiveness using Fourier Descriptors
Glass Bead
#1 Dry
Melt
Michigan Dune
Daytona Beach
Invariant Moments of Similar Images
Original Rotated and Resized
Invariant Moments of Similar Images
Invariant Moments
Image 1 Image 2 Difference
1 7.1164 7.1176 0.02%
2 15.2953 15.3027 0.05%
3 12.4116 12.1704 1.94%
4 25.1942 25.2073 0.05%
5 50.3498 49.0710 2.54%
6 32.9973 33.0157 0.06%
7 50.7640 50.7842 0.04%
Invariant Moments of Dissimilar Images
Image 1 Image 2
Invariant Moments of Dissimilar Images
Invariant Moments
Image 1 Image 2 Difference
1 7.1164 7.2694 2.15%
2 15.2953 16.7749 9.67%
3 12.4116 17.4857 40.88%
4 25.1942 26.9251 6.87%
5 50.3498 52.5509 4.37%
6 32.9973 35.5863 7.85%
7 50.7640 52.5528 3.52%
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2-D facets of 3-D particles in mix
3-D Descriptors2-D Descriptors from Mix 2-D Descriptors from Particle
2-D facets of Composite Particle
Composite 3-D “Reconstruction”
Overview of Approach
Statistics of Invariant Moment Descriptors
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Daytona Beach Sand Michigan Dune Sand
#1 Dry Sand Standard Melt Sand
Separability of Soil Mixes using Invariant Moment Descriptors
#1 Dry
Melt
Michigan Dune
Daytona Beach
Classification Effectiveness using Invariant Moment Descriptors
#1 Dry
Melt
Michigan Dune
Daytona Beach
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2-D facets of 3-D particles in mix
3-D Descriptors2-D Descriptors from Mix 2-D Descriptors from Particle
2-D facets of Composite Particle
Composite 3-D “Reconstruction”
Overview of Approach
Reconstruction of Projections from 3-D Descriptors
Original Image Reconstructed Image
Generation of Random Projections from 3-D Descriptors
Separability of Soil Mixes using Randomly Generated Projections
Comparison between Original and Generated Projections
1-Original
2-Generated
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2-D facets of 3-D particles in mix
3-D Descriptors2-D Descriptors from Mix 2-D Descriptors from Particle
2-D facets of Composite Particle
Composite 3-D “Reconstruction”
Overview of Approach
Extrusion Method
2nd Projection of Dry Sand
1st Projection of Dry Sand
3rd Projection of Dry Sand
All Projections in 3-D Space
Implementation of Extrusion Method on Dry Sand
Projections after Extrusion
Final “Reconstruction”
Effectiveness of Extrusion “Reconstructed” Composite Particle
#1 Dry
Melt
Michigan Dune
Daytona Beach
Rotate Into 3–D Method for Dry Sand
Effectiveness of Rotation into 3-D “Reconstructed” Composite Particle
#1 Dry
Melt
Michigan Dune
Daytona Beach
Tomographic Method
Effectiveness of Tomographic “Reconstructed” Composite Particle
#1 Dry
Melt
Michigan Dune
Daytona Beach
Results of Dry Sand “Reconstruction”
Reconstruction Method
Inter-ellipsoid Distance Percentages Dry Melt Daytona Beach Michigan Dune
Extrusion 31% 40% 100% 46%
3-D Rotation 60% 73% 100% 17%
Tomographic 45% 33% 100% 85%
Distance
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2-D facets of 3-D particles in mix
3-D Descriptors2-D Descriptors from Mix 2-D Descriptors from Particle
2-D facets of Composite Particle
Composite 3-D “Reconstruction”
Conclusion
Summary of Accomplishments Development of automated algorithms that can
estimate 3-D shape descriptors for particle aggregates Statistical combination of 2-D shape descriptors from multiple
2-D projections
Database containing a library of 2-D digital images for 5 aggregate mixtures
PCA and ellipsoid model to show consistency, separability and uniqueness of the algorithm
Composite 3-D shapes from multiple 2-D projections. Extrusion, Rotation and Tomographic reconstruction
Conclusions
Dissimilar soil mixes can be separated using the descriptor algorithms
Generation of random projections from the Fourier descriptors proves to be effective
Construction of a 3-D composite particle using a collection of 2-D projections appears feasible
Recommendations for Future Work
The optimal number and value of descriptors can be found, which allows the greatest separability
More work on Reconstruction Methods Extrusion – use more projections on more axes Tomographic – Rotate more images about
multiple axes and combine objects Apply composite particles created to a
discrete element model Algorithms can be applied to other
application areas (i.e. ink toner, industrial)
Acknowledgements
National Science Foundation, Division of Civil and Mechanical Systems, Geomechanics and Geotechnic Systems Program, Award #0324437
Dr. Shreekanth Mandayam, Dr. Beena Sukumaran, and Dr. Robi Polikar
Michael Kim and Scott Papson