Jonathan Corriveau Thesis Advisor: Dr. Shreekanth Mandayam

1. 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) 256-5330 http://engineering.rowan.edu/ Monday, October 27, 2014

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 Do Not Match Images Match

6. Application • Character Recognition Descriptor Database CharacterDescriptorsa b Phi 1: 1.0292 Phi 2: 2.5359 Phi 3: 8.917 Phi 4: 14.1381 Phi 5: 29.2098 Phi 6: 15.4456 Phi 7: 29.1866 Phi 1: 1.058 1.2377 Phi 2: 2.664 3.403 Phi 3: 9.4284 7.8057 Phi 4: 14.2453 13.702 Phi 5: 29.8432 27.6783 Phi 6: 16.0222 16.1285 Phi 7: 29.4245 28.2324

7. Inherent Particle Characteristics Hardness, Specific Gravity Distribution Shape and Angularity Particle Size and Size Distribution Motivation • Soil Behavior • Strong relationship between stress-strain behavior of soils and the inherent characteristics of its individual particles 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 • 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

16. Previous Work

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 ChordLength 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 • 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. [S1, S2, S3, S4,…… SN] [S1, S2, S3, S4,…….SN] f(s3) m3 s3 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

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: • Central moments:

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. 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