Transformation Module

Online. Transformation Module

Published on Sunday, January 5, 1997 by Gideon Ariel

Transformation Module

Dear Tomislav-

Gideon asked me to help you with your request for information regarding the
3D transformation algorithms the the APAS system uses.

Underlying the DLT is a physical model of the camera with 10 physical
parameters. These are:
       XYZ location of camera origin [3]
       Orientation of the camera axis [3]
       [u,v] offset if the intersection
           of the camera axis with the projection plane [2]
       “u” & “v” distances from camera origin to projection plane [2]

These 10 parameters define the transformation from lab xyz units to camera
digitizer units. Unfortunately the equations are non-linear and lead to a
rather messy solution when trying to calculate xyz from two or more sets of
(u,v) information. The DLT solution to the problem allows one to write a set
of linear equations with 11 DLT parameters which  make solving the problem of
reconstructing  xyz’s in the lab  much easier. These 11 DLT parameters can be
written in terms of the 10 physical parameters first discussed.

The problem with the 11 DLTs is that they are not all independent since there
are really only 10 degrees of freedom. The upshot of this is one can [and
actually always does] end up with a set of 11 parameters which describe an
inconsistent set of 10 physical parameters. This expresses itself in a
transformation  the orientation part of that does not preserve length which
is definitely not desired.

Our PPT [physical parameters transformation] sidesteps this shortcoming by
not calculating the DLT from the control point information directly but
rather calculates the 10 PPT parameters directly from the control point
information. This process finds the “best” set of 10 PPTs which minimize the
total square deviation of the (u,v)’s digitized for a camera view from the
projected (u,v)’s when run through the PPT given the known lab xyz’s for the
control points using standard non-linear minimization techniques. With the
PPTs known one can then generate the 11 DLTs in a completely consistent way.

When one uses a panning camera it is essential to work with the physical
parameters of the camera since these are changing in time.

I know your request was for source code. Unfortunately the actual
implementation of the algorithm is proprietary and something which we do not
distribute.

You will find a good  & detailed presentation on the relationship between the
DLT & camera parameters in at thesis by James Walton, 1979, “Close-Range
Cine-Photogrammetry”. Also I made a presentation on this topic about 6 years
ago at the ISB when it was at Poitiers, France. I cannot seem to find my copy
of the proceeding at the moment. If you are interested in that & have trouble
locating the proceedings let me know.

I hope this information helps.

Sincerely,
         Jeremy Wise
         Dir R&D

In a message dated 9/28/99 9:03:48 AM Eastern Daylight Time,
gideon@arielnet.com writes:

<<
 Dear Sir,
 
     it has been a week since I attended your lecture in Zagreb, Croatia.
You were nice to forward my questions, concerning some source codes, to your
associates. So far I haven’t heard from them . I do want to sound impatient,
but just to remind you in a case a message has got lost somewhere (Also if
there is any kind of problem I would like to be aware of it). Thank you again
for your time and patient. Looking forward to hear from you,
 
                     Tomislav.
 
 P.S. I do not mind to contact your associates directly with your permission.