We assume a number of points X," on straight line 
features of the "as is" object being measured. We only 
know the "blueprint" expressions for these straight lines: 
E,® s [E.? 4 $,.Cos o.Cosf],, 
N,? z [N,? « S,Sin a.Cosf],,,,, (2) 
HP = He * S,Sin fli, 
Where: 
S, -the distance of point A from reference point 0; 
a and B = the azimuth and vertical angles of the straight 
line feature, respectively (Fig. 2). In matrix notation this 
becomes: 
XP? =x? +S, TRIG (3) 
Where: 
Cos a.Cos p 
TRIG = Sin a.Cos B 
Sin p 
For every measured point A on a line of the "as is" model, 
three observation equations can be formulated for the 
seven unknowns, A, AX, R of the transformation (1), and 
the one additional unknown S, locating the point A on the 
line: 
X," = ARX,P + AX 
Where: 
X? - X? S A.TRIG 
The unknown transformation parameters, and the additional 
point unknown, S, may be determined from these 
equations by standard least squares adjustment, if sufficient 
measurements are available. 
How many measurements are needed to arrive at a unique 
solution to the problem? Let us assume we measure N 
points in the "as is" system, this gives 3N observation 
equations for the 7+N unknowns, leading to the inequality: 
3N>7+N. (4) 
At least four points have to be measured in the "as is" 
system in order to arrive at a unique solution for the 
system. These four points obviously need to be positioned 
on at least two linear features in order to avoid singularities. 
We may, for example, measure two points each on two 
linear features. 
J qu 
qu P. A o 
5, N y N 
fi 
line 1 
    
   
  
   
  
  
JI 
aman" 
Xo» 
ULL 
ine 2 
  
  
o denote measured points x, 
Figure 2. Absolute orientation: 
Determination of the transformation parameters from 
one coordinate system to another using straight line 
features only. 
We may verify our conclusions with the following example: 
We wish to find the transformation parameters between the 
coordinate system of a cube in object space and its blue 
print shape. 
line 2 
688 
  
Figure 3: Exemplification of Absolute orientation Problem 
Measuring only one line A on both cubes, does allow us to 
rotate the object cube into the blue print space, but we 
know neither its position along that line, its orientation with 
respect to the line nor its scaling. Measuring the two 
rectangular lines A and B, the object cube can now be 
positioned and oriented, but we cannot determine the scale 
factor between object and blue print space. Only when 
measuring two non-coplanar lines A and C, can the scale 
factor also be determined and the absolute orientation 
problem completed. 
4. RELATIVE ORIENTATION USING EDGES 
We start from the well known projective relationships 
(American Society of Photogrammetry, 1980) relating the 
object space to the image space: 
Xf! - P (X, , Ya) ) (5) 
Where: 
X, 2 vector of image coordinates (x,y) of point A in image 
(1); 
X, = vector of model coordinates (E,N,H) of point A; 
Ya 7 Vector of 6 parameters of image (1); being the tilts, 6, 
0, x and projection centre coordinates (E,N,H); 
P = projective relationship. 
In relative orientation using edges we assume that points 
on linear features are measured monoscopically both in the 
left and right image. Non homologous points are observed, 
that is, points on the linear feature measured in the left 
image differ from those in the right image. This will usually 
be the case in automatic industrial metrology, and no online 
image correlation is required. 
The following observation equations can be formulated for 
two points A and B measured on one and the same linear 
feature of the object: 
Left image: 
x" zP(X,; Y O) (6) 
X,= X, + S, TRIG 
Right image: 
X= P (Xs ; Yo ) (7) 
X, 2 X, * S,. TRIG 
The image parameters of the left image are set to zero. 
Five parameters of the right image are unknown after 
choosing a suitable model base. 
In addition, for every linear feature in model space, there 
are five unknowns: the three coordinate values X, of one 
reference point on the line, and two directions a, B (Fig. 4). 
Also, for every measured point, the distance S from the 
reference point X, is unknown. (For the first point 
measured on a new line, say point A, we choose S to be 
zero.)