The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part B4. Beijing 2008 
Schenk argued that lines can be taken as an alternative to points 
in photogrammetry, because ”1. A typical area of scenes 
contains more linear features than well defined points. 2. 
Establishing correspondences between [linear] features in 
different images and /or images and object space is more 
reliable than point matching” (Schenk, 2004). 
2.1 Line representation 
To solve fundamental tasks of orientation with straight lines, 
optimal line representations in 3D Euclidean space should meet 
the following requirements: 
• The number of parameters should be equal to the degree of 
freedom of a 3D line. 
• Representation should be unique and free of singularities. 
Parametric representation of manifolds in object space allows 
the specification of any point on the feature. A four parameter 
representation of the line is uniquely given by the 4- tuple 
(<P,0,x o ,y o ). 
R^e ~ Re ' R* ~ 
cos0cos</> cos#sin^ -sin# 
-sin^ cos^ 0 
sin# COS (f) sin# sin (f> cos# 
(1) 
A point on the straight line is represented with respect to the 
new coordinate system according to: 
(2) 
COS#COS^- x 0 -sin(Z)-y 0 +sin#cos^-s 
x' 
P = Ke ' P' = 
cos#sin^-x 0 + cos^-_y 0 + sin#sin^s 
= 
Y 
-sin#-x 0 + COS#-i 
z 
X, Y and Z are object coordinates of the point. 
2.3 Combination of collinearity equations and four 
parameter representation 
The parametric representation of the manifold in object space 
allows the specification of any point on the manifold. The 
collinearity model looks for the closest point in the manifold to 
the projection ray. Equation (3) defines a parametric 
representation of 3D straight lines, and expresses the 
relationship between an arbitrary point p on the 3D line and the 
four parameter representation (Schenk, 2004). By means of the 
known EOP derived from aerial triangulation and the four- 
parameter equation, the collinearity equations can be applied to 
the running point on the straight line in object space. This is 
achieved by inserting the parametric representation of the line 
into the collinearity equations. 
■u' 
f 
cos#cos^x 0 -sin^-+sin #cos^-s 
V 
= R T ■ 
cos# sin^ • x 0 + cos </>-y 0 + sin # sin^ • s 
- 
w 
-sin#-x 0 + cos0-s 
Z 0. 
J 
Figure 2. Illustration of the concept of 4-parameter 
representation (adapted from Schenk, 2004) 
, U V 
x = -f — y = —f 
W W 
(5) 
0 is the azimuth and 0 is the zenith angle of the straight line. A 
new coordinate system is defined in such a way that the new Z 
axis is parallel to direction of the straight line. The position of 
the line is defined by the point (x 0 , yo) where the X-Y plane in 
rotated coordinate system is intersecting the line. All the points 
on the straight line have a value namely S; as a point position 
parameter along the line. Thus the distance between the Sj and 
Sj+i is calculated as s i+t - Sj (the unit is in meter) (Schenk, 2004). 
2.2 Mapping between point representation and parametric 
representation of 3D straight line 
To find the rotation matrix between the original (initial) 
coordinate system and the line oriented coordinate system, the 
following expression is applied. As a first step, there should be 
a conversion of coordinates from Cartesian (X, Y, Z) to the 
spherical coordinate (0, #, p). p is not applied in this task 
because the procedure doesn’t deal with the radius. The rotation 
matrix which is necessary for this conversion is R^ 0 : 
Regarding to the equation (3) and general form of collinearity 
equations (5), there will be five unknowns in the final combined 
collinearity equations. As the EOP are considered to be known 
in the study, the unknown parameters are the four parameters of 
line representation (<j>,0, x 0 , y 0 ) and one additional unknown (s) 
for each point as an element of the straight line. 
2.4 Adjustment model to solve unknown parameters 
The general form of adjustment which will be used for 
nonlinear observation equations is represented below (least 
square adjustment): 
Sx = (A T A)' A(l-l 0 ) (6) 
In this formula A is the design matrix, / is observation vector 
and l 0 approximate values for the observations based on initial 
values for the unknown parameters. 
The residual vector v follows from