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NOTES ON THE DIRECT PROJECTIVE TRANSFORMATION OF GENERAL STEREO PAIRS
INTO THE RIGOROUS NORMAL CASE BY IMAGE CORRELATION
Gerhard Brandstätter, Prof. Dr.
Graz University of Technology, Austria
ISPRS Comm. III
Abstract: The problem of direct projective trans-
Formation from the general to the normal case of
stereophotogrammetry is treated by means of image
correlation. Therefrom result linear equations con-
taining optimal approximate values of relative
orientation, which are to be introduced into a
post-adjustment because of the redundancy of this
method. The resulting error propagation is dis-
cussed and finaly an example for a digital stereo
pair is given
KEY WORDS: Projective transformation, normal case,
image correlation, digital stereo images.
0. INTRODUCTION
In Vol.12, No.1(1990) of the photogrammetric jour-
nal of Finland H. Haggren and I. Niini published a
method for the 2-D projective transformation of
general stereo pairs into the strictly normal case
of photogrammetry. Their method is based on the
correlation of two overlapping projectivities of a
spatial object (Thompson 1968), from which the pa-
rameters of transformation can be derived. Since
the correlation refers to metric images, its effect
corresponds to the method of linearization by re-
dundant observations, because eight homologous
points are needed. This method is already known
from (Rinner 1963) as "unconditional conjunction of
Successive images" and delivers two components of
the base (bz,bs) and three rotations of the second
image.
The goal of the transformation to the normal case
is to obtain parallel epipolar lines in order to
facilitate the automatic search for homologous
points in the reconstruction of the object from
digital stereo pairs (Kreiling 1976). Thus the
parameters of Rinner’s method are not very useful,
because the normal case does not arise directly
therefrom. In contrast to this, the other possi-
bility of relative orientation, i.e. the use of
rotations only (Brandstätter 1991), delivers the
convergency and consequently the parameters of the
desired transformation.
1. THEORETICAL ASPECTS
1.1 Condition of intersection and projective trans-
formation
Using the analytical quantities
R- [, j, k] matrix of orientation (recon-
struction)
E unit matrix ( RTR = E )
XT z (x,y,-c) vector of centered image co-
ordinates
p-Rx projector in the model space
Xo (1) center of projection
b' - (bi,b2,b3) stereo base ( b - Xo"-Xo! )
À scalar coefficient (stretching
factor)
the reconstruction of a point X of the model space
from the coordinates x' and x" of the two images P’
and P" (condition of intersection) reads
X z Xo'* )'R'X! z Xo" + À"R"X" (4.1)
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and the coordinates in one of the two images arise
from the projection
AX = RT ( X - Xo ). (1.1.2)
If R does not yet contain the elements of absolute
orientation, its parameters 6’, K’, Q', 9", K' ( 2"
-difference of lateral tilts ) represent only the
relative orientation. The desired normal case ( de-
fined by the unit matrix E ) results analogously to
(1.1.2) from
ANXN - E( X - Xo ). (1.1.3)
Introducing X from (1.1.1) this relation converts
to
MXN = E ( Xo * ARX - Xo ) - AR x
and the direct projective transformation to the
normal case is given by
UN Rx, © = MW/N (1.1.4)
or after elimination of the unknown coefficient rc
by formation of the quotients -xn/c and -yn/c
i1x+jiy-kic e1.X
XN = -¢C — m — — —— = -C ;
isxtjsy-ksc e3.X
(4.1.5)
t2x+j2y-Kk2C e2.X
YN = -C ————— z -C ;
isxtjsy-ksc es.x
wherin the ei (i - 1,2,3) are the rows of R. These
equations correspond, of course, to the equations
of (Kreiling 1976) but also to those of (Haggren
and Niini 1990), disregarding the formal discre-
pancy that there the last number of the denominator
equals 1. The aim of this method is therefore, to
find the unknown orienations of the two images.
Knowing xn, the quotient t can be determined from
T2XNTXN = (Rx)T(Rx) = xTRTRx = XTX
regarding p? = p12+p22+p32 = x2+y2+c2 = x.x ( X.X
is equivalent to xTXx ), as
AKA p
T = = — ; (1.1.6)
4 XN « XN pN
the ratio of the two distances from the common
center of projection to the points x (original) and
XN (transformed).
1.2 Orientation from image correlation
Using b, the condition of intersection (1.1.1) can
also be written as
\’p’= b + À"p" (1.2.1)
from which follows after vector multiplication by
b and scalar multiplication by p' because of
