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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

General Linear Cameras with Finite Aperture

Andrew Adams and Marc Levoy

Stanford University

slices of ray space
Slices of Ray Space
  • Pushbroom
  • Cross Slit
  • General Linear

Cameras

Yu and McMillan ‘04

Román et al. ‘04

projections of ray space
Projections of Ray Space
  • Plenoptic Cameras
  • Camera Arrays
  • Regular Cameras

Ng et al. ‘04

Leica Apo-Summicron-M

Wilburn et al. ‘05

what is this paper6
What is this paper?
  • An intuitive reformulation of general linear cameras in terms of eigenvectors
what is this paper7
What is this paper?
  • An intuitive reformulation of general linear cameras in terms of eigenvectors
  • An analogous description of focus
what is this paper8
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
slices of ray space9
Slices of Ray Space
  • Perspective View
  • Image(x, y) = L(x, y, 0, 0)
slices of ray space10
Slices of Ray Space
  • Orthographic View
  • Image(x, y) = L(x, y, x, y)
slices of ray space11
Slices of Ray Space
  • Image(x, y) = L(x, y, P(x, y))
  • P determines perspective
  • Let’s assume P is linear
slices of ray space14
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

slices of ray space15
Slices of Ray Space
  • 0 < b1 = b2 < 1
slices of ray space21
Slices of Ray Space
  • b1 = b2 != 1, deficient eigenspace
slices of ray space22
Slices of Ray Space
  • b1 = b2 = 1, deficient eigenspace
slices of ray space23
Slices of Ray Space
  • b1, b2 complex
slices of ray space25
Slices of Ray Space

Real Eigenvalues

Complex Conjugate Eigenvalues

slices of ray space26
Slices of Ray Space

Real Eigenvalues

Equal Eigenvalues

Complex Conjugate Eigenvalues

slices of ray space27
Slices of Ray Space

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues

slices of ray space28
Slices of Ray Space

One slit at infinity

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues

projections of ray space32
Projections of Ray Space
  • Rays Integrated at (x, y) = (0, 0):

F

projections of ray space33
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

projections of ray space47
Projections of Ray Space
  • b1 = b2 != 1, deficient eigenspace
projections of ray space48
Projections of Ray Space
  • b1 = b2 != 1, deficient eigenspace
projections of ray space49
Projections of Ray Space
  • b1 = b2 = 1, deficient eigenspace
projections of ray space50
Projections of Ray Space
  • b1 = b2 = 1, deficient eigenspace
slices of ray space53
Slices of Ray Space

Real Eigenvalues

Complex Conjugate Eigenvalues

slices of ray space54
Slices of Ray Space

Real Eigenvalues

Equal Eigenvalues

Complex Conjugate Eigenvalues

slices of ray space55
Slices of Ray Space

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues

slices of ray space56
Slices of Ray Space

One focal slit at infinity

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues

projections of ray space57
Projections of Ray Space
  • Let’s generalize:
projections of ray space58
Projections of Ray Space
  • Let’s generalize:
projections of ray space59
Projections of Ray Space
  • Let’s generalize:
projections of ray space60
Projections of Ray Space
  • Let’s generalize:
projections of ray space62
Projections of Ray Space
  • Factor Q as:
  • M warps lightfield in (x, y)
    • warps final image
projections of ray space63
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)
conclusion65
Conclusion
  • General Linear Cameras can be characterized by the eigenvalues of a 2x2 matrix.
conclusion66
Conclusion
  • General Linear Cameras can be characterized by the eigenvalues of a 2x2 matrix.
  • Focus can be described in the same fashion.
conclusion67
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.
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