toe-in and vertical parallax?

Hi!
I've been taught that toe-in gives rise to something called "vertical parallax".
When I'm googling for the term, I can only find one source (several sites) where it is used in that context.
See for example
http://paulbourke.net/miscellaneous/stereographics/stereorender/

A definition of "vertical parallax" in this context is given towards the end of
http://www8.cs.umu.se/kurser/5DV051/HT11/lectures/1%20introduction.pdf
There it is a measure of the depth difference of homologous (corresponding, i.e the projected) points.

I have searched in "Foundations of the Stereoscopic Cinema - A Study in Depth" by Lenny Lipton for the term "vertical parallax" and he seem to use it as a measure of the vertical screen distance of homologous points.
To call the depth difference vertical parallax seems unintuitive.

This give me two questions:
1) Is there an agreed definition of what "vertical parallax" is?
2) Is there a name for the depth difference between homologous points in a toe-in stereo pair configuration?

Best regards Tobias N

PS.
Referring to the toe-in picture in the second link above (the pdf document).
Since the respective projection (homologous point) is at different distances along the viewing directions for the two cameras, it seems plausible that a vertical screen difference will be present at most points. My thought is that this could be the vertical parallax.
The depth difference would have a linear relationship to vertical parallax in that case, but it's not the same. Am I right?

 

If you "toe in" then each frame is slightly distorted the farther from the center of rotation that you go. I believe this is also known as "key stoning". On their own each frame is fine but by changing the angle of the camera objects on the sides of the scene are actually slightly closer on one side than the other. This means that you will have slightly more vertical information on one portion of one frame than the matching portion of the other frame. You can reduce this by either cropping vertically or using software to correct the distortion between frames.

 

Hello dear Tobias,

>1) Is there an agreed definition of what "vertical parallax" is?

Vertical Parallax is another name for "vertical disparity". Good stereoscopic pictures NEVER have disparities other than horizontal ones. "Horizontal disparity" is also called "parallax" and is what gives depth to a stereo image. Many good programs exist to remove unwanted disparities such as vertical disparity, zoom disparity, or even the worst of all : temporal disparities.

It happens I wrote 350 pges book about "La Stéréoscopie Numérique" (hazlas it is in French only for now) and in chapter 3 I descibe the problems related to "toe-in" (check for it on my StereoscopyNews web site - http://goo.gl/H0vfF ). In summary, when you toe-in your two cameras, you get a deformation in both images also called 'Trapeze', i.e. you images are no more exact rectangles, but become slightly higher on the left side of the left image and on the right side of the right image. If you don't correct this, you get homologous points at different heigths and thus "vertical parallax" as you name them. If this is larger than something around one pixel, it is enough to force our brain to compensate and after a while, it is a sure source of headaches.... So,
avoid toe-in, or correct in postproduction the unwanted trapeze deformation of your images.
The best convergence method is not to toe-in the cameras but to "shear deform" the cameras; I mean you get the same convergence effect without the trapeze deformation if you move the sensor latarally off-axis from its optics - this is done inside integrated cameras such as Panasonic Z1000 or AG3D1A or the Fuji W3 cameras (I own all of them and I checked - it works).

>2) Is there a name for the depth difference between homologous points in a toe-in stereo pair configuration?
the depth difference is usually named "Delta-Z" and is computed from the horizontal parallax and optical parameters. Formulas such as the Bercowitz formula compute the relationship between all those elements. This is also explained in my book; It is true I have to work on the English version of the book to allow for a quick cut-and-paste here...

>Best regards Tobias N
Best too !