Jason Miller is an associate professor of mathematics who studies visual perception and computation using techniques from geometry and topology. He gave a talk outlining his work using medial axes and relative critical sets to analyze medical images and segment objects. This involves translating assumptions about images into mathematical models and comparing implications to real data. His subsequent work has applied these methods to projects in biology and medical imaging.

1. Visual Perception,
Computation,
and Geometry
Jason Miller
Associate Professor of Mathematics
Truman State University
12 September 2009

2. Outline

3. Outline
• a bit about me

4. Outline
• a bit about me
• computers & sight

5. Outline
• a bit about me
• computers & sight
• medical imaging and medialness

6. Outline
• a bit about me
• computers & sight
• medical imaging and medialness
• relative critical sets

7. Outline
• a bit about me
• computers & sight
• medical imaging and medialness
• relative critical sets
• subsequent work

8. Me
• B.A. in math from small, private liberal arts
college
• Ph.D. in mathematics from University of
North Carolina
• area = differentiable topology & singularity
theory of René Thom
• “Relative Critical Sets in n-Space and their
application to Image Analysis.”

9. The miracle of appropriateness of the language of
mathematics for the formulation of the laws of [science] is a
wonderful gift which we neither understand nor deserve.
We should be grateful for it, and hope that it will remain
valid for future research, and that it will extend, for better
or for worse, to our pleasure even though perhaps also to
our bafflement, to wide branches of learning.
— Eugene Wigner, The Unreasonable
Effectiveness of Mathematics

10. Computers & Sight

11. Computers & Sight
Semi-Autonomous Vehicles

12. Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine

13. Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine
Automatic Annotation of
Digital Content

14. Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine
Automatic Annotation of Face Recognition,
Digital Content Motion Tracking, etc.

15. Computers & Sight
The secret is …

16. Computers & Sight
The secret is …
They Suck at it!

17. Computers & Sight
The secret is …
They Suck at it!
(they have no natural talent for sight)

18. Example: Captchas

19. Computers & Sight

20. Computers & Sight

21. Computers & Sight

22. Computers & Sight

23. Image Processing
• Challenges:
Segmentation and
Registration of Images
• Edge-based methods
• Medialness-based
methods

24. Medial Axis

25. Medial Axis

26. Medial Axis

27. Medial Axis

28. Medial Axis

29. Medial Axis

30. Medial Axis

31. Medial Axis

32. Medial Axis

33. Medial Axis
th
wid

34. Image Processing

35. Image Processing

36. Image Processing

37. Image Processing

38. Image Processing

39. Image Processing
• Digital images are
collections of pixels
• Each pixel has an
intensity
528 x 525 pixels
intensities: 0 ≤ I ≤ 255

40. Pixel intensity function

41. Pixel intensity function

42. Pixel intensity function

43. Pixel intensity function

44. Pixel intensity function

45. Pixel intensity function

46. Pixel intensity function
nsity values
Inte

48. Image
shapes

49. Image function
shapes geometry

50. Image
shapes
←→ function
geometry

51. Backstory: Why Me?
• high-powered computer science research group!
• they had algorithms computing medial axes of objects in
medical images
• dogged by some anomalous unexpected numerical
problems
• my advisor: “let’s figure out what should be happening”

52. Real Mathematical
World World
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data

53. Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data

54. Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data

55. Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data compare

56. Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
adjust assumptions
to improve
Logical Consequences
Real (Analyze Model)
Data compare

57. Relative Critical Sets
• They extended the concept of local extrema where
I=0
(vanishing derivative) to a higher dimensional set of
points.
• Let ei be the eigenvectors of the matrix of second
partials of I , and λi ≤ λi+1 be the eigenvalues.
I · ei = 0 for i < n
λn−1 < 0

59. Image
shapes

60. Image function
shapes geometry

61. Image
shapes
←→ function
geometry

62. Relative Critical Sets
• Used the following techniques to prove a
structure theorem for the CS’s group’s
medial axes
• wavelet theory (scale-space theory)
• Lie group actions
• transversality theorems
• semi-algebraic geometry
• combinatorics

63. Relative Critical Sets
• Used the following techniques to prove a
structure theorem for the CS’s group’s
medial axes
• wavelet theory (scale-space theory)
abstract
• Lie group actions mathematics in
service of
• transversality theorems
applied science
• semi-algebraic geometry
• combinatorics

64. Subsequent Work
• Undergraduate Research Project on
computing relative critical sets
• Applied wavelets to bat echolocation project
with Scott Burt (Biology)
• Use medialness methods in vascular network
project with Rob Baer (ATSU)

65. Subsequent Work
• Undergraduate Research Project on
computing relative critical sets ramming
Mathem atica prog
• Applied wavelets to bat echolocation project
with Scott Burt (Biology)
• Use medialness methods in vascular network
project with Rob Baer (ATSU)

66. Subsequent Work
• Undergraduate Research Project on
computing relative critical sets ramming
Mathem atica prog
• Applied wavelets to bat echolocation project
with Scott Burt (Biology) assific ation and
sta tistical cl ethods
cluster m
• Use medialness methods in vascular network
project with Rob Baer (ATSU)

67. Subsequent Work
• Undergraduate Research Project on
computing relative critical sets ramming
Mathem atica prog
• Applied wavelets to bat echolocation project
with Scott Burt (Biology) assific ation and
sta tistical cl ethods
cluster m
• Use medialness methods in vascular network
project with Rob Baer (ATSU)
grap h theor y
ramming
M atlab prog