General Linear Cameras with Finite Aperture

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General Linear Cameras with Finite Aperture

• Andrew Adams and Marc Levoy
• Stanford University

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Ray Space
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Slices of Ray Space

• Pushbroom
• Cross Slit
• General Linear
• Cameras

Yu and McMillan 04
Román et al. 04
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Projections of Ray Space

• Plenoptic Cameras
• Camera Arrays
• Regular Cameras

Ng et al. 04
Wilburn et al. 05
Leica Apo-Summicron-M
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What is this paper?
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What is this paper?

• An intuitive reformulation of general linear
  cameras in terms of eigenvectors

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What is this paper?

• An intuitive reformulation of general linear
  cameras in terms of eigenvectors
• An analogous description of focus

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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

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Slices of Ray Space

• Perspective View
• Image(x, y) L(x, y, 0, 0)

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Slices of Ray Space

• Orthographic View
• Image(x, y) L(x, y, x, y)

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Slices of Ray Space

• Image(x, y) L(x, y, P(x, y))
• P determines perspective
• Lets assume P is linear

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Slices of Ray Space
P
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Slices of Ray Space
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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

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Slices of Ray Space

• 0 lt b1 b2 lt 1

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Slices of Ray Space

• b1 b2 lt 0

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Slices of Ray Space

• b1 b2 1

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Slices of Ray Space

• b1 b2 gt 1

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Slices of Ray Space

• b1 ! b2

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Slices of Ray Space

• b1 ! b2 1

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Slices of Ray Space

• b1 b2 ! 1, deficient eigenspace

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Slices of Ray Space

• b1 b2 1, deficient eigenspace

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Slices of Ray Space

• b1, b2 complex

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

• Rays Integrated at (x, y) (0, 0)

F
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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

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Projections of Ray Space

• 0 lt b1 b2 lt 1

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Projections of Ray Space

• 0 lt b1 b2 lt 1

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Projections of Ray Space

• b1 b2 lt 0

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Projections of Ray Space

• b1 b2 lt 0

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Projections of Ray Space

• b1 b2 1

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Projections of Ray Space

• b1 b2 1

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Projections of Ray Space

• b1 b2 gt 1

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Projections of Ray Space

• b1 ! b2

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Projections of Ray Space

• b1 ! b2

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Projections of Ray Space

• b1 ! b2

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Projections of Ray Space

• b1 ! b2 1

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Projections of Ray Space

• b1 ! b2 1

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Projections of Ray Space

• b1 ! b2 1

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Projections of Ray Space

• b1 b2 ! 1, deficient eigenspace

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Projections of Ray Space

• b1 b2 ! 1, deficient eigenspace

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Projections of Ray Space

• b1 b2 1, deficient eigenspace

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Projections of Ray Space

• b1 b2 1, deficient eigenspace

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Projections of Ray Space

• b1, b2 complex

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

• Lets generalize

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Projections of Ray Space

• Lets generalize

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Projections of Ray Space

• Lets generalize

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Projections of Ray Space

• Lets generalize

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Projections of Ray Space

• Factor Q as

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Projections of Ray Space

• Factor Q as
• M warps lightfield in (x, y)
• warps final image

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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)

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Conclusion
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Conclusion

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

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Conclusion

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

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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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Questions