the directions 
s equivalent to 
1e first piece of 
ch concerns the 
t 
n the constant 
becomes UM, 
e 
== M; (8) 
e measurement 
id points. 
; invariant , and 
ation is 
(9) 
this together in 
n a sequence of 
1e situation is 
section 3 except 
squations (6) — 
a segment S 
with covariance 
| estimate of its 
it matrix. If we 
we assume that 
S is not moving 
ith segment at 
‚we consider the 
ions 
t matrix 
can compute the 
  
  
where each partial derivative is evaluated at 
(rhs bo). We then compute the Mahalanobis 
distances 
di = fáGS,bT CAT MG b 
for all segments in the 3-D frame at time |. 
Those segments with distances smaller than a 
fixed threshold are kept as matches. 
Each match defines a token, and we 
update the state as follows: 
ai zm bod Ki(zi — Hip) 
in which 
_ art 
Hy = 2: 5500) 
and 
= M HY (HM HT + By) 
The whight on a} is the matrix Pj= (Mo 
HTORI) Hi)! 
4. 2 Continuous processing 
Just as in the two-dimensional case ,we do the 
reasoning at time 2, but the generalization to 
an arbitrary stage follows. Let S be a token at 
time 1 represented by (7r,, C1) with state dy 
and with weight P, We make a prediction by 
computing the state ay =o, qas and its weight 
P,. We then determine the candidate 
segments at time 2. If S; is the ith segment at 
time 2 represented by (75,C5) ,we consider the 
"possible" measurement equations 
fiGia;)-—0 
where zi-[rf,r? |" has weight matrix 
: CO 
Ri = i 
Q ci 
and a; has weight matrix P,' Just as in the 
previous section, we select matches based on 
the Mahalanobis distance and update the 
state. We have 
di — aj -- Ki — Hia!) 
in which 
Hi == Ph ata) 
and 
Kim Pt HG SPL HR DS! 
The weight of the new state aj is the matrix 
PizPac KU. 
5. CONCLUSION 
The problem of tracking tokens in sequences 
of images or in sequences of stereo frames has 
received considerable attention in the last few 
years,and the use of the Kalman or extended 
Kalman filters or equivalently of recursive 
least-squares estimation theory has now 
become standard. The applications of these 
methods to the 2D-2D and 3D-3D tracking 
problems described in sections 3 and 4 have 
not been as numerous as their applications to 
the 2D-3D tracking problem in which the 
observations are made in the image and the 
tracking is done in three dimensions. 
6. REFERENCES 
H. Shariat and K. E. Price. Motion estimation 
with more than two frames. IEEE Trans. 
PAMI;12(5);417— 434,May 1990. 
Z. Zhang. Motion Analysis. from a Sequence of 
Stereo Frames and Its Applications. Ph. D. 
thesis ,1990. 
Z. Zhang and O. D. Faugeras. 3D Dynamic 
Scene Analysis: A Stereo Based Approach. 
Springer-Verlag, 1992.