187 
where 
coordl 
of B®. 
ON THE IMPORTANCE OF PROJECTIVE GEOMETRY FOR ANALYTICAL 
AND DIGITAL PHOTOGRAHMETRIC RESTITUTION 
Gerhard Brandstatter 
Institute of Applied Geodesy and Photogrammetry 
Graz University of Technology, Austria 
X» = ï 
gives 
The rc 
recipr 
vector 
systerr 
unit v 
yE mus 
Fig. 1 
vector 
of the 
y’-u 1 b 
Abstract 
Projective geometry is an already ancient scope of 
mathematics, generally treated synthetically, 
because during the period of analog restitution 
instruments there was no need for projective 
calculation techniques. But the use of analytic and 
digital methods in photogrammetry and videometry 
causes a new interest in those methods, in parti 
cular, when non-metric cameras are used. For prac 
tical purposes algebraic relations of projective 
methods based on linear transformations of homo 
geneous geometric eLements are required. The paper 
Will give a short treatment of this "algebraic 
projective geometry", of its application to photo- 
grammetric problems ( rectification and realtive 
orientation ) and the consequences regarding uni 
versity education in photogrammetry, videometry, 
and digital image processing. 
0. Preface 
Analytical and digital photogrammetry start from 
image coordinates, which are measured by means of 
optomechanical or optoelectronic instruments as 
monocomparators, stereocomparators, analytical 
plotters, digitizers, photodetectors, videocameras 
or CCD-cameras. None of these instruments can a 
priori ensure that its coordinates are related to 
an orthogonal and isometric system. Such systems 
are to be called here orthonormal and correspond, 
of course, to rectangular cartesian systems. Every 
manufacturer of instruments will try to approach 
this ideal state but will never attain it exactly 
because of too great expenses in production and 
quality checks. So, in order to simulate ideal 
conditions, usually the systems must be calibrated 
by the producer or by the user, and the corrections 
may be stored in tables or matrices. But often it 
1s more expedient to accept the fact, that in 
reality all -coordinates are oblique and hetero 
metric, or in one word, affine. In this case the 
working methods of photogrammetry may be adapted 
to the principles of algebraic projective geometry, 
which 1s based on linear transformations of homo 
geneous vectors. 
1. Projective transformations 
1.1 General linear homogeneous trans 
formation 
It is well-known, that a general 4x4-matrix 
■ 
• 
poo Poi PO 2 P0 3 
yo 
Po T y 
P10 P1 1 Pi 2 P13 
yi 
= 
pi T y 
P20 P21 P22 P23 
Y2 
P 2 T y 
P3 0 P3 1 P3 2 P3 3 
ya 
P3 T y 
defines a projective transformation (Hohenberg- 
Tschupik, 1972). If P is regular (rank(P)=4), the 
relation exists between two tridimensional (order 
n+1) vector spaces as e.g. for optical imaging of 
ideal lenses.If P is singular,the resulting vectors 
fill a projection plane P* (rank=3) or a straight 
line (rank=2). The linear image P 2 is the ideal 
foundation of all photogrammetric theories, the 
projective line sometimes appears in connection 
with architectural and engineering applications. 
(1.1.1) contains 15 Independent parameters so that 
five or generally n+2 noncomplanar spatial points 
define the relation between two projective spaces. 
These points may be four arbitrary points Pi and a 
unit point Pe, which determines affine units along 
the other four position vectors , yielding the 
base B(bo,bi,b2,b3) of the related vector space. 
1.2 Determination of transformat ion 
elements 
The basis vectors depend on the position vectors yi 
by the linear relation 
bi=Uiyi, B* = (poyo piyi U2 y2 H3 y3 ) (1.2.1) 
and must meet the conditions 
3 3 
I bi = I myi = yE or Yu = yE 
\ =0 1=0 
(Fuchs, 1988), from which the vector u T = (uo ,m ,M2 ,M3) 
of the unknowns results according to M = Y~ 1 yE with 
3 
Ui = (y*) T yE = I y 1 jyEJ , (1.2.2) 
j=0 
where the y 1 are the row vectors of Y _1 or the so- 
called reciprocal vectors of the y». Any other 
vector y may be combined by means of the basic 
vectors in the form 
3 
y = l X 1 bi = B*x B , (1.2.3) 
1 sO 
with u 
U = 
Fig. 1 
Becausi 
B® Uu = 
follows 
1 
u 1 
U2 
u 3 
In ordi 
Image s 
by Its 
wo 
Wo 
W2 
- = U 3 
Wo 
so that 
and the 
through 
The bas