[tech-3d] 3d equations

["bart kelsey" <attraxe@xxxxxxxxxxx>]

G’day

Recently, I have been engaging in a discussion on the photo-3d site 
regarding equations that appeared in an article titled “3-D or not 3-D?”, 
New Scientist 28 April 1984, written by Charles W Smith.  Both Chris 
Pickering and Andrew Woods agreed that the author had made a mistake in an 
initial equation which was then substituted to derive the rest of the 
equations in the article.

With this initial equation corrected, I have substituted it to derive the 
equations which followed. However, I seem to have a negative sign in some of 
these equations and am seeking a second opinion as to my correctness or 
otherwise. On just having learnt of this site I felt it more appropriate to 
seek  further assistance here, rather than make more demands of a few who 
have already kindly contributed.

The relevant excerpt from the article can be downloaded as a .pdf file 
(<1MB) which will open in Acrobat Reader 3 and 4. It can be obtained via 
your web browser at:
ftp://photo-alf.med.monash.edu.au/Production/Photo Pub/strangers/3d/
(Note that there is a space between Photo and Pub so be sure to copy the 
entire address into the browser’s URL site, rather than click on the link 
which doesn’t include the entire address)

If you have ftp software, enter the following:
Server field:	photo-alf.med.monash.edu.au
Path field:	/Production/Photo Pub/strangers/3d/
and leave the other fields blank.

In regard to differentiating using the new  equations for parallax, z = 
(sf/p) - 2h and
Z = M(sf/p - 2h), I have derived the following:

P = Vep/(ep+2Mhp-Msf)                          	equation I

Depth magnification
md = dP/dp = -VeMsf/(ep+2Mhp-Msf)squared        equation II
Note that md is negative. Is this right?

3D image width
Q = Meqf/(ep+2Mhp-Msf)

Image width magnification
mw = dQ/dq = Mef/(ep+2Mhp-Msf)          equation III

Shape ratio
md/ mw = -sV/(ep+2Mhp-Msf)              equation IV
Note the negative sign. Is this right?

Smith states that shape ratio will only be constant through the scene when 
2Mh-e = 0.
Is it correct that this will be so now when 2Mh+e = 0, because it seems
unlikely?

If the problem I have referred to in the article has come up before, could 
someone please inform me, rather than us reinventing the wheel. Any help 
would be much appreciated as my intent is to draw  up a table to refer to in 
the field.

Cheers

Bart
-----------------------------------------------------------------------

Chris Pickering wrote:
I think you are correct that z = 2h-(sf/p) is not the case.
Thinking logically, if h is the movement of each lens inwards, then
parallax (which begins at z = sf/p) would decrease by 2h.
So surely z = (sf/p)-2h.
The correct equations are, if I am correct:-
z = (sf/p) - 2h
Z = M(sf/p - 2h)
P=Vep/(ep-Msf+2hMp)
And likewise equations II, III, and IV  DO have to corrected, as you
originally wrote.

Andrew Woods wrote:

I agree with you that z=(sf/p)-2h not z=2h-(sf/p) as it says in the 
article).

BTW: the depiction of small 'z' (sensor parallax) in Figure 5 is also
misleading.

I also concur that the effect on Equation I is then as you & Chris describe:
  i.e. P=Vep/(ep-Msf+2hMp)  not P=Vep/[Mfs-p(2Mh-e)]

However I'd be careful using Smith's definition of 'h' ("If we now apply 
convergence, by axial offset of each lens inward by a distance 'h',...").
Using this definition, 's' (lens separation) also changes.
I think it's a bad idea to define 'h' (lens/sensor offset) in this way
since a change in 'h' results in a change in 's'.

The way I defined 'h' in my paper "Image distortions in stereoscopic
video systems" http://info.curtin.edu.au/~iwoodsa/spie93pa.html
(see Figure 2) is "the outward movement of the imaging sensor relative
to each lens".  i.e. a change in 'h' doesn't result in a change in 's'.
The lenses stay put, the imaging sensors move.

It essentially means the same thing as Smith, but the resultant equations 
become much simpler.  If indeed you do move the lenses inwards then you just 
change 's' accordingly.

So, defining 'h' my way ('s' doesn't change) and your observation becomes:
when:
h=0, z(h=0)=sf/p
h>0, z(h>0)=sf/p-2h

The fundamental check of whether Smith's equations are right is to
insert h=0 into Equation I - which does reveal a problem (as you already 
did).

It is also important to note that his only error is a minus sign.
Unfortunately this error does propagate through all of his following math.
It could be that this error (the minus sign) was caused by the
ambiguity of 'z' in Smith's Figure 5.




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