1. College of Engineering
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
Glassboro, NJ 08028
(856)
Monday, April 17, 2017

2. Outline Introduction Objectives of Thesis Previous Work Approach
Results
Conclusions

3. 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

4. Shapes
Rectangle
Circle
Triangle
Arbitrary Shape

5. Application Computer Vision Face Recognition Fingerprint matching
Image 1
Image 2
Image 1
Image 2
Images Match
Images Do Not Match

6. Application Character Recognition Descriptor Database
Character Descriptors a b
Phi 1:
Phi 2:
Phi 3:
Phi 4:
Phi 5:
Phi 6:
Phi 7:
Phi 1:
Phi 2:
Phi 3:
Phi 4:
Phi 5:
Phi 6:
Phi 7:

7. 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

8. Aggregate Mixtures #1 Dry Sand Michigan Dune Sand Daytona Beach Sand
Glass Beads

9. 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

10. 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

11. 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?

12. 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.

13. Desirable Descriptor Qualities
Fundamental Qualities
Uniqueness
Parsimony
Independent
Invariance
Rotation
Scale
Translation
Original
Rotation
Scale
Translation

14. Additional Qualities Reconstruction Interpretation
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

15. Previous Work Proponents Method Explanation Sebestyn and Benson
“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. Hu
2-D Invariant
Moments
2-D moments that invariant to translation, rotation, scale and reflection.
Ehrlich and
Weinberg
Radius Expansion
Introduced Fourier analysis for radius expansion into sedimentology.
Medalia
Equivalent Ellipses
Fits an ellipse to have similar properties to the actual shape. Does not need outline.
Davis and Dexter
Chord to Perimeter
Measures chord lengths between various points along an outline.

16. Previous Work Proponents Method Explanation Zahn and Roskies
Angular Bend
Introduced by Sebestyn, but made widely known by Zahn and Roskies. Discretize an outline into a series of straight lines and angles
Granlund
Fourier Descriptors
Uses x+jy from the coordinates of an outline to be analyzed by Fourier analysis.
Sadjadi and Hall
3-D Invariant Moments
3-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.

17. Radius Expansion R3 R2 R1 R4

18. Radius Expansion y
R2()
R1()  x

19. Angular Bend L2 L1 1 L3 2

20. Complex Coordinates y
(x1, y1) x

21. 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

22. 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

23. 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

24. Overview of Approach 2-D facets of 3-D particles in mix
Composite 3-D “Reconstruction”
2-D facets of Composite Particle
2-D Descriptors from Mix
3-D Descriptors
2-D Descriptors from Particle

25. Particles Orientation Regularity
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

26. Aggregate Mixtures #1 Dry Sand Michigan Dune Sand Daytona Beach Sand
Glass Beads

27. Statistics 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
[S1, S2, S3, S4,…… SN]
[S1, S2, S3, S4,…….SN] s3
f(s3) m3

28. From 2-D to 3-D
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)

29. 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

30. 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

31. Fourier Descriptors

32. Fourier Descriptors
Descriptors
Near Zero Values

33. 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)

34. 2-D Central Moments Equation of 2-D moment is given as:

35. Moments For a digital image the discrete equation becomes:
Normalized Central Moments are defined as:
where,

36. Invariant Moments

37. Overview of Approach 2-D facets of 3-D particles in mix
Composite 3-D “Reconstruction”
2-D facets of Composite Particle
2-D Descriptors from Mix
3-D Descriptors
2-D Descriptors from Particle

38. Creation of Composite Particle

39. “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

40. Extrusion Method

41. Rotation into 3-D Method

42. Tomographic Method

43. Implementation and Results
Experimental Setup
Normalization and Results of Complex Coordinate Fourier Analysis
Invariant Moment Results
Preliminary “reconstruction” results of the different methods introduced

44. Optical Microscope, Digital Camera, and Computer
Experimental Setup
Equipment
Data Samples
Optical Microscope, Digital Camera, and Computer
#1 Dry Sand
Daytona Beach Sand
Glass Bead

45. Preprocessing of Images
Original Image
Black and White
Inverted
Final Image
Cleaned

46. Obtaining Fourier Descriptors
Edge detection of the image
Plot of coordinates extracted from image
FFT of 1-D Signal
Plotted as a 1-D Function

