189 
The Inhomogeneous coordinates of the transformation 
p* —> p" are therefore 
Mi u* ’ 
u 1 " = . 
Mo + (mi~Mo)u 1 ’ + (m2~mo)u 2 ’ + (m3~mo)u 3 1 
1-Ums 1 -um3 2 
UM3 1 
UM3 : 
Mo- » 
Mi “ « 
M2- 
1 —U3 1 —U3 2 
U3 1 
U3 2 
and the affine coordinates 1n the map of an art) 
trary point of the image may be calculated from 
Those formulas hold 1f and only 1f all coordinates 
u 1 refer to the system of the basic points Gh . 
Therefore the measured coordinates v 1 must be 
reduced to this system using the spatial homo 
geneous affine transformation u=Av, where 
A = 
10 0 0 
aio ai 1 ai 2 ai3 
a2o a2i 322 a23 
a30 aai a32 a33 
and the aij can be computed from the measured 
coordinates of the basic points by 
Avi = ei 1=0,1,2,3 
ei 1s the homogeneous “unit vector" with 1 at the 
positions 0 and i. 
1.4 Transformation between protective planes 
The transformation between projective planes meets 
the well-known rectification of linear images of 
plane objects. The algebraic-projective procedure, 
which represents a very elegant proof of this 
method, may easily be derived from the relations of 
section 1.3 omitting all fourth components. Its 
spedai treatment is given in {Brandstatter, 1989). 
The dimension of the projective spaces "map" and 
"Image" is n=2, the required number of control 
points n+2=4. Three of them (fig.2) define the 
respective affine systems. As in the map all points 
have orthonormal coordinates, the homogeneous 
Fig. 2: The affine coordinate systems of map and 
linear image 
position vectors Xi T = (1,Xi 1 ,Xi 2) have to be 
converted to the affine system of the map by AmXì = 
= um . The components of Am result from AmXi=0ì, 
1=0,1,2, with the determinant 
D = j X11 X12 
- 
X01 X02 
+ 
X01 X02 
j X21 X22 
X21 X22 
X11 X12 
as 
1 ‘ D 0 0 
Am = ~ X02X21-X22X01 X22-X02 X02-X12 , 
D X01X12~Xi1X02 X01-X21 X11-X01 
• 
and the fourth control point now is um.i = AmX3 . In 
the image space the elements of the affine trans 
formation have quite the same shape replacing Xjk 
by vjk and the fourth point 1s U3 = AV3. According 
to (1.3.4) the factors pi are 
Miu 1 
um 1 = 
MO + (pi~Mo)u 1 +(M2-Mo)u 2 
(1.4, 
Flnaly its orthonormal coordinates are derived fr 
Am ' 1 um =X, where the components of Am -1 may 
determined from AM _1 ei=Xi as 
Am* 1 
1 0 0 
X01 X11-X01 X21-X01 
X02 X12-X02 X22-X02 
1.5 Optical rectification 
The procedure of 1.4 could be used for the comput 
tional rectification of digital images.But also t 
traditional optical rectification is a projectl 
method, where the transformation is realized by t 
laws of geometric optics. By means of the foe 
length f of an ideal lens the projection is defin 
by the regular projective matrix 
Pf=- 
f 0 0 1 
0 f 0 0 
-1 
or Pf _ 1 = 
1 0 0 -1/f 
0 10 0 
0 0 f 0 
0 0 G f 
f 
0 0 1 0 
0 0 0 1 
Introducing orthonormal coordinates X,Y,Z (= systi 
of rectifier, fig.3) in order to get metric settii 
elements, the homogeneous position vectors are i 
= (XO ,X1 ,X2 ,X3 ) , X0 = const. (X=X1 /xo , Y=X2/XO ,Z = Xj/X| 
and the projection is performed by x ’ =Pfl 
Therewith the contents of an image plane 
P : h T x = hoxo + hixi + 112x2 + h3X3 = 0 (1.5: 
are to be rectified into a map 
Pm: h’ T x’= ho’xo’+ hi’xi’+ h2’x2’+ h3’x3*= 0. 
The coefficients h’ depend on h because of x=Pr'i 
and therefore h T Pf 1 x’=0, by the simple relation 
h’ T = h T Pr ‘ 1 , (1.5.1 
yielding the equation 
fhoxo’+ fhixi’+ fh2X2*+ (fh3-ho)x3’= 0 (1.5.1 
of the plane Pm. For X3 = X3’=0 this equation ar 
(1.5.1) produce the same homogeneous line S 
hoxo + hixi + h2X2 = 0, 
in the principal plane H of the lens, proving ' 
the condition of Scheimpflug. In order to achiei 
sharp projection over the whole area of the recti 
fication this condition must be satisfied. The Hi 
S, which may be given by the equations 
B T u = 0 in P and Bm t um = 0 in Pm , 
must be parallel to : 
the image of the horizon (fig. 3), represented 
P by the homogeneous equation 
Mo ♦ (pi - mo )u 1 + (M2 - ¡Jo ) u 2 = a T u = 0 
from denominators in (1.4.1) and 
the var 
equatior 
1/Mo+O/ 
which r 
(1.3.2) 
two coef 
Bi =ai , 
Bm 1 = om 1 , 
In analc 
transfor 
an T =a T K' 
As the 
yields u 
Qm 0, qm 
from whi 
Bmo = 1 + 
By means 
plane cc 
can be 
con sum in 
section 
the X-ax 
setting 
performei 
usual 1 y 
1974). T 
lines 
G: ho+h2' 
and the ■