Lecture 5: Single Camera View Geometry

cross ratio
http://en.wikipedia.org/wiki/Cross_ratio
http://www.mathpages.com/home/kmath543/kmath543.htm
http://www.cut-the-knot.org/pythagoras/Cross-Ratio.shtml

measuring height: algebraic derivation from the cross ratio

Planar Perspective Map (homography) H
H Maps reference plane X-Y coordinates to image plane x-y coordinates
Fully determined from 4 known points on ground plane
Option A:  physically measure 4 points on ground
Option B:  find a square, guess the size
Option C:  Note  H = [avX  bvY  l]  (columns 1,2,4 of P )
              Play with scale factors a and b until the model “looks right”
Given x-y , can find X-Y by H-1 :

 

To compute camera projection matrix, we can measure

- Positions within a plane
- Height (More generally — distance between any two parallel planes)
- Camera position

Single View Metrology

criminisi_reid_zisserman_SingleViewMetrology.pdf

A. Criminisi, I. Reid and A. Zisserman
Department of Engineering Science, University of Oxford

> Assumptions
1)   3 orthogonal sets of parallel lines
2)   4 known points on ground plane
3)   1 height in the scene

> Complete approach
- Load in an image
- Click on parallel lines defining X, Y, and Z directions
- Compute vanishing points
- Specify points on reference plane, ref. height
- Compute 3D positions of several points
- Create a 3D model from these points
- Extract texture maps
- Output a VRML model

 http://en.wikipedia.org/wiki/VRML
VRML (Virtual Reality Modeling Language, pronounced vermal or by its initials, originally — before 1995 — known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphics, designed particularly with the World Wide Web in mind. It has been superseded by X3D.


3-D Metrology
http://research.microsoft.com/~antcrim/

single view reconstruction

http://en.wikipedia.org/wiki/Ground_truth

radial distortion
http://en.wikipedia.org/wiki/Distortion_(optics)