From: uy205@vtn1.victoria.tc.ca (Roy A. Fletcher)
Subject: Re: Pixel To World Coordinates Convertion
Date: 23 Sep 2000 03:59:03 -0800
Newsgroups: sci.math.num-analysis,comp.graphics.algorithms,sci.image.processing,alt.graphics.pixutils,comp.graphics.rendering.misc
Summary: [missing]

the quasar (thequasar1@yahoo.com) wrote:
with editing...
: Hi all,
:     I'm looking for a function that converts Image Pixel Coordinates to World
: Coordinates.

: I'm trying to scan a 3D object. To do this I grab the object by 4 digital
: cameras in the following way: I use a
: laser that illuminates a little object portion so the cameras can grab the
: illuminated area and then I rotate the
: object so it can be all illuminated  by the laser. Everytime the 4 cameras grab
: the illuminated portion, my input
: data are given by 4 lines in the pixel coordinates because the image is
: digital. I want to reconstruct the object
: profile so I want to have the image in the world coordinate system.

: Is there any book or material that deals with this stuff?
: Is this function unique?
: Does exist a closed form for this function or can we just solve it numerically?

Photogrammetrists have worked out similar problems.
The solution is called a bundle adjustment.

Quickly, here it is:
 You have 4 cameras with 3 positional parameters each - 12 parameters
 You have 4 cameras with 3 orientation parameters each - 12 more
     parameters.
 Then you use a co-planarity or a co-linearity equation set; to do this,
      you must know the focal length of each camera - 4 more parameters.
 But you've added a laser beam, so it has 6 parameters : 3 positional
      and 3 orientation.
 That's 34 parameters total. You'll have to know the position of 12
      points on the object to work out a solution ( 34 / 3 = 11.67 ).
 If you can approximate 12 positions, you can use least squares to
     refine your estimates and work out the 34 parameters.
 Then you make measurements to as many points as you need to obtain a
     smooth surface.

The shuttle undergoes this analysis after every flight to determine
   it's structural integrity; mappers use this etc.

Regards.		RAF