ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
195 
The plane defined by perspective center of image S and straight 
line I is called interpretation plane. According to coplanar 
coordination of perspective geometry, the straight line L in the 
object space is on the plane defined by the straight line I 
(projection of L) and Sp. So, L is perpendicular to the normal 
vector of plane defined by S and I in S-xyz .the dot-product of L 
and n is equal to zero 
-vyvX + v*-vY+ d - Vx .--^- + ^ y ~. JC ~. vZ = 0 (5) 
/ 
Equation (5) may be expressed by 
■Vy • a + Vx • ß + 
di - Vx • JA) + Vy XO 
/ 
= 0 (6) 
vA" 
where OC = 
vZ 
(7) 
When L is parallel to the X-axis in the object coordinate system, 
we can transform vector L from O-XYZ to S-xyz with the rotation 
matrix R and scale X.. 
vX~ 
c 
¿21 
bl 
C\ 
c 
¿21c 
vY 
= Â-'R-' 
0 
= r' 
¿22 
bi 
Cl 
0 
aie 
(8) 
vZ 
0 
¿23 
bi 
C 3 
0 
aie 
where c,0,0 are three coordinates of L on the X-,Y- and Z- axes 
in the object coordinate system respectively, 
substituting equation (8) into equation (7) 
a 1 
a 1 
A3 
¿23 
(9) 
<22 = k-(h2 + xo) 
ßi^kßgi + yo) (15) 
<23 = k • (/23 + xo) 
ßi = k-(gi + yo) 
where h2,g2,h 3 ,g 3 have meaning similar to h 3 ,gi,respectively. 
Considering orthogonal constrains of the nine elements in the 
rotation matrix R, we can get the following equations 
h-h+gi-gi-dh+h)-xo-d^+g2)-y)+X) 2 +y) 2 -\-f 2 =0 
h-h+gI • &Mh +/»)*AD+(gi +^)->o W +>o 2 +f= 0 (16) 
k-h+g2-&Mfa+h)-M-){gi+&)-y)+X) 2 +y) 2 +f 2 =0 
Obviously, in the linear equation above, the parameter x 0 ,yoand f 
can be solved. 
If we know the interior orientation parameters(x 0 ,yo,f)of camera, 
considering the equation (9),(12),(14) ,(15),we can get 
1 
<23 = —============ 
yj<21 2 + ß\ 2 — 1 
1 
C3 = . (17) 
yjcn 2 +/?3 2 -l 
y]oC2 2 + ¡3 2 2 —1 
Substituting equation (17) into equation (9) and (14), we can 
obtain nine elements of R 
where R = (cii,bi,Ci) 1 ,i=1,2,3 is the rotation matrix defining 
the camera orientation. ai,bi,ci are the direction cosines or the 
elements of the rotation matrix R. 
Suppose Hand I2 in the image plane are the projections of two 
straight lines parallel to the X axis in the object coordinate 
system. Equation (6) can be given by another expression of the 
following form respectively 
—Vy\ • <21+Vx\ • f3\+k • d\ - k ■ Vxi • yo+k ■ Vyi • xo = 0 
-Vy2-OC\+Vx2-/3\+k-d2~k-Vc2'yo+k-Vy2-XO=0 (10) 
7 1 
where K = — (11) 
/ 
According to equation (10) , we can obtain 
a\ = k-(h\ + xo) 
fi\ = k-(g\ + yo) (12) 
Vx2 • di — Vx\ • di 
where Vyl ’ Vx2 v >' 2 ' Vjrl (13) 
Vy2 ■ d\ - Vxl ■ d2 
gl = 
Vyl ■ Vx2 - Vyl ■ Vxl 
Similarly, when Li is parallel to the Y-axis in the object coordinate 
system or Li is parallel to the Z-axis in the object coordinate 
system we can get 
Oil - — 
bi 
to 
II 
Cl 
II 
CO 
a 3 = — 
C3 
b2 
bi 
C2 
C 3 
(14) 
According to collinear equations 
a\(X-Xs)+b\(Y-Ys)+c\(Z-Zs) 
X X °~ J a3(X-Xs)+b3-(Y-Ys)+C3(Z-Zs) 
a2(X-3S)+fe(y-K)+C2(Z-Z^) 
y yn ~ 3 cn(X-Xs)+bi{Y-Y)+a(Z-2s) 
If knowing the rotation matrix R, the camera interior orientation 
parameter(x 0 ,yo,f)and two control points, according to Taylor’s 
theorem, Eqs(18) may be expressed in linear form as 
fix fix fix 
X = (X) +—dXs + ——dYs + ——dZs 
dXs dYs dZs (19) 
y - 00+-¥-dXs + -^dYs + -^-dZs 
dXs dYs dZs 
where (x),(y) are approximates of function. dX s ,dY s ,dZs are 
corrections of X s ,Y s ,Zs respectively. Xs,Y s ,Z s are unknown 
parameters. x,y are observations. So, we can get the adjustment 
model and the least-square solution to equation (19). 
3 Computation of the distance ratios 
These lines in the image are assumed to correspond to lines 
in object space that are coplanar and parallel(see figure3-1). The 
image coordinates of the 4 corner points(i,j,k,l) of the 
parallelogram are arrived by measure manually or line 
intersection after edge detection. In the camera system, the 
image point can be expressed in vector built from the image 
coordinates^,y) and the focal length(f).This is the vector(x,y,-f)