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 Wilburn et al. ‘05 Leica Apo-Summicron-M

5. What is this paper?

6. 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

14. 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 < b 1 = b 2 < 1

16. Slices of Ray Space b 1 = b 2 < 0

17. Slices of Ray Space b 1 = b 2 = 1

18. Slices of Ray Space b 1 = b 2 > 1

19. Slices of Ray Space b 1 != b 2

20. Slices of Ray Space b 1 != b 2 = 1

21. Slices of Ray Space b 1 = b 2 != 1, deficient eigenspace

22. Slices of Ray Space b 1 = b 2 = 1, deficient eigenspace

23. Slices of Ray Space b 1, b 2 complex

24. Slices of Ray Space

25. Real Eigenvalues Complex Conjugate Eigenvalues

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

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

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

29. Projections of Ray Space

32. 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 < b 1 = b 2 < 1

35. Projections of Ray Space 0 < b 1 = b 2 < 1

36. Projections of Ray Space b 1 = b 2 < 0

37. Projections of Ray Space b 1 = b 2 < 0

38. Projections of Ray Space b 1 = b 2 = 1

39. Projections of Ray Space b 1 = b 2 = 1

40. Projections of Ray Space b 1 = b 2 > 1

41. Projections of Ray Space b 1 != b 2

42. Projections of Ray Space b 1 != b 2

43. Projections of Ray Space b 1 != b 2

44. Projections of Ray Space b 1 != b 2 = 1

45. Projections of Ray Space b 1 != b 2 = 1

46. Projections of Ray Space b 1 != b 2 = 1

47. Projections of Ray Space b 1 = b 2 != 1, deficient eigenspace

48. Projections of Ray Space b 1 = b 2 != 1, deficient eigenspace

49. Projections of Ray Space b 1 = b 2 = 1, deficient eigenspace

50. Projections of Ray Space b 1 = b 2 = 1, deficient eigenspace

51. Projections of Ray Space b 1, b 2 complex

52. Slices of Ray Space

53. Real Eigenvalues Complex Conjugate Eigenvalues

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

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

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

57. Projections of Ray Space Let’s generalize:

58. Projections of Ray Space Let’s generalize:

59. Projections of Ray Space Let’s generalize:

60. Projections of Ray Space Let’s generalize:

61. Projections of Ray Space Factor Q as:

62. Projections of Ray Space Factor Q as: M warps lightfield in (x, y) –warps final image

63. Projections of Ray Space Factor Q as: M warps lightfield in (x, y) –warps final image A warps lightfield in (u, v) –shapes domain of integration (bokeh, aperture size)

64. Conclusion

65. General Linear Cameras can be characterized by the eigenvalues of a 2x2 matrix.

66. Conclusion General Linear Cameras can be characterized by the eigenvalues of a 2x2 matrix. Focus can be described in the same fashion.

67. Conclusion General Linear Cameras can be characterized by the eigenvalues of a 2x2 matrix. Focus can be described in the same fashion. These matrices are a good way to analyze and specify linear integral projections of ray space.

68. Questions