The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2 QOS 
coordinates from the original coordinate system (X, Y, Z) 
to the line coordinate system (U, V, W), where the U axis 
is in the line direction (Figure 3.a). 
(3) Compute the variance-covariance matrix in the line 
coordinate system, Xuvw, for each of the two points A and 
B using the law of error propagation: 
follows: 
(1) Compute the weight matrix in the line coordinate system as 
follows: 
Pnw-R Pxrz R 
(7) 
(4) 
where Sxyz is the variance-covariance matrix in the (X, Y, 
Z) coordinate system (Figure 3.b,c), 
(4) Assign a large value for the variance in the line direction 
by applying a large scaling factor, m, (Figure 3.d): 
Where: 
p m is the weight matrix in the object coordinate system 
p mw is the weight matrix in the line coordinate system 
(2) Assign a zero value for the weights in the line direction; 
i.e.: 
a\= m <7„ 
Then, the new variance-covariance in the (U, V, W) 
coordinate system, £' rnf will be as follows: 
D 
0 
P'an,= 
0 
Pr 
Py,Y 
0 
PwY 
Pw\ 
(8) 
y 
La uvw 
(5) 
(3) Rotate the weight matrix to the original (X, Y, Z) system 
computing the new P' nz as follows, 
P'xrz^R P'uyhR 
(9) 
(5) Again, rotate the variance-covariance matrix to the 
original system (X, Y, Z) and compute the new X'*£S 
follows: 
(4) Apply a point-based solution using a least squares 
adjustment with the new P' xn • 
y 
La X) Z 
-r'Y\„r 
(6) 
(6) Apply a point-based solution using the two collinearity 
equations 2 and 3 with the new Ylm 
z 
a) Line coordinate system 
((/, V, tE) 
b) Original error ellipse in 
(X, Y, Z) coordinate system 
c) Error ellipse in ([/, V, fV) 
coordinate system 
d) Error ellipse in (U,V,W) 
coordinate system after expandinj 
the variance-covariance matrix in 
the line direction 
w 
2.2.3. Applications of point-based approaches for linear 
features 
In the this sub-section, the focus will be on the photogrammetric 
applications of the point-based approach for lines using both 
frame and line cameras by expanding the error ellipse, or 
restricting the weight matrix, in the image or object space. The 
incorporation of linear features in photogrammetric 
triangulation will proceed as follows: 
(11 Single Photo Resection Using Control Lines: 
In the case of single photo resection, the error ellipse expansion, 
or weight restriction, can be applied in object space or in image 
space. When expanding the error ellipse, or restricting the 
weight matrix, in object space (Figure 4.a), the image line will 
be represented by two end points with their variance-covariance 
matrices defined by the expected image coordinate 
measurement accuracy. On the other hand, the variance- 
covariance matrices of the end points of the object line are 
expanded to compensate for the fact that the image and object 
points are not conjugate. It should be noted that it does not 
matter if we use frame or line cameras. 
Figure 3: Expanding the error ellipse in the line direction 
2.2.2. Weight matrix restriction in the line direction 
This approach is similar to the previous one except that instead 
of a variance expansion a weight restriction is applied, i.e. the 
weights of points along the linear features are set to zero. The 
weight matrix restriction can be done in either image or object 
spaces. In image space, we use a 2x2 weight matrix, while in 
object space, weight matrix is 3x3. An explanation of the 3x3 
weight matrix restriction in object space is introduced as 
When expanding the error ellipse, or restricting the weight 
matrix, in image space (Figure 4.b), the object line will be 
represented by its end points, whose variance-covariance 
matrices are defined by the expected accuracy of the utilized 
procedure for defining these points. On the other hand, the 
variance-covariance matrices of the points along the image line 
are expanded along its direction. It should be noted that we can 
expand the error ellipse, or restrict the weight matrix, only in 
case of using calibrated frame cameras. For scenes captured by 
line camera, this approach is not appropriate since the image 
line orientation cannot be rigorously defined at a given point