Object tracking is a topic of great importance in the field
of computer vision, since its applications range from security
to autonomous vehicles. Object tracking has many practical
applications such as image processing, robotics, industrial
inspection, among others [1]–[3].For this reason, there is great
interest from various researchers to develop algorithms that
help improve object tracking processes.
The purpose of the object tracking process is to correctly
estimate the position of a moving object throughout its tra-
jectory in a scene. However, there are several factors that
affect the performance of an algorithm in the tracking process,
factors specific to the scene where the object is moving, such
as lighting and background clutter or camera movements. That
This research was supported by the Mexican CONACYT-SEP Project A1-
S-10287, Funding CB2017-2018
is, the target object is not followed exactly during the entire
trajectory, there may be variations between the real position
and the estimated position. These differences or variations
can be considered as coloured measurement noise (CMN)
[4]. Currently there is a great diversity of contributions to
the area and it has become a challenging task to reduce
estimation errors, but object tracking is still, but there is no
single approach that provides the best solution to all the factors
affecting the tracking task. That is why it remains a topic of
great research interest.
One approach to address estimation problems in object
tracking is the use of motion model and state estimators
(MMS) as a method to avoid large tracking errors. Various
investigations have shown that the MMS application provides
high precision in position estimations if the state-space model
of the moving object is correctly specified [5]–[10].
In this paper, we consider the frame variations as CMN,
and apply the filters Standard Kalman (KF),modified Kalman
for CMN(KF CMN, standard Unbiased Finite Impulse Re-
sponse (UFIR), and modified UFIR for CMN (UFIR CMN)for
estimate the state in object tracking process. The Filter test
was performed with simulated sequences setting different
conditions and benchmark data available on [11].To estimate
the trajectories and to be able to evaluate the performance of
the algorithms, the bounding box coordinates were used as
data for the state estimation. Each position of the object in
the trajectory is represented with a bounding box.
According to results, UFIR CMN and the KF CMN pre-
sented favorable results on simulated data with ideal conditions
and known process and data noise. However with no ideal
conditions noise values and with the the highest colored factor
Ψthe UFIR CMN showed better results than KF CMN and
the standard filter (KF, UFIR). Regarding the results obtained
with benchmark data, the UFIR CMN presented the best
performance followed closely by KF CMN.
Estimation of states with data under Colored Measurement
Noise (CMN)
1ELI G. PALE-RAMON, 1YURIY S. SHMALIY, 2LUIS J. MORALES-MENDOZA,
2MARIO GONZALEZ-LEE, 1JORGE A. ORTEGA-CONTRERAS, 1KAREN URIBE-MURCIA
1Electronics Engineering Department Universidad de Guanajuato, Salamanca, MEXICO
2Electronics Engineering Department Universidad Veracruzana, Poza Rica, MEXICO
Abstract: Object tracking is an area of study of great interest to various researchers, where the main objective is to improve
estimation of the trajectory of a moving object. This is due to the fact that in the object tracking process there are usually
variations between the true position of the moving object and the estimated position, that is, the object is not exactly
followed throughout its trajectory. These variations can be thought of as Colored Measurement Noise (CMN) caused by
the object and the movement of the camera frame. In this paper, we treat such differences as Gauss-Markov colored
measurement noise.We use Finite Impulse Response and Kalman Filters with a recursive strategy on the tracking: predict
and update. To demonstrate the filter with the best performance, tests were carried out with simulated trajectories and with
benchmarks from a database available online. The UFIR modified for CMN algorithm showed favorable results with high
precision and accuracy in the object tracking process with benchamark data and under no ideal conditions.While KF CMN
showed better results in tests with simulated data under ideal conditions.
Keywords: Colored measurement noise, Bounding box, estimation, object tracking, Unbiased FIR filters, Kalman filter,
Precision, F-score.
Received: July 17, 2021. Revised: May 29, 2022. Accepted: June 21, 2022. Published: August 3, 2022.
1. Introduction
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Object tracking is a topic of great importance in the area of
computer vision due to its wide field of application. Image
processing operations seek the best recognition of objects
in the track, which involves finding the right features to
differentiate the target from other objects and the background
of the scene. The content of an image can be described through
its properties. To do this, it is necessary to calculate the
properties of an image or region and use them as a basis for
further classification. Therefore, shape parameter extraction is
necessary for image representations. There are different ways
to locate a target object, for example, its contour or finding the
pixels of the object, one of the most used shape parameters
in object tracking is the bounding box [12]. In addition, there
is a diversity of databases that provide paths annotated with
bounding boxes [11], [13]–[16].
The bounding box (BB) is used to represent the target object
during an entire trajectory. The bounding box can be defined
as a rectangular box that encloses all target objects in a scene.
Information about the position of objects in is contained in an
array of bounding boxes. This information can be represented
by the coordinates of the upper left and lower right corners of
the bounding box [17]. The matrix consists of 4 columns and
n rows, the number of rows correspond to the total number of
detections, while the columns represent the dimensions of the
bounding boxes: coordinate ”x”, coordinate ”y”, width (xw),
height (yw).
