Lecture - Partial Scene Information

Reading Ma 2004:Ch 6.5

Parallel and Orthogonal Lines

  • Man-made constructs often display parallellism and orthogonality
    • even if not designed for the purpose of calibration
  • Parallellism and orthogonality can be assumed but not guaranteed
    • imagine a hyper-modernist architect

Two parallel lines and their vanishing point

  • Consider two lines \(\ell^1,\ell^2\in\mathbb{R}^2\)
    • represented by their co-images
    • i.e. the line is \(\ell^\bot\cap\text{image plane}\)
  • The vanishing point is \(v\sim\ell^1\times\ell^2\)
  • The vanishing point is the intersection of \(\ell^1\) and \(\ell^2\)
    • a point at infinity since the lines are parallel
    • hence \(v\) is orthogonal on both the co-images

Calibration from orthogonal lines

  • Consider three pair-wise orthogonal sets of parallel lines
  • Three vanishing points \(v_1,v_2,v_3\)
    • In 3D, these only make sense in homogenous co-ordinates
  • By orthogonality, and choice of world frame,
    • can assume that the directions co-incide with the principal directions \(e_1,e_2,e_3\)
    • \(v_i=KRe_i\)
  • Consider the inner product \[\langle v_i,v_j\rangle_S = v_i^TSv_j=v_i^TK^{-T}K^{-1}v_j = e_i^TR^TRe_j = e_i^Te_j =0 \quad\text{when }i\neq j\]
  • Three constraints and five degrees of freedom.
  • To get unique solution, assume
    • zero skew \(s_\theta=0\)
    • known aspect ratio (e.g. \(fs_x=fs_y\))

Calibration Rig

  • Known object points \(X_i\) as well as image points \(x_i\)
  • Single image suffices

Calibration with Planar Rig