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

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

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. \(fs_x=fs_y\))