1 NNT 
à = NIC IND 
IN ] IN IP 
Il 
HART nn VAN 
(NI 
From the definition of N, see section 3, we have || N || 
h the distance of the 3D line from the origin O. We use 
also the fact that for a unit vector n in $?, I3 — nn? — f?. 
After a little simplification we obtain the line motion field 
equation: 
T 
à - QA n4 — (An) (T) 
where n, L and h are defined as in section 2, and €? 
and V represent the angular and translational velocities. 
And they are all written in the camera coordinate system. 
If we define a vector H, passing through optical center 
and orthogonal to 3D line D. One may recover £2 only in 
the direction of H, which is parallel to n ^ L: 
ATL = QT (n AL) (8) 
And the rotation around an axis parallel to the 3D line is 
coupled with translational velocity: 
VT 
; ©) 
That is all the information about the motion that we 
can draw from a 3D line, its image on a camera and the 
motion field of its image. In the case of two calibrated cam- 
eras, one can write the same equations for both cameras. 
The equations are then expressed in the same coordinate 
system using the calibration data (R and T). In this case, 
for each 3D segment we may recover the angular velocity 
in the plane orthogonal to its direction define by n AL, and 
Rn’ AL. : 
  
Mn AL) =0TL+ 
5 Motion Estimation and Covari- 
ance analysis 
5.1 How to estimate the kinematic screw? 
Using equations (8), and (9), for each segment we obtain 
the following matrix equation for the first camera: 
Z=FX 
where X = | v | , in filtering terminology, is the state 
r- [^2 2| a 
is the transposed of the observation matrix and 
vector and 
Z-|27L h'(^n] (11) 
defines our measurement vector. À similar equation can 
be obtained for the second camera. As can be observed, 
the input of our system consists of the 2D line parameters 
n (resp. n' in the other camera), their covariance matri- 
ces, and the calibration data R and T, together with the 
stereo and temporal matchings of the 2D lines. The lines 
motion fields n (resp. n', 3D line directions L and their 
distances to the camera center h (resp. h') with their cor- 
responding covariance matrices, are then estimated from 
the input. They complete the input to our final system of 
    
equations. £2 and V form the output of this system. Using 
a Kalman filter, we begin with an estimation of f? and V. 
In our case a reasonable assumption is that the interframe 
motion is small, thus we set the initial values of © and V 
to zero. However, to take into account our lack of informa- 
tion about the real motion, we set the diagonal elements of 
their covariance matrices to rather large values. The next 
section makes explicit the covariance analysis done in the 
kalman filtering process in our case. 
5.2 Covariance analysis: Better Under- 
standing of the Kalman filtering pro- 
cess 
The Kalman Filtering process is very often used in com- 
puter vision. We use the estimations and covariance matri- 
ces, obtained by the kalman filtering process, to compute 
the Mahalanobis distances between the segments(see sec- 
tion 6.2). We show in this section that this is a good and 
justified choice for our purpose. 
The behavior of a dynamic system can be described by 
the evolution of a set of variables, called state variables. 
If we denote the state vector by s and denote the mea- 
surement vector by m’, a dynamic system (in discrete-time 
form) can be described by 
Si = gi(s:) "dT i= 0,1,- f 9 (12) 
f;(m';, s) = 0 ; i 0,1,--- ; (13) 
where r; is the vector of random disturbance of the dynamic 
system and is usually modeled as white noise: E[r;] = 0 
and E[r;rT] = Qi. 
Assume that the measurement system is disturbed by 
additive white noise, i.e., the real observed measurement 
m; is expressed as: m; = m’; + 7;. where 
Eln] = 0, 
Ay: 
Eni; | { o 
When g;(s;) is a linear function 
fori=3, 
fori. 
sit1 = Gisi + Ti 
and we are able to write down explicitly a linear relation- 
ship 
m; = Fis; +n 
from 
f;(m’;, Si) = 0 , 
then the standard Kalman filter is directly applicable, as 
follows: 
e Prediction of states: 
$jj1 — GiaSiaa 
e Prediction of the covariance matrix of states: 
Pii = Gi PjaAGT, 4 Qia 
e Kalman gain matrix: 
K; — PyaFI(AP4AFI TAS)! 
e Update of the state estimation: 
$; — By + Ki(m; — FiSgi-1) 
e Update of the covariance matrix of states: 
PR = (I- KiF)Pya 
e Initialization: 
Foo E Ag, 
Solo = Eso] 
In our case, see section 4, s; = X; and G;-1 = Ie, and 
H; O 
F- | h;L; ni |