c ÁáÓ 
X 
rectangular coordinates 
(156,65), AL 20 : d') 
l't! and t? are metric 
g A' 
N á 
N 
N / 
X 
VOU 
) "6 2 
AL A A” 
43 5 
e 
————————— Bn 
rectangular coordinates 
ie following steps: 
between the correspon- 
lex i21,2,3,4 
x and v' € 7 are found. 
responding linest € a = 
should be the trace of 7 
. Vienna 1996 
a:in plane a 
7 | I> ug 
   
b:in palne 7 
Figure 4:Homogeneous and related coordinates. 
4. The plane 7 is set to coincide with a, such that 
the congruent lines t and /' identically coincide. 
In this position, the space coordinates of the im- 
age points are determined, as though they belong 
toa 
5. The collineation between the two sets in o is de- 
termined through the perspectivity in the plane 
a: axis t and center So. 
6. The true position of the camera station S in 
space lies on the circle of rotation of Sy about 
the vanishing line v [1]. Its plane v is perpendic- 
ular to t. 
7. Through the collineation in step 5 above, a point 
T € a is found to correspond to As (still in a). 
Hence the point T € a and the control point As 
in space must be collinear in space with Aj in 
its true position in 7 in space. The line joining 
As and T' must therefore pass through S in the 
plane v. 
3.3 Transformation Equations between 7 and 
a 
Fig.(4a,b) show the four pairs of corresponding points 
Ailzo : zi : z2) — All(zo : x} : 22) with their 
homogeneous coordinates as given before. The equa- 
tions of such linear transformations can be expressed 
in the form: 
zo = 00% + any + 0222 
Zh. = mofo hi G% T 01222 (1) 
zh = agro + 2121 + 2222 
where a;; are constant coefficients. 
Substituting the coordinates of an object point in the 
3 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
R.H.S. of equation(1) and those of the correspond- 
ing image point multiplied by an unknown factor in 
the L.H.S., we get three equations. Totally, we get 
twelve equations in the nine coefficients and the four 
unknown factors. Since only the ratios between the 
factors are of interest, one of them can be arbitrarily 
chosen. The solution of these equations are simple 
and yield: 
ajo = 039—021 — 0415 — 0 
dos = be,cad(b'd' = c! d' Sed cb) 
ann = b'd'eica(bd — cid — cab) 9 
azo = b'd'cicy(bd — cid - cab) ( ) 
ao, = ca[b'e1d(c, — d') — bc, d'(ca — d)] 
a92 = c,[b'chd(b — c:) — bcad'(b' — c*)] 
Similar formulas are found for the inverse transfor- 
mation 
_ / / / / / / 
Top = 0070 + 40171 + 0272 
= / / 
Tye w= 01121 (3) 
— / / 
Zr» = A229 
with coefficients: 
/ ’ / 
foe y — c b, c 
dog = Zen, 9j — do, do2 = = 902, 
fours € p uc 
dj; — 72000, | 055 = 7 400 (4) 
4 
3.4 The Vanishing lines 
The vanishing line v in o corresponds to the line at 
infinity in v. It can be determined using the points 
at infinity of both z4 and z5 axes, whose coordinates 
are (0 : 1 : 0) and (0:0:1) respectively. 
Substituting in equation(3), we get the vanishing 
points: 
Vi(a), : 44) 70), V2(al, 0 ads): (5)