Revision 6c660349d4d968a779d8a22a35ec732d3c3ad1ac (click the page title to view the current version)
Changes from 6c660349d4d968a779d8a22a35ec732d3c3ad1ac to 3365b6f8dcca728b4c9e51ab6c14d32e207990ce
---
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
