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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bib. Beijing 2008 
its corresponding image point is x = P X, projecting a 3D point 
from object space to image plane i , the calculated image 
point (x'^y'i) can be obtained. 
The difference d between tracked image points (x,y) and 
corresponding calculated image point (x',y')can be expressed 
by 
fi \ 
b 
V e J 
Q 
\ L 6j 
(10) 
d = J(x-x') 2 +{y-y'f 
(4) 
Then the standard deviation a of image points corresponded to 
point (X,Y,Z) can be calculated by 
I4 2 
-,/ = 0, •••,« 
(5) 
The 3D coordinate of point (X,Y,Z) is estimated by 
intersecting all its viewing rays as 
x, 
( V '\ 
l 
y\ 
UY 
(X 
= A 
(6) 
Where, 
A = 
dx 
dX 
dy_ 
dX ) x 
dx 
ÔX 
dy_ 
dX 
dx 
~dY 
dy_ 
dY 
dx ■ 
ÔŸ 
dy_ 
dY 
dx 
~dZ y , 
dy^ 
dZ 
ax 
az 
dy_ 
dz 
(7) 
The partial derivatives are computed directly using the 
Euclidean interpretation of the projection matrix. So, the 
covariance matrix C for 3D point can be obtained by 
C = cr 
f T 4‘ 
(8) 
Then, the theoretical precision of the computed 3D point can be 
expressed as error (T 3D according to, 
If (J-, n is larger than a suitable threshold, the 3D point is not 
3 D 
accurate. 
The relation to the angle-distance form of a 2D line is given by 
a multiplication factor 1 / yja 2 +b 2 : 
Yos(#)' 
f°i 
/ = 
sin(6>) 
= \/yla 2 +b 2 
b 
l ~ d ) 
l C J 
(9) 
(11) 
If there are n lines matched across the image sequence, the 3D 
edge can be estimated by using Gauss-Markoff Model with 
constraints: N=3n observations 1 for U=6 unknown parameters L 
in Pliicker coordinates with H=2 constraints h . 
/ + v = /(¿) (12) 
h{L) = 0 (13) 
In order to get corrections A/ and AL , the following Jacobians 
are needed: 
A = 
df(L) 
dL 
\L = I? 
H = 
dh(L) 
dL 
L = L" 
(14) 
(15) 
An initial covariance matrix C n of the observed 2D edges can 
be calculated from the uncertainty of edge extraction result. 
More details are given in [Heuel, 2004]. So, 
(16) 
A r C,J I A 
H 
AL 
A T CH l Al 
¡53 
1 
0 
. ^ . 
C h 
v = -(Al - A AL) 
(17) 
Where, A 1 = 1-f(L°) , c h = -h(L°) and p is Lagrangian 
multiplier [McGlone et al., 2004]. 
Then, the covariance matrix for unknown 3D edge L and the 
estimated residuals v can be obtained 
C ii =C u -AC~A J 
(18) 
3.3 3D Edge Estimation 
With, C LL =M~ l -M-'H(H T M~'Hy { H T M-' 
The geometric construction can be described as an estimation 
task, where an unknown 3D edge has to be fitted to a set of 2D 
edges from different images. 
So a relation between a 3D line and 2D line can be defined as: 
And M = A T C;, X A 
The estimated variance factor cr 2 is given by 
(19) 
(20)