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# Full text

Title
Modern trends of education in photogrammetry & remote sensing

191
transition
n=2 point —> straight line
n=3 point —> plane
n>3 point —> hyperplane.
In an orthonormal vector space V 3 C can be derived
easily from the well-known condition of coplanarity
(f1g.4) of the position vectors yn, ys and the
vector yo (base vector or epipolar axis) between
the centers of projection Op and Os. Using (1.2.3)
with Br*=R, Bs*=S the position vectors read yR=Rx R ,
ys=Sx s and the condition turns to
(yn ,yo ,ys)=yR T
0
y0 3
— yo 2
-yos
0
yoi
yo 2
-yo 1
0
ys=XR T R T DoSxs
0.
Hence C results from the product
C = R T DoS, (2.1.2)
has the important attribute det(C)=0 because of
det(Do)=0 and contains the components of the base
vector in skew-symmetric form. Thus h R must equal
the null vector if Os is projected and vice versa,
because according to (1.2.3) XK 3 = -S _1 yo and there
fore
hx R =-CS~ 1 yo=-R T DoSS _1 yo=-R T yoxyo=0.
Consequently the epipoles follow from the homo
geneous systems
C T R _1 yo= C t xk r =0 in Pr (2.1.3)
CS _1 yo= Cxk 3 =0 in Ps,
which can be solved because of the already men
tioned attribute det(C)=0. By means of (2.1.3)
moreover
(h R ) t Xk r = (Cx 3 ) T x« R = (x 3 ) t C t xk r = (x s ) T 0=0
follows and # therefore the important fact, that the
coordinates of the epipole satisfy identically
every straight line h R =Cx 3 or, in other words,
every linear entity (h R ) T x R =0 contains the epipole
xk r and represents therefore an epipolar line. In
this way it is possible to calculate the matching
epipolar line in Pr/Ps directly for an arbitrary
point of Ps/Pr, i.e. the geometric locus of its
homologous point. Along this line the point must be
situated by non-geometric informations as the
profile of gray levels and its derivatives.
2.2 Dete.rnitnat.i on._and aggllQa.tlQn_.Qf-C
C consists of nine unknown components Cjk. with
regard to Its homogeneity, C can be normalized
dividing 1t by one component, so that eight
unknowns are left to be determined. The norming
component is not allowed to vanish and, for prac
tical computations, it should be the numerically
largest one.
Introducing now mensurable affine coordinates, the
condition (2.1.1) changes, because of (1.2.5), that
is x r -(1/o)Uur and x 3 =(1/o)Uus, to
(Ui R ) T Cui 3 = 0 (2.2.1)
with
C = U T R T DoSU . (2.2.2)
The procedure based on (2.2.1) requires coordinates
of eight homologous points of Pr and Ps 1n oppo
sition to conventional photogrammetry, where only
five orientation points are needed. One row of the
C00+Ufl 1 Cl0+UR 2 C20+US 1 C01+UR 1 U3 1 Cl1+UR 2 US 1 C21+
+US 2 C02+UR 1 US 2 C12+UR 2 US 2 C22=0.
In general it is not possible to predict the most
stable component, but in the majority of practical
photogrammetric applications with orthonormali zed
coordinates the values of C12 or C21 will be larger
then those of the other components (Rirmer, 1963).
Setting Zjk=Cjk/c2i a linear system Az=a is ob
tained, from which the normalized components follow
by z=A"'a. C consists therefor of the calculated
Zjk and Z2i=1.
coordini
elementi
2.3 Reli
The rei.
model s
structec
row
zoo
20 1
Zo 2
Z10
22 0
Z 1 1
Z1 2
Z22
-a
0
1
0
0
0
0
0
0
0
0
1
1
1
0
1
0
1
0
0
0
2
3
1
0
1
0
1
0
0
1
0
7
1
us 1
us 2
UR 1
UR 2
UR 1 US 1
UR 1 US 2
ur 2 us 2
UR 2 US 1
Tab. 1: Coefficients of the 8x8-system A for the
computation of the normalized components
Zjk=cjk/c2i. The absolute vector a con
sists of the coefficients of -C2l.
y = yOR
which a
orthonoi
and wh<
centers
homogen«
equivali
y = irTi
or else
TRXR =
Considering three of the homologous points as basic
points G, the affine coordinates may be referred to
the axis across them (fig.3).In this way three rows
of A contain very simple coefficients (tab. 1) and
yield the relations
ZOO=0, ZOI=-Zl0-Z11, Z02=-Z20-Z22,
arise.
Tr yo R “
because
yield tl
over, tl
whereby three unknowns* which indicate only
relations between the affine systems, may be
eliminated. The remaining five equations show the
well-known fact, that the intersection of five
corresponding rays of the two bundles of projection
satisfy the relative orientation of the images Pr
and Ps.
In order to avoid singularities of A, the coor
dinates of the correlation points do not dare to be
linear!ly dependent. Thus they should not be points
of planes passing three other points including the
centers of projection.
If C has the structure of (2.2.2),the first columns
of the 3x3-matrices (2.1.3) ace because of u T =
= (1,u\u 2 ) the absolute vectors and the solutions
of these linear systems give directly the affi Re
Tr yx =
of any
produce