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General Linear Cameras with Finite Aperture

General Linear Cameras with Finite Aperture. Andrew Adams and Marc Levoy Stanford University. Ray Space. Slices of Ray Space. Pushbroom Cross Slit General Linear Cameras. Yu and McMillan ‘04. Román et al. ‘04. Projections of Ray Space. Plenoptic Cameras Camera Arrays

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General Linear Cameras with Finite Aperture

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  1. General Linear Cameras with Finite Aperture Andrew Adams and Marc Levoy Stanford University

  2. Ray Space

  3. Slices of Ray Space • Pushbroom • Cross Slit • General Linear Cameras Yu and McMillan ‘04 Román et al. ‘04

  4. Projections of Ray Space • Plenoptic Cameras • Camera Arrays • Regular Cameras Ng et al. ‘04 Leica Apo-Summicron-M Wilburn et al. ‘05

  5. What is this paper?

  6. What is this paper? • An intuitive reformulation of general linear cameras in terms of eigenvectors

  7. What is this paper? • An intuitive reformulation of general linear cameras in terms of eigenvectors • An analogous description of focus

  8. What is this paper? • An intuitive reformulation of general linear cameras in terms of eigenvectors • An analogous description of focus • A theoretical framework for understanding and characterizing linear slices and integral projections of ray space

  9. Slices of Ray Space • Perspective View • Image(x, y) = L(x, y, 0, 0)

  10. Slices of Ray Space • Orthographic View • Image(x, y) = L(x, y, x, y)

  11. Slices of Ray Space • Image(x, y) = L(x, y, P(x, y)) • P determines perspective • Let’s assume P is linear

  12. Slices of Ray Space P

  13. Slices of Ray Space

  14. Slices of Ray Space • Rays meet when: ((1-z)P + zI) is low rank • Substitute b = z/(z-1): ((1-z)P + zI) = (1-z)(P – bI) • Rays meet when: (P – bI) is low rank

  15. Slices of Ray Space • 0 < b1 = b2 < 1

  16. Slices of Ray Space • b1 = b2 < 0

  17. Slices of Ray Space • b1 = b2 = 1

  18. Slices of Ray Space • b1 = b2 > 1

  19. Slices of Ray Space • b1 != b2

  20. Slices of Ray Space • b1 != b2 = 1

  21. Slices of Ray Space • b1 = b2 != 1, deficient eigenspace

  22. Slices of Ray Space • b1 = b2 = 1, deficient eigenspace

  23. Slices of Ray Space • b1, b2 complex

  24. Slices of Ray Space

  25. Slices of Ray Space Real Eigenvalues Complex Conjugate Eigenvalues

  26. Slices of Ray Space Real Eigenvalues Equal Eigenvalues Complex Conjugate Eigenvalues

  27. Slices of Ray Space Real Eigenvalues Equal Eigenvalues Equal Eigenvalues, 2D Eigenspace Complex Conjugate Eigenvalues

  28. Slices of Ray Space One slit at infinity Real Eigenvalues Equal Eigenvalues Equal Eigenvalues, 2D Eigenspace Complex Conjugate Eigenvalues

  29. Projections of Ray Space

  30. Projections of Ray Space

  31. Projections of Ray Space

  32. Projections of Ray Space • Rays Integrated at (x, y) = (0, 0): F

  33. Projections of Ray Space • Rays meet when: ((1-z)I + zF) is low rank • Substitute b = (z-1)/z: ((1-z)I + zF) = z(F – bI) • Rays meet when: (F – bI) is low rank

  34. Projections of Ray Space • 0 < b1 = b2 < 1

  35. Projections of Ray Space • 0 < b1 = b2 < 1

  36. Projections of Ray Space • b1 = b2 < 0

  37. Projections of Ray Space • b1 = b2 < 0

  38. Projections of Ray Space • b1 = b2 = 1

  39. Projections of Ray Space • b1 = b2 = 1

  40. Projections of Ray Space • b1 = b2 > 1

  41. Projections of Ray Space • b1 != b2

  42. Projections of Ray Space • b1 != b2

  43. Projections of Ray Space • b1 != b2

  44. Projections of Ray Space • b1 != b2 = 1

  45. Projections of Ray Space • b1 != b2 = 1

  46. Projections of Ray Space • b1 != b2 = 1

  47. Projections of Ray Space • b1 = b2 != 1, deficient eigenspace

  48. Projections of Ray Space • b1 = b2 != 1, deficient eigenspace

  49. Projections of Ray Space • b1 = b2 = 1, deficient eigenspace

  50. Projections of Ray Space • b1 = b2 = 1, deficient eigenspace

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