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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

- Pushbroom
- Cross Slit
- General Linear
Cameras

Yu and McMillan ‘04

Román et al. ‘04

- Plenoptic Cameras
- Camera Arrays
- Regular Cameras

Ng et al. ‘04

Leica Apo-Summicron-M

Wilburn et al. ‘05

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

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

- 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

- Perspective View
- Image(x, y) = L(x, y, 0, 0)

- Orthographic View
- Image(x, y) = L(x, y, x, y)

- Image(x, y) = L(x, y, P(x, y))
- P determines perspective
- Let’s assume P is linear

P

- 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

- 0 < b1 = b2 < 1

- b1 = b2 < 0

- b1 = b2 = 1

- b1 = b2 > 1

- b1 != b2

- b1 != b2 = 1

- b1 = b2 != 1, deficient eigenspace

- b1 = b2 = 1, deficient eigenspace

- b1, b2 complex

Real Eigenvalues

Complex Conjugate Eigenvalues

Real Eigenvalues

Equal Eigenvalues

Complex Conjugate Eigenvalues

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues

One slit at infinity

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues

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

F

- 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

- 0 < b1 = b2 < 1

- 0 < b1 = b2 < 1

- b1 = b2 < 0

- b1 = b2 < 0

- b1 = b2 = 1

- b1 = b2 = 1

- b1 = b2 > 1

- b1 != b2

- b1 != b2

- b1 != b2

- b1 != b2 = 1

- b1 != b2 = 1

- b1 != b2 = 1

- b1 = b2 != 1, deficient eigenspace

- b1 = b2 != 1, deficient eigenspace

- b1 = b2 = 1, deficient eigenspace

- b1 = b2 = 1, deficient eigenspace

- b1, b2 complex

Real Eigenvalues

Complex Conjugate Eigenvalues

Real Eigenvalues

Equal Eigenvalues

Complex Conjugate Eigenvalues

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues

One focal slit at infinity

Real Eigenvalues

Equal Eigenvalues

Equal Eigenvalues,

2D Eigenspace

Complex Conjugate Eigenvalues

- Let’s generalize:

- Let’s generalize:

- Let’s generalize:

- Let’s generalize:

- Factor Q as:

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

- 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)

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

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

- 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.