193
t3= 0 Indicates that T Is singular respectively,
T*=T-1 of (2.3.1) cannot exist and Illustrates the
Impossibility of reconstruction of a spatial object
from one single Image. But because of det(T) = 0
the homogeneous systems (2.3.3) become resolvable
and produce the affine coordinates uom of the two
centers of projection.
Using (2.3.5) the relations
Ti UM*
To + tri-TojuM^ÍTz-To )um 2 -toUM 3
and, consequently, for each Image the two equations
{U 1 (tl-tO)~T1lUM 1 +U 1 (T2 —to)UM 2 ~U 1 tOUM 3 =-U 1 to
u 2 (tl-to)ynL+{U 2 (T 2 *t0)-t2)UM 2 -U 2 tOUM 3 =-U 2 tO,
arise. These are, just as usual, four equations for
the three unknown spatial affine coordinates um 1 .
In this way every pair of homologous points of P« ,
Ps delivers a point of the model space Pm, from
which the desired orthonormal coordinates may be
found by an inverse affine transformation.
3. Final remarks.
This paper intended to show by means of two
examples, how projective geometry influences the
conventional methods of photogrammetry. The most
important difference is, that the number of control
points or correlation points and, consequently, the
expenditure of work increases. On the other side
non-metric cameras or measuring devices can be used
and all relations are linear. Therefore, projective
methods will also be very useful 1 for photogram-
metry by digital images and stereovideometry.
From the educational point of view, the disadvan
tage of a rather abstract geometry exists. However
on the other hand a more general insight is gained,
which enables also the deduction of the special
relations of or thononnal ized metric systems. More
over, a lot of problems will result from practical
applications and will give new stimulations to
photograrmnetri c i nvest i gat i on.
References:
Brandstätter G.: Fotogrametria proyectiva, pro
cesamiento de imágenes digitales, métodos de tele-
deteccíon. VI curso de geodesia superior, instututo
de astronomía y geodesia, Universidad Complutense
de Madrid, octubre 1989
Fuchs H.: Projektive Geometrie; Anwendungen in
Photogrammetrie und Robotik. Mitteilungen der geo
dätischen Institute der Technischen Universität
Graz, Folge 63, August 1988
Hohenberg F. u. J. Tschupik: Die geometrischen Grundla
gen der Photogrammetrie - in Handbuch der Vermes
sungskunde Jordan/Eggert/Kneissl, Band IIIa/3, J.ß,
Metzlersche Verlagsbuchhandlung, Stuttgart 1972
Rinner K.: Studien über eine allgemeine vorausset
zungslose Lösung des Folgebildanschlusses.
Österreichische Zeitschrift für Vermessungswesen,
Sonderheft 23, Wien 1963
Wolf P.R.: Elements of Photogrammetry. International
Student edition, Mc Graw-Hill Kogakusha ltd., 1974
