The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
223 
Figure 6: Point-based incorporation of planar patches 
3.2.1. Variance-covariance expansion for planar patches 
a) Patch coordinate 
b) Original error ellipse in 
(X, Y, Z) coordinate system 
c) Error ellipse in (U,V,W) coordinate 
system after expanding the variance in 
the patch direction 
Figure 7: Expanding the error ellipse in the patch plane 
Variance expansion in object space is done as follows: 
(1) Define the patch by three points, A, B, and C belonging to 
it in object space. 
(2) Compute the rotation matrix, R, to relate the variance co- 
variance matrix in the original coordinate system (X, Y, Z) 
to the variance co-variance matrix in the patch coordinate 
system (U, V, IV), where the U and V axes are within the 
patch plane and the W axis is normal to it (Figure 7.a). 
(3) Compute the variance-covariance matrix in the patch 
coordinate system, Zww, for the three points A, B, and C 
using the law of error propagation: 
St)» - R I™* (ii) 
where Z X yz is the variance-covariance matrix in the (X, Y, Z) 
coordinate system (Figure 7.b). 
(4) Assign a large value for the variance in the plane direction 
by applying a large scaling factor, m (Figure 7.c): 
<j\, = m <7u' cr'i =OT cr, • 
Then, the new variance-covariance matrix in the (U., V, W) 
coordinate system, YJww > will be as follows: 
G r r 
(Tur 
y = 
JLu unv 
Gw 
G'r 
Gvw 
G h i/ 
O wv 
tr,j 
(12) 
(5) Rotate the variance-covariance matrix to the original 
{X, Y, Z) system computing the new S' A1Z = R YSwwR ■ 
(6) Apply a point-based solution using the two col linearity 
equations 2 and 3 while considering the modified 
variance-covariance matrix, S\rz> f° r the points. 
3.2.2. Weight restriction for planar patches 
This approach is similar to the one in sub-section 3.2.1 except 
that instead of a variance expansion a weight restriction is 
applied, i.e. the weights for points along areal features are set to 
zero. The procedure is as follows: 
(1) Define an areal feature by any three points A, B, C lying on 
it. 
(2) Compute the weight matrix along the planar patch as 
follows: 
PuVW~R PxrzR (13) 
Where P xyz'Puyw are the weight matrices in the object and 
patch coordinate systems, respectively. 
(3) Assign a zero value for the weights along the patch plane: 
jo 0 
|o 0 
| o o 
|o 0 
P» 
Therefore, 
FxrZ=R P'rwR (15) 
(4) Apply a point-based solution using a least squares solution 
with the modified weight matrix, P' m . 
4. EXPERIMENTAL RESULTS 
Experimental work was conducted to validate the feasibility and 
applicability of the above approaches using simulated data. The 
simulation model consists of a group of buildings covering an a 
rea of 7000 x 7700 square meters. All buildings have planimetri 
c dimensions of 10 x 10 meters with different heights and roof a 
ngle of 20 degrees (Figure 8). The spacing between LiDAR foot 
prints was chosen as 0.500 meters in the planimetric direction ( 
X, Y directions). 
Figure 8: Isometric view of a sample building 
A simulated camera was assumed to capture all the photographs 
(Table 1). Six synthetic photographs with normal geometric con