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ints from the vehicle 
uld be tracked at least 
» Tracked 
eo images, points to be 
the successive video 
. It is well known that 
oints which have large 
tions. Since the corner 
. endpoints), aperture 
most frequently used 
(Harris and Stephens, 
oint by a matrix which 
. These derivatives are 
an image and are 02x, 
r derivatives of pixels 
d. This new image is 
ssian” arises from the 
a point (Dogan, et. al, 
defined by: 
'asonable criterion for 
envalue of the spatial 
predefined threshold. 
itioned and above the 
ts inverse does not 
in a certain critical 
etric information from 
und within a sub-pixel 
late pixels around the 
mallest eigenvalues at 
represent the spatial 
les of the maximum of 
location for being a 
e obtained in subpixel 
ra begins for image 
isly in real time from 
ts are selected and on 
^d and instantaneous 
ted. 
ach selected point on 
ars for the first time, 
ime. In the ideal case, 
be the same point on 
onding point, there is 
t itself. If we assume 
    
  
that each image in the each frame is flowing by the very short 
time period and thus changing the position during the flow, then 
4 modelling approach which models this flow event can be 
used. These kinds of flow models are called “optical flow”. 
3.2.1  Lukas-Kanade (LK) Optical Flow Method: When 
only one video camera is used, there is no information other 
than themselves of the selected points to find their 
correspondences on the next frame. For this reason, it is not 
possible to know exactly where the corresponding points are on 
the next frame. But however, by investigating the nature of the 
problem, some assumptions may be made about the possible 
locations where the corresponding points might be located. In 
order to ensure these assumptions are as close to the physical 
reality as possible, there must exist a theoretic substratum at 
which these assumptions are supported. Furthermore, this 
theoretic substratum must be acceptable under some certain 
situations. Lukas and Kanade have cleverly given three 
assumptions for the solution of the correspondence problem in 
their paper (Lukas and Kanade, 1981). The assumptions of 
Lukas-Kanade Method are: 
i Intensity values are unchanged: This 
assumption asserts that the intensity values of a 
selected point p(t) and its neighbours on the frame 
image I(t), do not change on the next frame I(t + At), 
where At is too short time period less than one 
second. When the time interval At that passed 
between two successive image frames is too short, it 
can be seen that really the possibility of the 
occurrence of this assumption is too high. Because, in 
a very short time period which is measured in 
milliseconds, the effects such as the lighting 
conditions of the scene medium etc. that cause the 
intensity values to be changed must not lead to 
meaningful change effects since the time is too short. 
2 Location of a point between two successive 
frames changes by only a few pixels: The reasoning 
which the assumption is based on, is similar to the 
reasoning of the first assumption. Between the frame 
images I(t) and I(t + At), when At is getting smaller, 
then the displacement amount of the point also gets 
smaller. According to this observation, a point p(t) at 
(x,y) coordinates of image I(t) will be at the 
coordinates (x + Ax, y + Ay) on image I(t + At) and 
these new coordinates will be closer to the previous 
coordinates within a few pixels. Thus the positions of 
the corresponding points on both images will be very 
close to each other. 
$ A point behaves together with its neighbours: 
The first two assumptions, which are assumed to be 
valid for a point must also be valid for its neighbours. 
Furthermore, if that point is moving with a velocity v, 
then its neighbours must also move with the same 
velocity v, since the motion duration At is too short. 
The three assumptions above help develop an effective target 
tracking algorithm. In order to track the points and to compute 
their speeds by using the above assumptions, it is necessary to 
express those assumptions with mathematical formalisms and 
then velocity equations must be extracted by using these 
formalisms. For this purpose, the first assumption can be written 
in mathematical form as follows: 
Kp(x y,t), 9 2 l(p(x y, 0), t + At) (2) 
where I(p.t) is the intensity value of a point p on the image I(t) 
which was taken at the time instant t. Note that the geometric 
location of the point is expressed by its position vector p € R? 
(i.e., in 2D space). Here I(p,t) expresses the intensity value of a 
pixel at the point p on the frame image I(t). In similar way, the 
right side of the equation expresses the intensity value of the 
corresponding pixel at the point p + Ap on the frame image I(t + 
At). Accordingly, Equation (2) says that the intensity value of 
the point p on the current image frame does not change during 
the time period At that passed. In other words, it expresses that 
the intensity I(p,t) does not change by the time At. In the more 
mathematical sense, change rate of I(p,t) iz zero over the time 
period At. This last situation is formally written as follows: 
pe y Dt) _ 
et 
0 (3) 
If the derivative given in Equation (3) is computed by using the 
chain rule of derivative, we obtain: 
a(pxy90,9- Ol(p, t) op(t) De 2 d 
& p ot a 
  
(4) 
In Equation (4), the derivative @Z /@p is spatial derivative at 
point p on the image frame I(t). Thus it can be expressed by 
oI / op = VI . We can write this expression in explicit form as 
follows: 
ol ol 
VI-—ic—jzlyilyj (3) 
ox Oy 
The derivative @p()/ ot can also be written in a more explicit 
form: 
Bl) de y OY 
mt ey) = i=] (6) 
ot ot x À 
If Equation (6) is investigated carefully, it can easily be seen 
that the vector (@x / @t)i is equal to the velocity of the point p 
in the x-axis direction. In other words, it is the x component 
namely v, component of the velocity vector v. In similar way, 
the vector (@y/ @t)j is the y component namely vy, component 
of the velocity vector v. Now we can rewrite the Equation (6) as 
follows: 
Op(O 
— = V= Vit Vy] (7) 
ot Ya 
If again Equation (4) is investigated, it is seen that the 
derivative Of / Ot = I, expresses the change rate of the intensity 
values at point p, between the frame images I(t) and I(t + At). 
Thus, Equation (4) can be rewritten as follows: 
VIV +1, =0 (8) 
where: