Computer Vision, Computation, and Geometry

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A slide deck for an overview of my research interests.

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  • digital pictures are messy
    object boundaries are not well defined
  • digital pictures are messy
    object boundaries are not well defined
  • digital pictures are messy
    object boundaries are not well defined
  • big problems in computer vision
  • differential calculus
  • differential calculus
  • differential calculus
  • there are problems when the eigenvalues are equal or vanish

    (I put these here because a sophomore mathematics major can understand them)
  • but mostly I just retool myself, learn new mathematical tools
  • but mostly I just retool myself, learn new mathematical tools
  • but mostly I just retool myself, learn new mathematical tools
  • Computer Vision, Computation, and Geometry

    1. 1. Visual Perception, Computation, and Geometry Jason Miller Associate Professor of Mathematics Truman State University 12 September 2009
    2. 2. Outline
    3. 3. Outline • a bit about me
    4. 4. Outline • a bit about me • computers & sight
    5. 5. Outline • a bit about me • computers & sight • medical imaging and medialness
    6. 6. Outline • a bit about me • computers & sight • medical imaging and medialness • relative critical sets
    7. 7. Outline • a bit about me • computers & sight • medical imaging and medialness • relative critical sets • subsequent work
    8. 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. 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. 10. Computers & Sight
    11. 11. Computers & Sight Semi-Autonomous Vehicles
    12. 12. Computers & Sight Semi-Autonomous Vehicles Descriptive and Diagnostic Medicine
    13. 13. Computers & Sight Semi-Autonomous Vehicles Descriptive and Diagnostic Medicine Automatic Annotation of Digital Content
    14. 14. Computers & Sight Semi-Autonomous Vehicles Descriptive and Diagnostic Medicine Automatic Annotation of Face Recognition, Digital Content Motion Tracking, etc.
    15. 15. Computers & Sight The secret is …
    16. 16. Computers & Sight The secret is … They Suck at it!
    17. 17. Computers & Sight The secret is … They Suck at it! (they have no natural talent for sight)
    18. 18. Example: Captchas
    19. 19. Computers & Sight
    20. 20. Computers & Sight
    21. 21. Computers & Sight
    22. 22. Computers & Sight
    23. 23. Image Processing • Challenges: Segmentation and Registration of Images • Edge-based methods • Medialness-based methods
    24. 24. Medial Axis
    25. 25. Medial Axis
    26. 26. Medial Axis
    27. 27. Medial Axis
    28. 28. Medial Axis
    29. 29. Medial Axis
    30. 30. Medial Axis
    31. 31. Medial Axis
    32. 32. Medial Axis
    33. 33. Medial Axis th wid
    34. 34. Image Processing
    35. 35. Image Processing
    36. 36. Image Processing
    37. 37. Image Processing
    38. 38. Image Processing
    39. 39. Image Processing • Digital images are collections of pixels • Each pixel has an intensity 528 x 525 pixels intensities: 0 ≤ I ≤ 255
    40. 40. Pixel intensity function
    41. 41. Pixel intensity function
    42. 42. Pixel intensity function
    43. 43. Pixel intensity function
    44. 44. Pixel intensity function
    45. 45. Pixel intensity function
    46. 46. Pixel intensity function nsity values Inte
    47. 47. Image shapes
    48. 48. Image function shapes geometry
    49. 49. Image shapes ←→ function geometry
    50. 50. 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”
    51. 51. Real Mathematical World World Assumptions Mathematical about Phenomena Model Logical Consequences Real (Analyze Model) Data
    52. 52. Real Mathematical World World translate Assumptions Mathematical about Phenomena Model Logical Consequences Real (Analyze Model) Data
    53. 53. Real Mathematical World World translate Assumptions Mathematical about Phenomena Model Logical Consequences Real (Analyze Model) Data
    54. 54. Real Mathematical World World translate Assumptions Mathematical about Phenomena Model Logical Consequences Real (Analyze Model) Data compare
    55. 55. Real Mathematical World World translate Assumptions Mathematical about Phenomena Model adjust assumptions to improve Logical Consequences Real (Analyze Model) Data compare
    56. 56. 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
    57. 57. Image shapes
    58. 58. Image function shapes geometry
    59. 59. Image shapes ←→ function geometry
    60. 60. 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
    61. 61. 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
    62. 62. 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)
    63. 63. 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)
    64. 64. 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)
    65. 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) 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

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