---
title: Lecture - Partial Scene Information
categories: lecture
---

**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