Bounding box stabilization for visual object tracking using Kalman and
FIR filters
ELI G. PALE-RAMON1, YURIY S. SHMALIY1, LUIS J. MORALES-MENDOZA2 AND MARIO
GONZÁLEZ-LEE2
1 Electronics Engineerig Dept.
Universidad de Guanajuato
Km 3.5 +1.8 Salamanca-Valle Santiago road, Salamanca, Guanajuato, 36885
MEXICO
2 Electronics Engineerig Dept.
Universidad Veracruzana
Ing. Guillermo Alvizouri, Poza Rica, Veracruz, 93390
MEXICO
Abstract: -In visual object tracking, the estimation of the trajectory of a moving object is a widely studied
problem. In the object tracking process, there are usually variations between the real position of the objet in the
scene and the estimated position, that is, the object is not exactly followed throughout its trajectory. These
variations can be considered as color measurement noise (CMN) caused by the object and the camera frame
movement. In this paper, we treat such differences as Gauss-Markov coloring measurement noise. We use
Finite Impulse Response filters and Kalman filter with a recursive strategy in tracking: predict and update. To
demonstrate the best performance, tests were carried out with simulated trajectories and with benchmarks from
a database available online. The OUFIR and UFIR algorithms showed favorable results with high precision and
accuracy in the object tracking task.
Key-Words: - Object tracking, state estimation, FIR filters, Kalman filter, bounding box
Received: February 26, 2021. Revised: December 17, 2021. Accepted: January 22, 2022. Published: February 11, 2022.
1 Introduction
The visual object tracking is a topic widely studied
by various researchers, mainly due to multiple
practical applications such as video surveillance and
security, autonomous vehicle navigation, robotics,
etc. [1] [2] [3]. Visual object tracking is a topic of
interest in signal and image processing, in which the
coordinates of the frame sequences are considered
input data for the trajectory estimation [3] [4] [5].
Visual object tracking can be defined as the
process of estimating the object trajectory in the
image plane as it moves around a scene [5] [6].
However, during the tracking process the object is
not followed exactly. There are variations in the
estimated, that is, there is a discrepancy between the
real position and the estimated position. These
variations can be considered as colored
measurement noise (CMN) which is not white [7].
An example of differences between real and
estimated position is shown in Fig. 1 for the
“Human2” benchmark [8], where a desirable frame
is shown red and estimation errors in yellow, in this
case, the target is the person dressed in blue.
It has been demonstrated that the use of a motion
model and state estimators is a effective in avoiding
large tracking errors [7] [9] [10] [11] [12] [13] [14].
Fig. 1 Example Human tracking in a video sequence
In various investigations, it has been shown that
if the model is correctly specified in the state space,
it can represent the object dynamics for different
movements with great precision [7] [9] [10] [11]
[12] [13] [14].
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DOI: 10.37394/232014.2022.18.2
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee
E-ISSN: 2224-3488
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For that reason, in this paper, we use the Kalman
filter (KF), Optimal Finite Impulse Response
(OFIR), Unbiased FIR (UFIR), and Optimal
Unbiased FIR (OUFIR) are used in object tracking
process to stabilize the bounding box trajectory. The
state estimation method was developed is two steps:
predict and update.
The FIR and Kalman filters are tested on
simulated trajectories and benchmark data available
on [8]. Based on these tests, we show that FIR and
KF algorithms showed favorable results on
simulated data with low data and process noise
values, but with larger values, OUFIR and UFIR
produced lower errors than KF and OFIR. Whereas,
the results using the benchmark data showed a better
performance of OUFIR.
2 Image processing
In order to carry the visual object, it is necessary to
identify the object to be tracked. Image processing
operations look for the best object recognition in
tracking task, which involves finding the correct
features to differentiate the target from other objects
and the scene background. The image is divided into
regions and the discontinuities are known as the
boundaries between the regions [15] [16] [17].
An image can be described through its
properties. To do this, it is necessary to calculate the
mathematical properties of an image or region and
use them as a basis for further classification [16].
Therefore, shape parameter extraction is necessary
for image representation. One of the most
commonly used shape parameters in object tracking
is bounding box.
2.1 Bounding box
The Bounding box is a rectangular box that enclose
an object in an image or scene. It can be represented
by the coordinates of the upper left and lower right
corners of the box [18]. When using the bounding
box (BB) as a shape parameter in target tracking,
information about the position of objects is
contained in an array of the minimum and maximum
vertices of the box that encloses the detected object
within the scene. The distribution of pixels within a
frame starts at the upper left corner and ends at the
lower right corner [19]. The bounding box matrix is
distributed over n rows and 4 columns, the rows
represent the number of recognized objects and the
columns contain the measurements for each
bounding box located as follows:
( 1)
Where Xc, Yc, Xw, and Yh are the coordinates of 4
corners of the bounding box: corners, weight, and
height. The algorithm to generate the bounding box
in the tracking process must predict the four
coordinates, X corner, Y corner, width, and height,
for each bounding box.
