| Qi Dj in (3.1).
l| points of the
quadric surface.
, must be points
rding to (1.2), all
> and hence the
adric surface) is
are well-known
famous "Vienna
nainly by Josef
ter Wunderlich
re achieved by
jased on a pro-
| projective geo-
gital photogram-
ntial ancl hence,
al situations of
braic analysis of
Q's in equation
| using the m, of
. They show that
Q22D22
=
ome
contains one significant component -0, that is qoo.
It represents the constant part of the tridimensional
equation of second order. Hence, apart of the
centers of projection the surface must pass the
origine Gg of the spatial affine coordinate system
as shown in Fig. 2. Therefrom follows that all
quadrics which can be seen from the base in the
mode of closed concave surfaces represent critical
loci of projective stereo correlation. The well-
known traditional critical quadrics cylinder, cone
and hyperbolic paraboloid form a subset of this
group.
Figure 2: Quadric surface and coordinate system
The coefficients in Tab. 2 depend on the para-
meters of relative orientation. They can be sub-
stituted by expressions depending on the co-
ordinates yg of the centers of projection if a pre-
determination of critical situations is required.
Regarding the structure (1.3) of M, equation (1.2)
can be solved with respect to the y. The results
read by means of the abbreviations U3p=1-U34-U32
and Yoo=1-Yo1-Y02"Y03
., Ua1Yo0 _ U32Y00 ye Yoo
U30Yo1 U30Y02 U30Y 03
(Brandstátter 1996). By their use, for any arrange-
ment base-to-object the critical locus of stereo
correlation is predictable.
For each column additionally exist individual
critical quadric surfaces which are caused by the
possibility that all its components become simul-
taneously zero. This case occurs if all points out of
the subject (3..7) satisfy one of the equations
y'Q,y-0. These surfaces must also pass the
origine and the centers of projection. Critical
surfaces not passing the origine will be defined by
proportional columns caused by dyadic products
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
resulting in constants Ci. that is vy Quy = Cj. If
two columns show that behaviour, critical loci exist
as well.
4. Final Remarks
Because of the above mentioned statements it will
not be difficult, to avoid critical situations of pro-
jective stereo correlation. First of all, there must
exist two points clearly outside the subset of the
others. Further, the internal coordinate system of
the model space must be defined by one of those
two and three non-collineatory points of the sub-
set. At last and as usual, all points of the subset
should not be located on surfaces similar to
quadrics passing origine and centers of projection.
Finally it should be pointed out that all methods of
projective relative orientation based on the matrix
of correlation (Rinner 1963, Haggren & Niini 1990,
Brandstátter 1992) also must obey these
guidelines in order to guarantee stable numerical
conditions.
References:
Brandstétter G. Zur relativen Orientierung projektiver Bündel.
Zeitschr. f. Photogrammetrie und Fernerkundung, 59. Jg. Heft 6,
Karlsruhe 1991, pp. 199-212
Brandstátter G. Notes on the Direct Projective Trans-
formation of General Stereo Pairs into the Rigorous Normal Case
by Image Correlation. XVII ISPRS Congress, Comm. Ill,
Washington 1992, pp. 701-706
Brandstátter G. The Inverse Problem of Algebroprojective
Photogrammetry. ^ Proceedings of the International Conference
on Inverse Problems, Potsdam August/September 1993,pp. 87-99
Brandstátter G. Fundamentals of Algebroprojective Photo-
grammetry. Session Report of the Austrian Academy of
Sciences, May 1996, in edition
Fuchs H. Projektive ^ Geometrie; Anwendungen in
Photogram- metrie und Robotik. Mittlgn. d. geodátischen
Institute d. TU Graz, Folge 63, Graz 1988
Haggren H. & I. Nini Relative Orientation Using 2-D
Projective Transformation. The Photogrammetric Journal of
Finland, Vol. 12, No. 1, 1990, pp-22-33
Krames J. Über die bei der Hauptaufgabe der Luftphoto-
grammetrie auftretenden "gefährlichen Flächen". Bildmessung
und Luftbildwesen, 1/2, 1942, pp.1-18
Rinner K. Studien über eine allgemeine, voraus-
setzungslose Lósung des Folgebildanschlusses.
ÖZfV, Sonderheft 23, Wien 1963
Rinner K. in Handbuch der Vermessungskunde J/F/K, Band llla/1
Photogrammetrie. —J. B. Metzler, Stuttgart 1972, pp. 422-427
Thompson E.H. The Projective Theory of Relative Orientation.
Photogrammetria, Vol. 23, 1968, pp. 67-75
Wunderlich W. Zur Eindeutigkeitsfrage der Hauptaufgabe
der Photogrammetrie. Monatshefte für Mathematik und
Physik, 50. Bd. 1941, pp. 151-184
