(3) 
1 and Sp we obtain the 
yz as: 
(4) 
-xyz to O-XYZ with the 
or à we get N as: 
0 
9 65:) 
yosin 6 
'ee components of N on 
ely. 
(6) 
direction cosines or the 
R. 
ally use six parameters 
ctor ) to describe it. Take 
and (a, B, y ) as the 
oquation can be written 
(79 
te of any one point on L, 
or of (X, Y, Z). 
om, C is often chosen the 
B,y) one unit vector. 
' of plane S-I-L, we have 
iship: 
(8) 
Nz (Zs-Zc) - 0 
it line in the object space 
4. From last section we 
e straight line L’ and its 
re in one plane. N' is the 
s the nearest point on L' 
ld coordinate system O- 
by the both planes S-L 
vector of L". From the 
we know that L" passes 
| the image plane, which 
is defined by all parallel lines of L. As mentioned above, 
N and N’ are both perpendicular to L and L’. So the 
vector N" can be obtained by the cross-product of N and 
N° (ie. N” = N x N’ see figure 2). 
S 
NA 
L' 
/ MI 
Figure 2. Geometric Constrains Between Two Lines 
Now take the normalization forms of these direction 
vectors we get three unit vectors of L, L' and L^ as: 
Ix=Px=Px 
IysIyzry (9) 
IzzIz-zI"s 
Where I, I' and I" are the unit direction vectors of L, L' 
and L” respectively. 
In ( 9 ) there are three unkonwns @, ®, K and four 
observed values 0, op, 60',p'. Now we get the basic 
mathematical equations to calculate the focal length 
and the attitude of camera as following: 
I”=F,-(0, op, 0’, p’, 9, œ, x) (10) 
N'-Fe(0,0,0' p', 9, ®, K) (11) 
Where, F = function of0, p, 0',p', P,œ and x. 
Horizontal Situatà 
When L and L’ are horizontal lines, their components 
projected on the Z-axis of the world coordinate system 
equal zero. This case is true because these horizontal 
lines abound in the man-made scene. Thus ( 11 ) can be 
written as: 
N°7(0, p, 0°, p’, 9,œ,K)=0 (12) 
Vertical Si ; 
Similar to ( 12 ), when L and L’ are vertical lines, we 
will get two equations as: 
N°x(0, p, 0’, p’, 9,0, x)=0 
(13) 
N”’y(0, pop, 0’, p’, Q,®, K)=0 
2.3. Calculation of the Focal Length and the Attitude of 
Camera 
To keep the mathematical model as general as 
possible, it is assumed that the unknown parameters 
25 
have a priori measured values and approximated 
values. This will contribute to the following: 
L’+V=L 
(14) 
X? - AX-X 
where, L?, V, and L = observations, residuals, and 
adjustments matrices for the observations (0, p, 8°, 
p»; 
X?, AX, and X - approximates, corrections, and 
adjustments matrices for the unknown parameters ( @, 
®, €, and f). 
After linearizing ( 12 ) and ( 18 ) by Newton's first- 
order approximation, and assuming that more than 
three pair of parallel lines, which can be horizontal or 
vertical, are available in the object space, the 
mathematical model can be written in the following 
matrix form by combining ( 12 ), ( 13 ), and ( 14 ): 
AV+BAX+W=0 (15) 
where, A = residual coefficient matrix of V; 
B = correction coefficient matrix of AX; 
W = the constant column matrix. 
The least-square solution to this model results in the 
following normal equation: 
AX = -(B"(APA")"B)!B"(AP-AT)"W (16) 
Where P = weight matrix corresponding to the 
observations L. 
The calculation must be done iteratedly until to the 
allowance error. 
2.4. . Calculation of the Position of Camera 
After the attitude of camera have been obtained the 
normal vectors of all planes are known. From equation 
( 8 ) we see, only the camera position parameters are 
unknown. To calculate the three unknown parameters 
we only need three control lines, which are all not in 
the same plane. The solution equations are: 
Nx,(Xs-Xc,) + Ny,(Ÿs-Yc,) + Nz, (Zs-Zc,) = 0 
Nx,(Xs-Xc,) + Ny,(Ÿs-Yc,) + Nz,(Zs-Zc,) = 0 (17) 
Nx,(Xs-Xc;) + Ny,(Ÿs-Yc,) + Nz,(Zs-Zc,) = 0 
3. RELATED IMAGE PROCESSING STRATEGY 
As previously mentioned, the systemic errors such as 
lens distortions and scale difference of two directions of 
one pixel must be corrected before the determining task. 
The calibration method requires completion of following 
major tasks: 
€ Image smoothing filter.