In the tracking process there may be errors in the
position estimation, an effective method to reduce
them is to apply a filtering method. A filtering
method us used to predict the coordinates of a point
of bounding box. The aim of using prediction and
correction methods is to mitigate the noise present
in the object tracking process, the CMN. The
prediction indicates the posterior position of the
bounding box based on its previous position. The
update is a correction step. It includes the new
tracking model measurement and helps improve
filtering [20] [21].
3 Tracking performance evaluation
The performance of tracking algorithms can be
evaluated using metrics called precision and
accuracy. Accuracy is the percentage of correct
object detections, and the accuracy can be measured
from the F-score which is an option metric to
measure accuracy.
3.1 Precision
The precision is calculated from the other
parameter, intersection over union (IoU). The
equations for calculating precision and IoU are (2)
and (3), respectively. The variables used in the
calculation of the precision are obtained from the
comparison of the IoU result with an established
threshold [22].
IoU= IA
(TBB- PBB)-IA
(2)
Where IA is the intersection area of the true
bounding box (TBB) and the estimated/predicted
bounding box (PBB). The IoU value is calculated in
each position of the bounding boxes.
Precision=
TP
(
TP +
FP )
(3)
Where TP is true-positive, and FP is false-
positive.
To determine the correct object detection of the
target, a threshold value of IoU must be established.
IoU is generally set to 0.5 [22]. Assuming the IoU
threshold is 0.5, if the value is greater or equal to
0.5, the detection is classified as True Positive (TP).
If IoU value is less than 0.5 it is considered as a
wrong detection and classified as False Positive
(FP). The IoU threshold can be set to a value of 0.5
or more, such as 0.75. 0.9, 0.95, or 1.
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Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee
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3.2 Accuracy
One metric to measure the accuracy in object
tracking model is F-score [23] [24]. The F-score use
the precision and recall and it can provide important
information about the model performing at various
threshold values. Recall can be calculated as the
number of correct detected objects divided by the
total number of detections in the ground truth. This
metric is based on the bounding box overlap
obtained between the algorithm and the real
trajectory to calculate the accuracy with which the
algorithm operates on an object displacement
sequence, the F-score is computed by equation (4).
(4)
4 State-space model
We consider a moving object represented in
discrete-time state-space with the following state
and observation equations (5) and (6).
( 5)
( 6)
Where xn k is the state vector, n∈ M is
the vector of observation. is the model of the state
transition, which is applied to project the previous
state xn1 to xn. E is the input control model, un is
the input control, is the noise matrix. is the
observation model. is the process noise
vn ∈ M is the colored Gauss-Markov noise with
white Gaussian with zero mean wn~N0, Qn ∈
Pand vn~N(0, Rn) ∈ M have the covariances
and , and the property Ewnvk
T=0 for all n and k.
Under the assumption that the two noise sequences
and the initial state are uncorrelated and
independent of each testing instant [25].
5 Kalman filter
The Kalman filter uses the equation of state of the
linear system to estimate the state of the system
through observation of input and output. The KF
requires knowledge of the system parameters, initial
values, and measurement sequences. The KF can
estimate the state sequences of the system iteratively
[26].
The Kalman filter calculate the optimal state
estimate by recursively combining previous
estimates with new observations. It consists of two
steps: predict, where the optimal state
previous
to observing
is calculated and update, where after
observing
the optimal posterior state
is
calculated. Additionally, it computes the prior
estimation error
, the posterior
estimation
, a priori estimate error
covariance
, and posterior estimate
error covariance
.
The a priori error covariance matrix is produced
in the predict step. Since the process noise wn is
assumed white Gaussian with zero mean, the a
priori state estimate is given by (7), and the a priori
error covariance matrix is estimated by (8).
( 7)
Then, in the update stage, the current a priori
predictions are combined with the current state
observation to redefine the state estimate and the
matrix of error covariance. The current observation
is used to improve the estimation, and it is called a
posteriori estimation of the state.
The measurement yn is corrupted by the noise vn.
Since vn is assumed white with zero mean, this
becomes (9), and the measurement residual (10).
