Re: Twin Camera Questions: Toe-In

[bercov@xxxxxxxxxx (John Bercovitz)]

Yesterday I talked about scale error from toe-in.  You'll get both 
vertical height errors and horizontal width errors due to toe-in.  
Let's look at what happens if you take a picture of a ruler with 
the left lens in a normal stereo camera and also the left lens of 
a toed-in pair of cameras.  I've put the ruler in a plane which is 
one of many planes parallel to a line between the two lenses of 
the camera(s).  These are the planes which are common to both 
cameras and are the "depth planes" which you will see when the 
image is reconstructed.
 
Using shift, as a normal stereo camera does, left lens shown:
 
                                  ruler
                         __________________________
                        |        |        |        |      
 
 
 
 
 
 
                             .  <- rectilinear lens or pinhole
 
                     |__|__|__|
 
                        film
 
 
Using toed-in camera(s), left lens shown:
 
                         ruler
              __________________________
             |        |        |        |      
 
 
 
 
 
 
                 
          \       .  <- rectilinear lens or pinhole
            \ /
              \/
                \/                                
        film ->   \     
                    \/
                      \ 
                        
Do you see how the tick marks are bunched together on the left 
side of the film plane of the toed-in camera and spread out on the 
right?  The left side of the film plane of course represents the 
right side of the object plane (the ruler) so don't get confused 
on that point.  The right lens of the toed-in camera pair is going 
to have these same scale errors except the image is going to get 
bigger as you go from right to left instead of from left to right.  
Then when you turn the images right side up to view them you'll 
have:
 
              left                     right
         _______________           _______________
        |       |    |  |         |  |    |       |
 
 
 
So what happens when you look at a pair like this?  Well, your 
eyes/brain have to make some sense of them and here's how to see 
what they construct.  Take a 3x5 card and hold it up in front of 
your face so you're looking squarely at the flat of the card and 
the long dimension of the card is running from left to right.  Now 
push the left and right edges of the card towards each other so 
that the card buckles and bulges toward you.  Notice what you see: 
to the left eye, the left side of the card is quite a bit taller 
than the right side, which is as it should be (qualitatively), but 
if you had tick marks across the card you'd see them spread out on 
the left and bunched up on the right.  This is in fact the geometric 
solution to the two views presented above.  What you're 
doing is looking at the vertex (pointy end) of a hyperbola from 
in the plane of, but outside of the curve (your perspective point 
is not between the "arms" of the hyperbola.
 
As I said a couple of days ago, if you want a quantified look at 
this phenomenon, check out Andrew Woods' paper in bobcat.  It's a 
truly beautiful paper that you will remember for years to come.
 
 
                      Viewing the ruler
 
 
 _______________________         / \                    / \
|_____|_____|_____|_____|        \  \                  /  /
                                    \/ \             / \/
                                       \    \___/    /
                                           \__|__/
 
 
 
 
           *-*                               *-*
          /   \                             /   \
        )|     |(                         )|     |(
          \___/                             \___/
 
 
       This is you.             This is you on toed-in views.
 
 
John B


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