Direct Linear Transformation from Comparator
Coordinates into Object Space Coordinates in
Close-Range Photogrammetry*
*An interim report on a study sponsored by the National Science Foundation as a part of research grant GK-11655
Y.I Abdel-Aziz and Dr. H.M. Karara
Abstract
A method for photogrammetric data reduction without the
necessity for neither fiducial marks nor initial approxima-
tions for inner and outer orientation parameters of the camera
has been developed. This approach is particularly suitable for
reduction of data from non-metric photography, but has also
distinct advantages in its application to metric photography.
Preliminary fictitious data tests indicate that the approach
is promising. Experiments with real data are underway.
1. INTRODUCTION
In analytical photogrammetry, measurements of image points
are normally done on comparators. The transformation of
comprarator coordinates into object space coordinates is usu-
ally performed in two steps:
a) Transformation from comparator coordinates into im-
age coordinates, and
b) Transformation from image coordinates into object
space (ground coordinates)
For the transformation from comparator coordinates into
image coordinates, it is necessary to calibrate and measure
fiducial marks. For the transformation from image coordinates
into object space coordinates, an iterative solution is gener-
ally used, for which one needs initial approximations for the
unknown parameters (elements of outer orientation and in
some cases also elements of inner orientation of the camera).
In working with hand-held non-metric cameras, neither of
the above two requirements are satisfied. In view of the ever
increasing use of non-metric cameras in close-range photo-
grammetry, particularly in cases of medium to low accuracy
requirements, it was deemed desirable to develop a method
suitable for data reduction from non-metric photography.
The proposed method involves a direct linear transforma-
tion from comparator coordinates into object space coordi-
nates. In a sense, it is a simultaneous solution for the two
aforementioned transformations. Since the image coordinate
system is not involved in the approach, fiducial marks are
not needed. Furthermore, the method is a direct solution and
does not involve initial approximations for the unknown
parameters of inner and outer orientation of the camera.
The proposed method is thus particularly suitable for re-
duction of data in non-metric photogrammetry. When applied
to metric photography, the proposed approach yields at least
the same accuracy as the conventional methods, but is easier
to program (no linearization necessary) and uses less comput-
er memory and executing time.
2. Mathematical Basis of the Proposed Method
As mentioned above, the proposed method involves a simul-
taneous solution of two transformations which are usually
done separately in conventional analytical photogrammetry.
The transformation of comparator coordinates into image
coordinates is generally done in the following forms:
x– = a
1
+a
2
x + a
3
y
(1)
y– = a
4
+a
5
x + a
6
y,
where:
x– , y– are image coordinates
x, y are comparator coordinates
Such a transformation takes into account errors in perpen-
dicularity between the x and y comparator coordinate axes,
and possible differential linear distortions along the x and y
comparator coordinate axes (due to lens distortion, film defor-
mation and comparator unadjustment).
Photogrammetric Engineering & Remote Sensing
Vol. 81, No. 1, February 2015, pp. 103–107.
0099-1112/14/812–103
© 2014 American Society for Photogrammetry
and Remote Sensing
doi: 10.14358/PERS.81.2.103
When does a scientific paper become a “classic”? In the case
of the below article, there is a following that has born the test
of time. Each year at the beginning of the school semester,
we receive many requests from professors for reprints of this
article to use in class. The original article was published
in 1971 as part of committee work within ASP (the original
name for ASPRS), but it has not appeared in
PE&RS
or other
journals, as far as we know. Unfortunately, one of the authors
has passed away and we have not been able to reach the oth-
er, but if you were a student or colleague of either we would
appreciate hearing from you. Dr Karara was an important
member of ASPRS and there is a lengthy In Memoriam in the
July 2001 issue of
PE&RS
, but we have located little further
information about Y.I. Abdel-Aziz. In light of the continuing
demand for their paper, it makes its
PE&RS
debut in this
issue. We hope you enjoy this “classic.”
–Dr. Michael Hauck,
ASPRS Executive Director
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
February 2015
103