( 9)
(10)
The residual covariance matrix is given by:
(11)
The optimal Kalman gain:
(12)
A posteriori state estimate:
xn=xn
+Kn(znC
xn
)
(13)
A posteriori error covariance matrix:
(14)
A pseudo-code of the Kalman filter is listed as
Algorithm 1.
Algorithm 1: Kalman Filter
Data:
Result:
Begin
for
n
=
1,2,
do
end for
End
( 8)
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6 Optimal Finite Impulse Response
(OFIR)
For faster processing, the OFIR algorithm is used in
its iterative form. Its iterative computation on
horizon [m, k] for given and provided by
Kalman filter (Algorithm 1) if we change the
auxiliary time-index i from m +1 to n and take
output when i =n. The iterative form for the OFIR
filter has been developed and tested in [27] [28].
The pseudo-code of the a posterior iterative
OFIR filter is listed as Algorithm 2. Given and
, this algorithm is iteratively updates values
from i = m +1 to i = n using optimal recursions of
KF (Algorithm 1) and obtains and Pn, when i =
n. The number of iterations can be limited by
optimal horizon length, Nopt of the Unbiased Finite
Impulse Response (UFIR) filter.
Algorithm 2: Iterative OFIR filter
Data: Pm,Qn, Rn
Result:
Begin
For
n=1,2, … do
if and
otherwise
For =m+1,m+2,…, n do
Algorithm 1:P
end for
end for
Pn
End
7 Unbiased Finite Impulse Response
(UFIR)
Unlike the KF and iterative OFIR, the Unbiased
FIR does not require any information about initial
conditions and noise, except for the zero mean
assumption [9] [14] [29] [30] [31]. Therefore, the
Unbiased FIR filter is more suited for object
tracking, where measurement and process noises are
not exactly known. However, the UFIR filter
requires an optimal horizon length,
Nopt
, from
, to minimize the Mean
Squared Error, and cannot ignore the CMN
,
which violates the zero-mean assumption on short
horizons.
Since the UFIR algorithm does not require noise
statistics, the prediction phase calculates only one
value, a priori state.
(15)
In the update step, the state estimate is combined
with the current observation state to refine the state.
The estimate is iteratively updated to the a posteriori
state estimate using (16) -(19).
Generalized noise power gain:
(16)
The measurement residual:
(17)
The UFIR gain:
(18)
The a posteriori state estimate is given by:
(19)
A pseudo-code of the UFIR algorithm is listed as
Algorithm 3. To initialize iterations, the algorithm
requires a short measurement vector
and matrix (20).
(20)
Algorithm 3: Unbiased FIR filter
Data:
Result:
Begin
For
n= N
do
For
l=s+k
: do
end for
end for
End
Where and are given by (21) and
(22) respectively,
is de Kth row vector in
(21).
=
(21)
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m,s=diag(
(22)
8 Optimal Unbiased Finite Impulse
Response (OUFIR)
Generally, in real object tracking applications, not
all information about the initial conditions of the
model is available. Since the OUFIR filter is very
indifferent to the initial conditions. In this case, we
apply an Optimal Unbiased iterative filter (OUFIR)
[32].
The iterative OUFIR algorithm is stated below. The
prediction phase calculates a single value, a priori
state, considering input (u) equal to zero the a priori
state is computed by (23).
(23)
In the update stage, the state estimated is
combined with the current observation state to
refine the state. In the same way as UFIR, the
estimate is iteratively updated to the a posterior state
estimate using equations (24) -(28).
The residual covariance matrix is given by:
Sn=CnPn
Cn
T+Rn
(24)
The OUFIR filter gain:
Kl=Pn
n
TSn
(25)
=(I Kl) NSn
(26)
Gain=Kl
(27)
The a posteriori state estimate:
(28)
A pseudo-code of the Optimal Unbiased FIR
filter is listed as Algorithm 4.
9 Object tracking tests
The KF and FIR algorithms were tested on
simulated data and benchmark data available in [8].
9.1 Test on simulated data
In this section, we test the algorithms numerically
by different simulated data. Our main goal is to
evaluate the performance in object tracking using
precision and F-score metrics. We consider the two-
state model and suppose an object is disturbed by
white Gaussian acceleration noise with a given
value of standard deviation. The model of a moving
target in a two-dimensional space was specified by
(5) and (6) with matrices:
=
, .
In simulation data two scenarios will be
considered:
Algorithm 4: Iterative OUFIR filter
Data: Qn, Rn
Result:
Begin
For
, if and
otherwise
Compute and
For l=m+2, m+3, … n do
SClPn
Gl=IPn
Cl
TSlCl
T)Gl-1
Sl=ClPCT+R
N=
Kl=Pn
Cn
TSn
=(I Kl) NSn
Gain=Kl
end for
end for
End
1) Simulated data 1. An object target is
disturbed by white Gaussian acceleration noise
with a standard deviation of . The
for the data noise originates from white Gaussian
. The simulation of the trajectory was
1000 points with sample time T= 0.05 seconds,
,
, , on a short horizon Nopt
= 15.
