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Research of Moving Targets Tracking Algorithm
Based on Kalman Filtering
Huiying Dong1 , Shengfu Chen1 School of Information Science and Engineering
Shenyang Ligong University Shenyang, China
Jiayu Zhu2 Shenyang Normal University
Shenyang, China
Abstract—The purpose of moving targets tracking
is to determine the trajectories of moving targets,
the key of which is to establish the corresponding
relationships between the detected prospect targets
and the tracked moving targets. The established
corresponding relations are equivalent to the
matching problem of target features in consecutive
image frames. In the paper, Kalman filter is used
as the motion model of the targets. The recursive
filtering method is adopted to calculate and predict
the location of moving targets. Through the
experiment, the results show that the algorithm not
only track well the moving targets ,but also can
effectively overcome the obstacles in the process of
tracking and so on.
Keywords-target tracking; feature matching;
Kalman filter
I. INTRODUCTION
Moving targets tracking is to use the effective features of moving targets and adopt appropriate matching algorithm on the base of moving targets detection and recognition to find the candidate targets object’s position that is most similar to the targets template in the sequential images. It simply says to locate for targets in the sequence of each image. Filtering problem is to restore a time series interfered by the noise as much as possible and can also be regarded as a prediction problem. Mathematically speaking, the prediction is from past data of a time series to estimate the statistical parameters of the entire sequence. This estimate obtains the average value of the statistical parameters, which has certain
differences from objective reality. In general case, the best prediction should make these errors to least under the least mean-square error criterion.
II. KALMAN FILER PRINCIPLE
Kalman filter[1, 2] is an efficient recursive filter. It can estimate dynamic system state from a series of measurement of incomplete included noises. It can take any point as a starting point to begin to observe. Using recursive filtering method has the features of small amount of calculation and real-time computing. Kalman filter uses the state equation and observation equation to describe a dynamic system.
Kalman filter reduces the tracking of error
covariance matrix of each time point K to least, which consists of two steps to complete:
a) Prediction procedures: including state prediction and error covariance prediction.
b) Modification procedures: including Kalman gain calculation and behind state and error covariance modification.
Kalman filter estimates motion state by using the feedback control system, filter estimating the state of a certain time, and obtaining the prediction value of this state.
III. TARGET TRACKING BASED ON KALMAN
FILTER
Kalman filter, as the motion model of targets, predicts the location of moving targets. When matching the predicted moving targets and the current foreground targets, image centroid and
2010 Third International Conference on Intelligent Networks and Intelligent Systems
978-0-7695-4249-2/10 $26.00 © 2010 IEEE
DOI 10.1109/ICINIS.2010.104
20
window size are used to locate precisely position of targets. First, calculating eigenvalues, then using the Kalman filter to build motion model ,at the same time, the next step movement is predicted through the moving targets that have been extracted on the current which will narrow the scope of target-matching and accelerate the speed of target matching. Second, target association is established by doing the target-matching in the next frame within the specified bound. Last, update motion model and form the target motion tracking chain to get moving target trajectory [3, 4].
A. Calculating Kalman filter tracking Eigen values
In the case of smaller interval, it can be considered the tracking window size and position of center of mass of a moving target in the adjacent two images change little, which means the movement of moving targets have continuity characteristics. Therefore, the Centroid tracking and tracking window size are chose as the characteristic value[4] to track targets. The tracking window size is slightly larger than the target image, and target image is trapped tightly by the tracking windows, thus, targets haven’t been affected by window background and noise. At the same time, due to the large amount of image data of the entire field of view, in order to shorten the time, tracking window can be used to reduce the size of the processing image and only process interesting partial image in real-time. For many emerging targets in the view field, it can also set up a few tracking windows to track moving targets respectively. After marking well in all tracking windows, the centroid of the targets in the window is sought respectively. Each window centroid coordinates can be calculated by the following formula:
∑∑∑∑
==
i j
i j
jif
jiif
mm
x),(
),(
00
10
(1)
∑∑∑∑
==
i j
i j
jif
jijf
mm
y),(
),(
00
01
(2)
In the formula, ),( jif is gray value of the target image in the tracking windows, ),( ji is the point of targets area in the tracking windows. Centroid coordinates of each tracking window is an important state parameter in moving targets tracking process, in the subsequent tracking process, the centroid coordinates is one of the key state variables.
B. Motion estimation model
Supposing the state vector 1+ks of time point 1+k in the model is composed of the transfer
function of vector ks at time point k and
noise. While the observation vector is determined
by the observation function of vector 1+ks at time point 1+k and noise.
Equation of state is as follows:
kkk wAss +=+1 (3)
Observation equation is as follows:
111 +++ += kkk vCsz (4)
Where: kw、 1+kv are noise. The
introduction of noise, first, rely on experience to determine, second, rely on studying statistics to obtain. Supposing dynamic noise and observation noise are normal white noise whose mean is zero.
ks is the state vector, and is composed of an
eight-dimensional vectors
] [ ykxkykxkkkkkT
k LLLLyxyxs = (5)
where: kx and ky are respectively target
cancroids coordinates; kx, ky are respectively
the unit displacement of the cancroids
coordinates in the x, y direction; xkL , ykL are
respectively the width of tracking windows in the
21
x, y direction; xkL , ykL are respectively the unit
displacement of tracking windows width in the x, y direction.