47. Reconstruction of 2-D Projections
Reconstruction using all descriptors
Reconstruction using 20 descriptors

48. Frequency Normalization Process
Original Image
Half-Sized Image

49. Original Functions and FFTs
Original Image
Half-Sized Image

50. After Normalization
Original Image
Half-Sized Image

51. Overview of Approach 2-D facets of 3-D particles in mix
Composite 3-D “Reconstruction”
2-D facets of Composite Particle
2-D Descriptors from Mix
3-D Descriptors
2-D Descriptors from Particle

52. Statistics of Fourier Descriptors
#1 Dry Sand
Standard Melt Sand
Daytona Beach Sand
Michigan Dune Sand

53. Ellipsoid Model for 3-D Shape Characterization
Center of Ellipsoid –
3-D Coordinate of Descriptor Means
Radius in X – Variance of
First Descriptor
Radius in Y - Variance of Second Descriptor x z y
Radius in Z – Variance of
Third Descriptor

54. Separability of Soil Mixes using Fourier Descriptors
#1 Dry
Melt
Daytona Beach
Glass Bead
Michigan Dune

55. Classification Effectiveness using Fourier Descriptors
#1 Dry
Melt
Daytona Beach
Glass Bead
Michigan Dune

56. Invariant Moments of Similar Images
Original
Rotated and Resized

57. Invariant Moments of Similar Images
Difference 1
7.1164
7.1176
0.02% 2
0.05% 3
1.94% 4 5
2.54% 6
0.06% 7
0.04%

58. Invariant Moments of Dissimilar Images

59. Invariant Moments of Dissimilar Images
Difference 1
7.1164
7.2694
2.15% 2
9.67% 3
40.88% 4
6.87% 5
4.37% 6
7.85% 7
3.52%

60. Overview of Approach 2-D facets of 3-D particles in mix
Composite 3-D “Reconstruction”
2-D facets of Composite Particle
2-D Descriptors from Mix
3-D Descriptors
2-D Descriptors from Particle

61. Statistics of Invariant Moment Descriptors
#1 Dry Sand
Standard Melt Sand
Daytona Beach Sand
Michigan Dune Sand

62. Separability of Soil Mixes using Invariant Moment Descriptors
#1 Dry
Melt
Daytona Beach
Michigan Dune

63. Classification Effectiveness using Invariant Moment Descriptors
#1 Dry
Melt
Daytona Beach
Michigan Dune

64. Overview of Approach 2-D facets of 3-D particles in mix
Composite 3-D “Reconstruction”
2-D facets of Composite Particle
2-D Descriptors from Mix
3-D Descriptors
2-D Descriptors from Particle

65. Reconstruction of Projections from 3-D Descriptors
Original Image
Reconstructed Image

66. Generation of Random Projections from 3-D Descriptors

67. Separability of Soil Mixes using Randomly Generated Projections

68. Comparison between Original and Generated Projections

69. Overview of Approach 2-D facets of 3-D particles in mix
Composite 3-D “Reconstruction”
2-D facets of Composite Particle
2-D Descriptors from Mix
3-D Descriptors
2-D Descriptors from Particle

70. Extrusion Method All Projections in 3-D Space
1st Projection of Dry Sand
2nd Projection of Dry Sand
3rd Projection of Dry Sand

71. Implementation of Extrusion Method on Dry Sand
Projections after Extrusion
Final “Reconstruction”

72. Effectiveness of Extrusion “Reconstructed” Composite Particle
#1 Dry
Melt
Daytona Beach
Michigan Dune

73. Rotate Into 3–D Method for Dry Sand

74. Effectiveness of Rotation into 3-D “Reconstructed” Composite Particle
#1 Dry
Melt
Daytona Beach
Michigan Dune

75. Tomographic Method

76. Effectiveness of Tomographic “Reconstructed” Composite Particle
#1 Dry
Melt
Daytona Beach
Michigan Dune

77. Results of Dry Sand “Reconstruction”
Distance
Reconstruction Method
Inter-ellipsoid Distance Percentages
Dry Melt Daytona Beach Michigan Dune
Extrusion
31%
40%
100%
46%
3-D Rotation
60%
73%
17%
Tomographic
45%
33%
85%

78. Conclusion 2-D facets of 3-D particles in mix
Composite 3-D “Reconstruction”
2-D facets of Composite Particle
2-D Descriptors from Mix
3-D Descriptors
2-D Descriptors from Particle

79. 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

80. 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

81. 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)

82. Acknowledgements
National Science Foundation, Division of Civil and Mechanical Systems, Geomechanics and Geotechnic Systems Program, Award #
Dr. Shreekanth Mandayam, Dr. Beena Sukumaran, and Dr. Robi Polikar
Michael Kim and Scott Papson