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| Archives of 
Vol 28, No.4. 
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AT: Frankfurt (1983- 
a 1996 
ON CRITICAL CONFIGURATIONS OF PROJECTIVE STEREO CORRELATION 
Gerhard Brandstátter 
Professor, Institute of Applied Geodesy and Photogrammetry 
Technical University Graz, Austria 
Commission Ill 
KEY WORDS: Projective photogrammetry, stereo correlation, critical configurations 
ABSTRACT: 
Stereocorrelation in projective photogrammetry is performed by means of eight irregularly distributed points and 
yields eight linear equations for the determination of 
configurations are characterized by vanishing of the determinant of those eight equations, wherefrom critical 
point distributions may be derived. Of course, the well-known critical configuration of five orientation points will 
belong to it but with respect to eight points, additional conditions of singularity will arise. The following 
considerations contain investigations on this subject using projective relations in homogeneous vector notation. 
KURZFASSUNG: 
the components of the correlation matrix. Critical 
Stereokorrelation der projektiven Photogrammetrie erfolgt mittels acht unregelmäßig verteilter Punkte und 
beruht auf acht linearen Gleichungen für die Bestimmung der Komponenten der Korrelationsmatrix. Kritische 
Anordnungen sind charakterisiert durch das Verschwinden der Determinante jener acht Gleichungen, woraus 
die kritischen Punktanordnungen abzuleiten sind. Natürlich wird auch die wohlbekannte kritische Konfiguration 
der fünf Orientierungspunkte dazugehören, bei acht Punkten müssen aber noch weitere Singularitäts- 
bedingenen auftreten. Die nachstehenden Ausführungen enthalten einige Entwicklungen zu diesem Problem 
unter Verwendung projektiver Beziehungen in homogener Schreibweise. 
0. INTRODUCTION 
Algebroprojective photogrammetry deals with pro- 
jective images of completely unknown interior 
orientation. The coordinate system of such images 
cannot be referred to a more or less orthogonal and 
isometric external coordinate system, but to an 
(affine) internal system defined by suitable object 
points which are projected to homologous points in 
the images. In this case, stereo correlation results in 
the determination of eight parameters, consisting of 
the usual set of relative orientation and an additional 
set of three parameters related to an interior affine 
coordinate system defined by three non-collinear 
homologous image points (Brandstätter 1991). These 
two affine systems are corresponding images of the 
first coordinate plane of a tridimensional affine 
coordinate system in the object space and produce 
most uncomplicated projective transformations. 
The parameters of stereo correlation are the eight 
significant components of a 3x3-matrix C, the core of 
the homogeneous coplanarity condition. Applying this 
condition to eight pairs of homologous points, a 
system of 8 linear equations results wherefrom the 
components mentioned above may be computed. The 
solution depends on the regularity of its matrix of 
International Arch 
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ives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
coefficients exclusively composed of the homologous 
affine image coordinates and hence on the spatial 
distribution of the corresponding object points in the 
tridimensional model space. The matrix will become 
singular if the positions of those points of correlation 
correspond with a critical configuration. The intention 
of the following treatise is, to discover this critical 
situation from the geometric relations and to find out 
a strategy to avoid it. For this aim the following 
symbols in homogeneous notation are used: 
homogeneouse affine image 
coordinates 
affine coordinates of the 
object space 
Yo center of projection 
M= {my} j- 0.12 
k=0,123 
P= {ow} j=k=0,1,2,3 regular projective matrix 
C= {ci}, i=j=0,12 matrix of correlation 
A = {an} |=-M=3...7 matrix of coefficients 
Q = {au} j=k =0,1,2,3 matrix of a quadric surface 
local stretching coefficient 
u! - (1,u,,u;) 
y! -(1y« Yo. Y3) 
singular projective matrix 
M eere SEN Bonner anne