1+kz is the observation vector, which is
composed of a four-dimensional vectors
] [ LLyx 1yk1xk1k1k +++++ =zTk
(6)
As the sampling time t =0.04 seconds, time is very short, so it can be approximately thought that the moving targets move at a constant speed, moreover tracking window size is changed little,
then the state transition matrix A is:
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
100000000100000004.00100000004.00100000000100000000100000004.00100000004.001
A
Observation matrix C is:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
00001000000001000000001000000001
C
After defining the state equation and observation equation of motion model, Kalman filter is used to estimate the position of target cancroids and window size, and it can search cancroids within a small area in the next frame.
C. Target feature matching and model updating
For a group of motion image sequences,
{ }nfffF ,......,, 21= ,creating the same cartesian
coordinate system for each of these images, in
the k th frame image kf , m targets are
recorded as { }mrrrR ,......, 21= , centroid coordinates and window width of the i th targets are
recorded respectively as ikx ,
iky,
ixkL
,
iykL .
First defining the i th target’s centroid of
the k th frame and the centroid distance function
of the j th target of the 1+k th frame
*
1
1
max),(
+
+=k
ik
jk
ik
cc
ccjiD
(7)
Where:
( ) ( )21
211
jk
ik
jk
ik
jk
ik yyxxcc +++ −+−=
, 1),( ≤jiD
Subsequently defining area differences
function, that is comparing the i th target’s
windows area of the k th frame with the j th
target’s windows area of the 1+k th frame
*1
1
max),(
+
+
−
−=
kik
jk
ik
aa
aajiA
(8)
Where:
jyk
jxk
iyk
ixk
jk
ik LLLLaa 111 +++ ×−×=−
, 1),( ≤jiA
Defining similarity function
),(),(),( jiAjiDji ξγ +=Δ (9)
Where γ , ξ are weighting values, and
meeting the condition that ξγ > , 1=+ ξγ ,
1),( ≤Δ ji if the smaller the ),( jiD is, the closer
target it indicates, while the smaller the ),( jiA is, the more similar target shape it indicates,
while the smaller the ),( jiΔ is, the more probability of the two targets similar it indicates.
As a result, the threshold ΔT of similar function is setted to judge that the targets are the same goal or not. If all the goals on a frame and the smallest value of the results of estimating and matching on the fore frame are over a threshold, which shows that the frame has not the same follow-up targets; If it is below this threshold, which shows that the targets with the smallest value is the subsequentce of the fore frame
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targets. While the minimum value of similar function is found, the follow-up of the same
targets has been found, that is say, the j th
target of the 1+k th frame can be seen as the
follow-up of the i th target of the k th frame, it
shows that both of the two targets are the same target. At this time, the characteristic value of the
j th target of the 1+k th frame is used as the
input of motion model estimating the next frame, and so on, completing the model updates.
IV. ANALYSIS OF EXPERIMENTAL RESULTS
Target tracking algorithm based on Kalman filter is adopted to track the two moving cars on the road. In the experiment , the 320×240 pixels video image sequence have been processed, target moving along the x direction of the right
bottom of the road, taking γ =0.7,ξ = 0.3,
ΔT = 0.6. Figure 1 shows the eight images that have been extracted from the tracking experimental results respectively and Table 1 shows the centroid position of moving targets and tracking window size.
Figure 1 Target tracking results map based on Kalman filter
From Table 1, it can be shown that the algorithm can be better for moving targets tracking from the simulation results above. In the process of target motion, the tracking window size of each object can do the state transfer based on the actual state, adjusting itself in the process of the next frame tracking automatically.The exist of normal white noise of mean zero has been taken into account in the process of motion state transition. Because in the process of
tracking moving targets, the Eigen value’s centriod coordinates of the target and tracking window size can conduct the state transition in the process of the next frame tracking, such tracking algorithm can also effectively overcome the obstacles in the process of tracking and so on.
TABLE 1 THE CENTROID COORDINATES OF CONSECUTIVE FRAME MOVING TARGETS AND THE VALUES OF WINDOW SIZE
V. CONCLUSIONS
Kalman filter tracking model is divided into four sub-modules: Eigen value calculation, to calculate the Eigen value of moving targets, centroid, tracking windows. Motion model, use Kalman filter to establish the systematic motion model, define the state vector and predict possible position of moving targets in the next frame. Feature matching, define the similar function of targets, use the changes of targets in the relative frame targets, besides, apply the Eigen value to calculate similar function values, then, determine whether the targets are the same tracking targets. Model updating, update motion model, to take as the input of Kalman filter of the next motion model.
REFERENCES [1] Kalman,R,E.and R.S.Bucy.New results in filtering and
prediction theory,T-ransactions of the ASME Journal of Basic Engineering,Vol.83,1961.95
[2] Meditch,J,S..Stochastic Optimal Linear Estimation and Control,McGraw-Hi-ll,1969.188-196
[3] S. Julier, J. Uhlmann, and H.F Durrant-Whyte. A new method for the nonl-inear transformation of means and covariances in filters and estimators. IEEE Transactions on Automatic Control, 2000. 43(3):477-482
[4] Ribaric S.G, Adrinek, Segvic S.. Real-time active visual tracking system. IEEE IEEE Proceedings Mediterranean Electrotechnical Conference. 2004, 5,vol.1:231-234
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