2) Simulated data 2. A trajectory is simulated
at 1000 points , . With
sample time T= 0.05 seconds, ,
,
, on a short horizon Nopt = 10.
9.1.1 Test on simulated data 1
We examine the results of algorithms in the object
tracking simulation using the bounding box
coordinates as a metric of evaluation. In Fig. 2 the
true trajectory of the object and the estimations
made by the algorithms are shown, where the black
line is the true trajectory, the blue line is KF, the red
line is UFIR, the yellow line is OUFIR, and the
green line is OFIR. Given that the Nopt for the FIR
filters was 15, the estimates started from this.
With low values of white Gaussian acceleration
noise and data noise, the OUFIR and UFIR filters
showed similar behavior. To provide a more
complete view, we calculate the root mean square
error (RMSE). The RMSE values were 929.98 for
KF, 929.93 for OFIR, 444.55 for OUFIR, and
478.21 for UFIR. According to these results, we
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consider that the OUFIR algorithm presented the
best performance with the lowest RMSE value.
The test precision results of simulated data 1 are
shown in Fig. 3. The OFIR and KF algorithms
produced a low precision with similar behavior on
the all thresholds range. According to the results, it
can be inferred that in each detection the overlap of
the Predicted Bounding Box (PBB) on the True
Bounding Box (TBB) was poor. It can be inferred
that each detection of the Kalman and OFIR filters
covers less to 50% of the TBB area. Since the most
used threshold values are 0.50% and 0.75% [22].
Therefore, we consider that Kalman and FIR filters
algorithms gave poor results in the most widely used
threshold IoU range. On the other hand, the OUFIR
and UFIR filters showed good results with an
average precision of 0.87 and 0.85, respectively,
that is, they cover at least 80% of the TBB area.
Fig. 3 Precision of simulated data 1
The F-score metric was used to measure the
accuracy. This metric is based on the bounding box
overlap obtained between the algorithm and the true
trajectory to calculate the accuracy with which the
algorithm operates on an object trajectory. The
results of the F-score for simulated data 1 are shown
in Fig. 4. The OUFIR and UFIR algorithms
produced a high accuracy from 0 to 0.9 threshold,
from which to decay. The OUFIR filter showed the
highest accuracy with average of 0.87, closely
followed by UFIR with 0.84. OFIR and KF
algorithms presented lowest values with average of
0.50 and 0.49 , respectively.
Fig. 4 Accuracy of simulated data 1
9.1.2 Test on simulated data 2
Next, we analyze the simulated data 2 test. In Fig. 5
the real trajectory of the object and the estimations
made by the algorithms are shown, where the black
line is the true trajectory, the blue line is KF, the red
line is UFIR, the yellow line is OUFIR, and the
green line is OFIR. Given that the Nopt for the FIR
filters was 10, the estimates started from this.
With high values of acceleration noise and data
noise, the OUFIR and UFIR filters showed similar
behavior. To analyze this, we calculate the root
mean square error (RMSE). The RMSE values were
6963.56 for KF, 6962.39 for OFIR, 3149.15 for
OUFIR, and 3149.15 for UFIR. Therefore we
consider the OUFIR and UFIR showed best
performance.
The precision results of simulated data 2 test are
shown in Fig. 6. OUFIR and UFIR presented a
better performance, these produced a precision over
77% from 0 to 0.9 threshold, from which to decay.
It can be inferred that each detection covers at least
77% of the TBB area. While OFIR and KF
presented low precision values below 40% in the
threshold range. The average precision for OUFIR
was 0.79, for UFIR was 0.79, for OFIR was 0.29,
and for KF was 0.29. As already mentioned, usually
the threshold used is 0.5, we consider that the
OUFIR and UFIR algorithms gave favorable results.
Fig. 2 Trajectory estimation of data 1 using Kalman
and FIR filters.
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Fig. 5 Trajectory estimation of simulated data 2
using Kalman and FIR filters.
Fig. 6 Precision of simulated data 2
In Fig. 7 the accuracy results of simulated data 2.
The OUFIR and UFIR algorithms produced
accuracy values over 0.6 from 0 to 0.8 threshold,
from which to decayed. The accuracy value
towards the 1 threshold is close to 0.2. The OFIR
and KF showed a poor performance from 0.1 to 1
threshold, with an accuracy below 0.3. The accuracy
value decreases as the threshold value increases.
