Computer Vision, Computation, and Geometry
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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.
![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](https://image.slidesharecdn.com/090912facresconftalk-090912155633-phpapp02/85/Computer-Vision-Computation-and-Geometry-9-320.jpg)
Computer Vision, Computation, and Geometry
- 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
- 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)
- 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
- 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
Editor's Notes
- #21: digital pictures are messy object boundaries are not well defined
- #22: digital pictures are messy object boundaries are not well defined
- #23: digital pictures are messy object boundaries are not well defined
- #24: big problems in computer vision
- #56: differential calculus
- #57: differential calculus
- #58: differential calculus
- #67: there are problems when the eigenvalues are equal or vanish (I put these here because a sophomore mathematics major can understand them)
- #72: but mostly I just retool myself, learn new mathematical tools
- #73: but mostly I just retool myself, learn new mathematical tools
- #74: but mostly I just retool myself, learn new mathematical tools