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