Camera from F

1
Camera from F
Extracting camera parameters HZ6.2
Camera matrix HZ6.1
Camera matrix from F HZ9.5
Computing K from 1 image HZ8.8
IAC and K HZ8.5
IAC HZ3.5-3.7, 8.5
Calibration using Q HZ19.3 Hartley 92
2
F does not uniquely identify a camera pair

• a fundamental matrix F relates two images, and
  therefore two cameras
• F already gives indirect information about the
  camera through the epipole (the null vector of F)
• we will now see that much fuller information
  about both cameras can be extracted from F, in
  particular the two camera matrices
• but a fundamental matrix does not uniquely
  identify two camera matrices
• Theorem Let H be a homography of 3-space. The
  fundamental matrix associated with camera pair
  (P,P) is the same as the fundamental matrix
  associated with camera pair (PH, PH).
• Proof if (x PX, x PX) is a point
  correspondence in the 1st camera system arising
  from 3d point X, then (x (PH)(H-1X), x
  (PH)(H-1X)) is a point correspondence in the 2nd
  camera system arising from 3d point H-1X
• thus, upon finding F, we only know the camera
  matrix up to a homography
• analogous to affine rectification extra work is
  required to remove this projective freedom and
  get to metric structure of the camera (stay
  tuned!)
• luckily this is the only degree of freedom if
  two camera pairs have the same F, then they must
  differ by a homography (Thm 9.10)
• HZ254

3
F does not depend on world frame

• camera matrix depends on image frame and world
  frame, since x PX is dependent on the image
  coordinates x and the world coordinates X
• but the above result is telling us that the
  fundamental matrix is independent of the world
  frame
• we are happy to relax to within a similarity of
  the truth, but we are relaxing to within a
  projectivity of the truth, which pleases us less
• we will need calibration techniques
• (these calibration techniques may be interpreted
  as ways of protecting the absolute conic)
• HZ253

4
Camera matrix from F

• lets see what we can get from F
• we assume that the first camera matrix is always
  I 0
• we can always normalize so that it is, using some
  homography
• we are solving for the relative offset of the
  2nd camera from the 1st
• Lemma F is fundamental matrix of camera pair
  (P,P) iff PtFP is skew-symmetric.
• Proof PtFP is skew-symmetric ?? Xt (PtFP) X
  0 for all X ?? xt F x 0 where x PX and x
  PX, the images of X ?? F is fundamental matrix of
  (P,P)
• Theorem The camera matrices associated with F
  may be chosen to be I 0 and SF e where
• et F 0 (2nd image epipole)
• S any skew-symmetric matrix
• proof just check that SF et F I 0 is
  skew-symmetric
• Corollary The camera matrices associated with F
  may be chosen to be I 0 and ex F e.
• choose S ex to guarantee a rank 3 camera
  matrix P
• see proof on 256
• ironic to guarantee rank 3 P (necessary, so
  good), we force rank 2 M (a bit bad)
• HZ255-6

5
Pseudoinverse P

• the camera matrix P is rectangular (3x4) so it
  does not have an inverse
• but we want to use the notion of inverse to talk
  about the preimage of an image point
• a rectangular matrix P has a pseudoinverse P
• P (Pt P)-1 Pt
• note that Pt P is square
• note that P degenerates to P-1 when P is square
  and nonsingular
• x Ab is the least-squares solution of Axb if
  A is full rank (Trefethen 81)

6
F from P

• two camera matrices do uniquely specify a
  fundamental matrix
• let P and P be two camera matrices, and e the
  epipole in the 2nd image
• note e can be calculated as PC (image of other
  camera center)
• the associated fundamental matrix of these 2
  cameras
• F ex P P
• note PP is the homography H in the earlier
  definition of F ex H
• proof idea epipolar line of x is built from the
  epipole e and the camera-2-image of a typical
  point of the camera-1-preimage of x (or P Px)
• proof Fx L e x P (P x) so F ex P
  P
• note how the fundamental and camera matrices are
  independent of the actual contents of the image
  only dependent on the camera setup
• HZ244