The OUFIR filter showed the highest accuracy with
an average of 0.72, closely followed by UFIR with
0.71, the average for OFIR was 0.29, and for KF
was 0.28. With the given conditions, and according
to the results, we can determine that the OUFIR and
the UFIR present a good performance in the object
tracking process.
9.2 Test on benchmark trajectory
Then, we realized a test with a benchmark
trajectory, called “SUV” , available on [8]. The
coordinates of the SUV trajectory are measured by a
visual object tracking system. The SUV moves and
maneuvers on a highway road.
Fig. 7 Accuracy of simulated data 2
For the test, we considered that an object is
disturbed by white Gaussian acceleration noise with
the standard deviation of . The for the
data noise (CMN) originates from white Gaussian
. With sample time
T
= 0.05
seconds,
,
, , and the model of a moving
target in a two-dimensional space can be specified
by (5) and (6) with:
F
,
.
Since the UFIR filter requires an optimal
averaging horizon [m, n] of Nopt points. Following
[33], we determine:
Finally, we analyze the results obtained for the
“SUV” trajectory. The true trajectory and the
estimates by FIR and Kalman algorithms are shown
in Fig. 8. The identification colors remain as already
mentioned previously in this paper. The Nopt, as
already mentioned for the FIR filters, was 49.
In this test, the estimates of OUFIR and UFIR
were the best compared with those obtained by KF
and OFIR. Using a quantitative measurement, we
calculate the root mean square error (RMSE). The
RMSE values were 2316.90 for OUFIR, 2361.90 for
UFIR, 4962.39 for OFIR, and 4963.56 for KF.
According to these results, we consider that the
OUFIR and UFIR algorithms presented a good
performance, where OUFIR and UFIR present a the
best performance in the object tracking task under
the given conditions
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Analyzing the precision metric, it is observed
that the performance of the algorithms was higher
than in the previous simulated tests. The results are
shown in Fig. 9. Again, the OUFIR and UFIR
showed higher precision compared to OFIR and KF.
The precision mean is 0.84 for OUFIR, 0.83 for
UFIR. Taking as a reference that the Threshold most
used to evaluate the precision is 0.5, in this value the
precision is over 80% we can determine that the
performance of OUFIR and UFIR were good. In the
same way, the performance of KF and OFIR was
good, although lower than those already mentioned,
with an average precision of 0.68 and 0.66,
respectively.
The F-score values, accuracy, are shown in Fig.
10. The OUFIR and UFIR algorithms produced
accuracy values over than 0.7 in the 0.1 to 0.8
threshold range. OFIR and KF algorithms produced
accuracy values over 0.7 from 0.1 to 0.5 threshold
range, from which to decay. The OFIR and KF
presented a lower performance than UFIR and
OUFIR. According to the results, we consider that
OUFIR and UFIR present the best performance in
the object tracking process under the given
assumptions in a benchmark trajectory.
Fig. 10 Accuracy of “SUV” benchmark
10 Conclusion
The KF and OFIR estimation algorithms seem
to be less efficient than the OUFIR and UFIR in
object tracking process under the conditions
given in this paper. On the other hand, the
algorithms OUFIR and UFIR showed
favourable results in object tracking tests and
provided state estimation with higher precision
and accuracy, which can be useful in many
visual tracking applications such as video
surveillance and security, robotics, autonomous
vehicle navigation, etc. Remarking that, UFIR
does not require noise information and to know
the initial position. Likewise, the OUFIR is
highly insensitive to initial conditions.
According to the accuracy and precision
results, the OUFIR filters showed better
performance for tracking objects. Consequently,
the OUFIR filter in general shows higher
robustness against initial conditions and noise
statistics than UFIR, OFIR and KF.
Therefore, we conclude that the
incorporation of state estimators and the use of
OUFIR and UFIR filtering can provide further
development of object tracking algorithms for a
wide variety of application areas.
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Fig. 8 Trajectory estimation of “SUV” benchmark
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Fig. 9 Precision of “SUV” benchmark
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WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2022.18.2
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee
E-ISSN: 2224-3488
19
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Eli G. Pale-Ramon has written, reviewed, edited the
paper, and implemented the Algorithms in Matlab.
Yuriy S. Shmaliy has developed the methodology;
creation of models, and the project administration
Luis J. Morales-Mendoza has supervised and
reviewed the paper, was responsible for the
validation metrics used
Mario González-Lee was responsible for the
validation of the computational methods applied.
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2022.18.2
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee
E-ISSN: 2224-3488
20
Volume 18, 2022
