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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

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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

  • Regular Cameras

Ng et al. ‘04

Leica Apo-Summicron-M

Wilburn et al. ‘05


What is this paper?


What is this paper?

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


What is this paper?

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

  • An analogous description of focus


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 Space

  • Perspective View

  • Image(x, y) = L(x, y, 0, 0)


Slices of Ray Space

  • Orthographic View

  • Image(x, y) = L(x, y, x, y)


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 Space

P


Slices of Ray Space


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 Space

  • 0 < b1 = b2 < 1


Slices of Ray Space

  • b1 = b2 < 0


Slices of Ray Space

  • b1 = b2 = 1


Slices of Ray Space

  • b1 = b2 > 1


Slices of Ray Space

  • b1 != b2


Slices of Ray Space

  • b1 != b2 = 1


Slices of Ray Space

  • b1 = b2 != 1, deficient eigenspace


Slices of Ray Space

  • b1 = b2 = 1, deficient eigenspace


Slices of Ray Space

  • b1, b2 complex


Slices of Ray Space


Slices of Ray Space

Real Eigenvalues

Complex Conjugate Eigenvalues


Slices of Ray Space

Real Eigenvalues

Equal Eigenvalues

Complex Conjugate Eigenvalues


Slices of Ray Space

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues


Slices of Ray Space

One slit at infinity

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues


Projections of Ray Space


Projections of Ray Space


Projections of Ray Space


Projections of Ray Space

  • Rays Integrated at (x, y) = (0, 0):

F


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 Space

  • 0 < b1 = b2 < 1


Projections of Ray Space

  • 0 < b1 = b2 < 1


Projections of Ray Space

  • b1 = b2 < 0


Projections of Ray Space

  • b1 = b2 < 0


Projections of Ray Space

  • b1 = b2 = 1


Projections of Ray Space

  • b1 = b2 = 1


Projections of Ray Space

  • b1 = b2 > 1


Projections of Ray Space

  • b1 != b2


Projections of Ray Space

  • b1 != b2


Projections of Ray Space

  • b1 != b2


Projections of Ray Space

  • b1 != b2 = 1


Projections of Ray Space

  • b1 != b2 = 1


Projections of Ray Space

  • b1 != b2 = 1


Projections of Ray Space

  • b1 = b2 != 1, deficient eigenspace


Projections of Ray Space

  • b1 = b2 != 1, deficient eigenspace


Projections of Ray Space

  • b1 = b2 = 1, deficient eigenspace


Projections of Ray Space

  • b1 = b2 = 1, deficient eigenspace


Projections of Ray Space

  • b1, b2 complex


Slices of Ray Space


Slices of Ray Space

Real Eigenvalues

Complex Conjugate Eigenvalues


Slices of Ray Space

Real Eigenvalues

Equal Eigenvalues

Complex Conjugate Eigenvalues


Slices of Ray Space

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues


Slices of Ray Space

One focal slit at infinity

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues


Projections of Ray Space

  • Let’s generalize:


Projections of Ray Space

  • Let’s generalize:


Projections of Ray Space

  • Let’s generalize:


Projections of Ray Space

  • Let’s generalize:


Projections of Ray Space

  • Factor Q as:


Projections of Ray Space

  • Factor Q as:

  • M warps lightfield in (x, y)

    • warps final image


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)


Conclusion


Conclusion

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


Conclusion

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

  • Focus can be described in the same fashion.


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.


Questions


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