y,) and (0,0) are 
o the straight line; 
'eprojection errors, 
oints to the closest 
Q) 
e pose parameters, 
nates that are to be 
PX, 
max 
edge 
stance and its 
stimation using the 
to find the three- 
jections belong to 
points lying on the 
rays coming from 
nd finding their 
at pose parameters 
s represented by a 
id 3D coordinates 
era (or observer) 
. The angles of 
‚and x. The 
ith homogeneous 
cording to 
3) 
(4) 
(5) 
PO -W[2 0 
= —-H/2 0 
p.192 / (6) 
0. 0 z, Je, 2.) -z,2, Je, - z.) 
0 0 -1 0 
where V — view matrix 
P = projection matrix 
f= focal distance in pixels 
W, H = width and height of the image 
z, , z, = coordinates of the near and far clipping 
planes. 
The 3x3 rotation matrix R in (5) is fully specified by three 
angles, and depends on the system of angles used. Since the 
screen coordinates of any 3D point can be computed 
analytically using formulas (3)-(6), it becomes possible to 
obtain the expressions for partial derivatives of pixel 
coordinates with respect to pose parameters: 
ox Hou war 
m, tV Om, 1 6m, 
(7) 
Oy (2 y = 
Om, t Om, 
em, A 
where m, is the i-th component of m. Partial derivatives of the 
signed distance are given by 
Od Bx Oy 
——— cos 0 4 
0 
sinQ . (8) 
om, om, m, 
2.3 Iterative minimization of contour-based distance 
In order to find the six unknown parameters that minimize the 
function (2) we use Levenberg-Marquardt method. Assuming 
that an initial guess m" exists, on every iteration the residual 
r -d(m') and the Jacobian matrix J(m') are computed and 
used to update the estimate according to 
à,--U'J-Adig(J 3) Jr, — (9) 
m" -m'«ó, (10) 
The scalar parameter 40 is changed dynamically over the 
course of the iterations. The second term in (9) allows taking 
into account the non-uniform curvature of the misfit function. 
At the first iteration F(m^), r(m^) and J(m^) are computed. 
Using the initial value of À =107 a trial solution m is found 
according to (9) and (10). If at iteration i the condition 
F(M)< F(m'") is true, then m'=m, À = 1/10. In this case 
the Jacobian is recomputed, along with the new value of the 
residual r(m'), and a new iteration starts. Otherwise, 
À :-104 ; the computation of the trial solution m is performed 
until either the misfit function decreases, or any other 
termination condition is satisfied. 
False edges are frequent in images. Such edges may have a 
higher absolute value of the directional derivative than true 
image edges. It is therefore necessary for each inspected 
contour point (x;, y,) to take into account a// the local maxima 
of the image derivative along direction 6, that are within the R- 
neighborhood of it. This can be done by using a modified misfit 
function: 
F(m)=3"p(d,) (11) 
where N. is the number of pixels lying on the projected 
contour, and the function p(x) penalizes outliers, i.e. points 
located at a distance that exceeds some threshold (Vacchetti, L., 
2004). In our work we used the Tukey function 
= (1-2) EET 
Pru (X) 71 6 C (12) 
c^[6 otherwise. 
The minimization of (11) may be reinterpreted as a least 
squares minimization with reweighting (Lepetit, V., Fua, P., 
2005). Weights are chosen according to the formula 
wm J^ (d,) (13) 
i d. 
i 
  
The misfit function then becomes 
F(m)- | wa]. (14) 
  
where W = diag(w) is a matrix of weights, w= Whores Wi, l. The 
Levenberg-Marquardt step (9) is rewritten as 
à,2-(WJeAdiagQ7W^J) J7W'r. — (15) 
2.4 Contour-based algorithm and Kalman filter 
Contour-based algorithm for estimation of exterior orientation 
elements of ISS based on appropriate contour templates is very 
time-consuming. It doesn't manage to process each video frame 
in real time. Therefore parallel algorithm of processing and 
tracking on the basis of widely known Kalman filter was 
realized. The algorithm under consideration uses the results of 
contour based one as input data. 
l. Atreceipt of the frame with number n on the basis of the 
previous data of estimation algorithm Kalman filter is 
initialized. 
2. Kalman filter predicts data at the moment of n+1. 
3. These data move to contour algorithm as new position 
estimation. 
4. As contour algorithm works the values for n+1, n+2, ..., 
n+k are defined by Kalman filter as a forecast in the mode of 
absence new input data. 
5. At the moment of time of n+k new data are arrived from 
contour algorithm from the moment number n. 
6. According to these data Kalman filter corrects values of 
parameters for time n, and then data for n+k in are recalculated 
in the mode of absence new input data. 
7. Step 2 is repeated. 
This technique including fusion of contour-based algorithm and 
Kalman filter was tested on modelling and real data/