The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
its projection qi' (x qi ', y qi ', z qi ') onto the surface patch. In 
Pothou et al., 2006b, details for the algorithm were presented (it 
is called algorithm B). Also in Pothou et al, 2007, analysis and 
performance of this algorithm, for the boresight parameters 
estimation was proposed. 
In Equation 1, R (co, <p, k) is the orthogonal rotation matrix, 
defined in Equation 2, while b x , b y , b z are the elements of the 
offset vector. 
x„ 
[bl 
p 
q 
X 
y P 
= R • 
yq 
+ 
b y 
_ Z P. 
_ Z q _ 
b z 
0) 
COSCpCOSK 
-COSCpsinK 
sirup 
COS CO sin K + sin CO sin (p COS K 
COS CO COS K - sin CO sin cp sin K 
-sin CO cos cp 
sin CO sin K - cos CO sin cp COS K 
sin CO COS K + cos co sin cp sin K 
COS CO cos cp 
(2) 
Figure 1: Transformation between qi points and control 
surface P 
The parameters of the plane’s equation (which passes from the 
3 known points (p m , p k , p ; )), are given by the derivatives in 
Equation 3. 
y pm 
Zpm 1 
X pm 
^pm 
1 
y pk 
Z pk 1 
B = 
V 
Z »k 
1 
y P < 
Z P < 1 
Z P < 
1 
Xpm 
y pm 1 
X pm 
y pm 
^pm 
c = 
X P k 
y P k 1 
D = 
V 
y P k 
Z P k 
V 
y* 1 
V 
y P < 
V 
Based on Equation 3, the coordinates of q{ (x qi ', y qi ', z qi '), 
which correspond to projection of point q, (x qi , y qi , z qi ) on the 
plane (p m , p k , p # ), can be calculated as described by Pothou et 
al., 2006a. In order to perform the transformation between the 
two datasets, the transformed coordinates of point q, have to be 
used in the calculation of x qi ', y qi ', z qi \ So, after the input of 
Equation 1 to the equations for the calculation of x^', y qi ', z qi ', 
(based on that the difference of each point of dataset Q and its 
projection on the P surface should be equal to zero), the 
Equation 5 is derived in which the T and L are the matrices of 
Equation 4 that depend on the plane’s parameters of Equation 3. 
il a2 
BA 
CA 
AD 
a 2 +b 2 +c 2 
a 2 +b 2 +c 2 
a 2 +b 2 +c 2 
a 2 + b 2 +c 2 
AB 
1- 
CB 
BD 
a 2 + b 2 +c 2 
a 2 +b 2 +c 2 
a 2 +b 2 +c 2 
a 2 + b 2 +c 2 
AC 
BC 
1 ° 2 
CD 
a 2 + b 2 +c 2 
a 2 +b 2 +c 2 
a 2 +b 2 +c 2 
a 2 +b 2 +c 2 
f 
- 
\ 
X Qi 
b x l 
R 
yqi 
+ 
by 
\ 
'5* 
N 
. b d 
/ 
Equation 5 is the observation equation for each point Q, 
therefore it is the base for producing the system of Equation 6 
which is provided through the Taylor expansion linearization 
for the parameters (b x , b y , b z , co, q>, k). After having performed 
least squares estimation the solution of Equation 6 and the best 
estimation of the vector x , is provided by Equation 7. 
A8x = b£ + v (6) 
x = x° + (a t Wa)"‘ A T W8f (7) 
In Equations 6 and 7, A is the design matrix, W is a diagonal 
weight matrix of the observations, x° is the vector of the 
approximated parameters, 6£=£-£° is the second part of the 
observation equation in which £=0 and £° is the result of 
Equation 5 using x°, and finally v is the residual vector. 
Applying this algorithm for each building, which has been 
photogrammetrically restituted (P surface), the best estimation 
of the six parameters (offsets and rotations) is provided. In this 
process, photogrammetric restitution and LiDAR strips which 
both are available, the algorithm takes place, setting the 
proximity between each LiDAR point and each TIN from the 
building as a choice criterion. 
The corresponding covariance matrix of x and the a’ posteriori 
variance of unit weight a 2 are calculated as following where r 
is the degree of freedom. 
v, =s;(a t wa)-‘ 
„2 v^Wv 
o z = 
O r 
Applying the algorithm for each building, in combination with 
available LiDAR strips, a number of independent estimations of 
transformation parameters are provided. The redundancy of 
estimations provides the ability of chosen desirable number and 
type of estimations i,..., j, in order to provide a unique 
estimation of the parameters. This can be achieved if a set of 
chosen solutions is assumed as observations creating a new 
linear system of observation equations such as Equation 10. 
The confrontation of the solution of this system is the same as 
Equations 6 and 7. The matrix of observations £ includes the 
best estimations of each solution i, ..., j which have been 
chosen from the total of the solutions. 
(8) 
(9) 
Aî = f + v (10) 
The solution of Equation 10 by Equation 11 is same as that of 
Equation 7, for the case of linear equations. 
(11) 
303