The Camera Projection

\[\lambda x = K\Pi gX\]

1. Euclidean transformation \(g\) from world frame to camera frame
2. Projection \(\Pi\) from 3D to 2D
3. Camera intrinsic transformation \(K\)

Ambiguities

There are three ambiguities

\[\lambda x = (KR^{-1})(R\Pi H^{-1}) (H gg_w^{-1})g_wX\]

Due to the arbitrary choice of frame, \(R\) and \(g_w\) are inconsequential

Equivalence classes of the calibration matrix

• \(K\) can be chosen to be upper triangular without loss of generality

• QR decomposition

• The special linear group \(\mathsf{SL}(3)\)
  • invertible matrices with determinant \(+1\)
• The group of rotations \(\mathsf{SO}(3) < \mathsf{SL}(3)\) (subgroup)
  • self-orthogonal \(R^{-1}=R^T\)
• Equivalence classes \[\frac{\mathsf{SL}(3)}{\mathsf{SO}(3)}\]
  • equivalent because they induce the same iner product \(\langle\rangle_S\)
  • \(S=K^{-T}K^{-1}\)
  • \((KR)^{-T}(KR)^{-1}=K^{-T}R^{-T}R^{-1}K^{-1}=K^{-T}RR^{-1}K^{-1}=K^{-T}K^{-1}=S\)
  • one-to-finite correspondence between \(S\) and upper triangular \(K\)
  • usually only one \(K\) is a valid camera calibration matrix
• We cannot distinguish \(K\) from \(KR_0^T\)
  • and \(g=[R,T]\) from \(\tilde g=[R_0R,R_0T]\)

Intrinsic and Extrinsic

• intrinsic \(K\)
• extrinsic \(g\)

\[\Pi X = (\Pi H^{-1})(H X) = \tilde \Pi \tilde X\]

• \(X\) in the true world
• \(\tilde X\) in the distorted world

Stratified Reconstruction

• Projective \(\to\) Affine \(\to\) Euclidean