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

resulting 8x9-system reads (without i)

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

already

Al 1 th<

orienta

determ1

Their s

the horn

buted i

hedron