I V
Vi=| * and V=|.* (9)
ly Vy
Then Equation (8) can be written as:
Vx
("EM vie (10)
The values of I, I, and I, in Equation (10) can easily be
computed from the frame images. The variables v, and v, are
two unknown components of the velocity vector v and these are
respectively the components in the directions x and y axes of
the image coordinate system. In Equation (10), we have two
unknowns to be solved, but we only have one equation. Since
only one equation is not enough for unique solution of the
unknowns, at the moment it seems not possible to solve these
unknowns. In order to solve these two unknowns, we need more
independent equations. For this purpose, the third assumption
of the LK algorithm is used. That is, point p behaves together
with its neighbours. So its neighbours must also satisfy the
Equation (10). In other words, neighbour points (or pixels) of
point p must move with the same velocity v(v,,v,). According to
these explanations, the same equations as (10) are written for 3
x 3 or 5 x 5 neighbourhood of the point p. In this case, we
totally have 9 or 25 equations with the same unknowns v, and
vy. Now the unknowns can be solved with overdetermined set of
Equations (10) by using least squares or total least squares
estimation method (Dogan, et. al, 2010).
During the real time tracking, some selected points may not be
seen on the next frame. This situation may arise because of
different reasons. Especially, when the vehicle is entering into
or exiting from the FOV of the camera, the possibility of
occurrence of this situation is too high. In order to prevent such
situations, we have interpreted the algorithm with the image
pyramid approach, which uses coarse to fine image scale levels.
For details of the image pyramid approaches, we refer the reader
to (Bouget, 2000) and (Bradsky and Kaehler, 2008).
3.3 Estimation of Speed
To find the vehicle speed, successive frame images of the
camera can be used. In this case, only the instantaneous speed
can be found. This instantaneous speed is computed as follows:
N
es (11)
At
where v is instantaneous velocity vector of a point and v € R?
(i.e., in 2D space since one camera is used), Ap is displacement
vector of that point and Ap € R?. The displacement vector
expresses the spatial displacement of a point during the time
interval At. Here the time interval At is equal to the time which
passes between two successive video frames and is equal to the
frame replay rate (or frame capture rate) of the camera. In the
experiments given in this paper, At is 33.3 milliseconds, which
is the frame capture time of the camera that we used. Equation
(11) gives the instantaneous speed (or velocity) of a point which
is marked on the vehicle and selected for tracking. To find the
velocity of the vehicle, only one point is not enough. During the
selection of the points from the image of the vehicle, local
approaches are used. If some errors occur during this selection
step, the computed velocity vector will be affected by those
errors and so the computed speed will be erroneous. For this
reason, to estimate the speed of a vehicle, many more than one
point should be selected and all of their instantaneous velocity
vectors should be computed. Then by averaging the
instantaneous velocity vectors of the whole selected points, the
instantaneous velocity vector of the vehicle is found. For the
formal expression, let us assume that n points are selected from
the vehicle to be tracked and let v; (t) (i 1, ..., 1) represent the
instantaneous velocity vectors of each of n points at time
instance t. Then by using those instantaneous velocity vectors,
we can find the instantaneous velocity vector of the vehicle by:
1
Viger
Ts
Vy:
TD (12)
where vi, is the instantaneous velocity vector of the vehicle at
time instance t, v; is the instantaneous velocity vector of i^ point
on the vehicle and n is the number of the selected and tracked
points. Here, it should be noted that, if some of the vi vectors
are erroneous, then v;, will also be erroneous. So, before
computing the instantaneous velocity v;, of the vehicle, the
erroneous v; vectors must be eliminated. Then the value of n
also changes, i.e., number of the points decreases. For the
elimination of the erroneous vectors, standard deviation of the n
velocity samples can be used for fast evaluation:
IMIs]v - 5] (13)
In order to find absolute values of displacement vectors or
velocity vectors in object space, the vectors computed in video
image coordinate system should be transformed to the object
coordinate system which is in the object space. For this
purpose, at least the length of a line joining two points within
the field of view of the camera and on the road and aligned
along the velocity vectors, must be measured precisely. In this
paper, we measured the lengths of two lines along the road by
geodetic measurements using a simple measurement tape,
within a precision of +1 millimetre.
4. EXPERIMENTS AND RESULTS
In this paper, we propose a method for real-time estimation of
the speed of moving vehicles by using uncalibrated monocular
video camera. Since it is not possible to extract 3D geometric
information with one camera, in order to solve the speed
estimation problem, some geometric constraints are required
and the images should be taken under these constraints and the
processing procedures should also be performed with those
restrictions. For example, we assume that the imaged scene 15
flat. Perspective distortions on the acquired images must be
either very small or of a degree that they can easily be rectified.
We have used a camera with a frame rate of 30 fps and with an
effective area of 640 x 480 pixel The pixel size which
corresponds to the effective area of the camera is 9 microns.
The focal length of the camera is 5.9 mm. We capture images in
grey level mode at 30 fps (frames per second), meaning that a
frame is captured within 33.3 milliseconds after the previous
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