What is a Camera?What is a Camera?

Jean Ponce (ponce@di.ens.fr)http://www.di.ens.fr/~ponce

Willow project-teamLaboratoire d’informatiqueENS/INRIA/CNRS UMR 8548ENS/INRIA/CNRS UMR 8548

Ecole normale supérieure, Paris, FRANCE

Computational photographyComputational photography

© M. Levoy, Stanford

© R. Raskar, MIT

© S. Nayar, CU

(Pajdla, 2002)(Seitz and Kim, 2002)( Yu and McMillan, 2004)

The two-plane parameterization of lines

(Durand et al., 1996; Gortler et al., 1996; Levoy & Hanrahan, 1996; Gu et al., 1997)

© Gu et al. (2007)

What is a What is a (regular) (regular) camera?camera?

x

cξ

ry

x

(Veblen & Young, 1910; Pottman and Wallner, 2001). Illustrations © H. Havlicek, VUT.

x

ξyr x

yr

ξ

© E

. Mol

zcan

© L

eica

??

© T. Pajdla, CTU

Nondegenerate linear

congruences

x

cξ

ry

c

x

cξ

ry

x

c

ξ

x

cξ

ry

x

c

ξ

ξ

x

cξ

ry

x

ξ

ry

Linear familyof lines

x

ξ

x

c

ξ

ξ

ξ

Rank-3 reguli

© H. Havlicek, VUT

Rank-4 (nondegenerate) linear congruences

© H. Havlicek, VUT

© H. Havlicek, VUT

© H. Havlicek, VUT

© H. Havlicek, VUT

x

ξ

yr

x

yr

ξ

Hyperbolic linear congruences

© H. Havlicek, VUT

© E

. Mol

zcan

© L

eica

Hyperbolic linear congruences

© H. Havlicek, VUT

© T. Pajdla, CTU

Elliptic linear congruences

© H. Havlicek, VUT

© H. Havlicek, VUT

© H. Havlicek, VUT

y ≈ P xξ ≈ Q yB ( y1, y2 ) = 0T ( y1 , y2 , y3 ) = 0

Rank-4 (nondegenerate) linear congruences

x

ξ

x

Ax

ξ

The Essential Map (Pajdla, 2002)

x ! » = x Ç Ax

x

ξ

p1

±1

±2

a1

b1

a2

b2

z1

z2

p2

Hyperbolic linear congruences

x ξ

±

a2

p1

z

p2

p

a1

Parabolic linear congruences

±

s°

T

TO DO…

• “Regular” construction of parabolic congruences• A 4£4 fundamental matrix for these?• Understanding the interplay between cameras and the essential map A• Understanding the interplay between lines common to two cameras and epipolar loci• Intrinsic parameters, natural retinal parameterizations, and calibration• What about non-linear essential maps?• What about the light field?• Implementation