In the tracking process, the ”x” and ”y” coordinate infor-
mation of each bounding box is used to estimate the position
of the object during the entire trajectory. However, there
may be errors in the position estimation. To reduce these
errors, an effective method is the application of a filtering
method. A filtering method is used to predict the coordinates
of a bounding box point. The estimation method consists
of 2 stages: prediction and correction, whose objective is to
mitigate the noise present in the object tracking process. The
prediction indicates the next position of the bounding box
based on its previous position. The update is a correction step,
which includes the new measurement of the follow-up model
and helps to improve the estimate of the filteringl [18], [19].
Bounding boxes are useful in object tracking algorithms
when intersecting objects must be processed [20].In other
words, the overlapping of the predicted bounding box (PBB)
with the true bounding box (TBB) that contains the informa-
tion of the target object along its trajectory in a sequence.
Taking this into account, the performance of the tracking
algorithm can be analyzed through the information provided
by the bounding boxes. The most common ways to evaluate
the performance of an object tracking algorithm, using the
information provided by the bounding box matrix, are the
precision and the F1-score.
The evaluation of the performance of the tracking algo-
rithms can be done using the standard metrics, precision and
accuracy, for calculate the accuracy we use the metric F-score.
Precision can be defined as the percentage of the number of
correct predictions over the total number of predictions. F-
score, is a metric that combines precision and recall, this last
term is calculated as the number of correct detected objects
divided by the total number of detections in the ground truth
[21]–[25].
To calculate the precision it is necessary to first calculate
another metric, intersection over union (IoU), which indicates
the percentage of overloap of the predicted bounding box
over the True Bounding box (TBB). The variables used
in the calculation of the precision are obtained from the
comparison of the IoU result with an established threshold.
The variables used for computed the precision are obtained
from the comparison of the IoU result with an established
threshold [21]–[23], [26]. The equations for calculating IoU
and precision are (1) and (2), respectively.
IoU =IA
(T BB −EBB)−IA (1)
P recision =ΣT P
ΣT P + ΣF P =ΣT P
All detections (2)
Where the IA is the area of intersection between the
bounding box of the target object, the true bounding box
(TBB), and the estimated bounding box (EBB). The TP is
true positive, and FP is false positive.
The IoU metric allows establishing the degree or percentage
of EBB overloap over TBB, for which it is necessary to
establish an IoU threshold that works as the comparison
parameter to establish whether it is a correct or incorrect
detection. Generally, the IoU threshold is set to 0.5.
Considering a single object tracking, many measures to
evaluate the performance of the tracking algorithm are based
on the overlap comparison of the EBB versus the TBB. The
possible qualification of the bounding box overloap in object
tracking compared to a given threshold is shown below [21]–
[23]:
•True Positive (TP). It is a correct detection of a bounding
box, that is, the IoU between the EBB and TBB is greater
than or equal to the established threshold value.
•False positive (FP). It is an incorrect detection of an
object or an off-site detection. The IoU is less than the
given threshold value, but greater than zero
•False negative (FN). It is an undetected TBB.
Most benchmarks use an axis-aligned bounding box as the
Ground Truth and estimate the accuracy commonly by the IoU
criterion between the EBB and the TBB, i.e. based on the
2. Object Tracking
2.1. Bounding Box
3. Performance Metrics of
Tracking Algorithm
3.1. Precision
3.2. F-score
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comparison of the EBB versus the ground truth . One of the
position-based evaluation parameters to measure accuracy is F-
measure or F-score. Which is defined as the harmonic mean of
the precision and recall. Where, Recall can be calculated as the
number of correct detected objects divided by the total number
of frames in the ground truth [21]–[23]. Recall is computed
as shown below:
Recall =T P
All ground truths (3)
F-score metric is delimited to the interval [0,1], the value
will be 0 when the value of precision and recall is 0 and
1 when both values are equal to 1. Remembering that F-score
considers both precision and recall [22], one way to calculate
is given by:
F−score = 2 P recsion ×Recall
P recision +Recall (4)
Another way to calculate f-score directly considering the
overloap qualifications between the EBB and the TBB given
a given threshold value is computed by [23]:
F−score =T P
T P +T P
F N +F P
(5)
If we consider the total number of frames of the target’s
trajectory, the F-score result in both equations (4) and (5)
would consist of a sum of its values in each frame divided
by the total number of frames.
The tracking evaluation parameters, F-score and precision,
are calculated concerning a threshold value. Which ranges
from 0 to 1. Depending on the degree of overloap that we want
to achieve, it will be the IoU comparison parameter, generally
being a value of 0.5, that is, the algorithm is expected to be
able to identify at least 50% of the area of the TBB. Also,
the scale from 0 to 1 can be used and in this way evaluate at
which IoU level the algorithm performs best.
We consider a moving object with observation corrupted
by CMN can be represented in discrete-time state-space using
with the following state and observation equations:
xn=Anxn−1+Bnwn,(6)
vn= Ψnvn−1+ξn,(7)
yn=Cnxn+vn,(8)
where xn∈RKis the state vector, yn∈RMis the observation
vector, vn∈RMis the colored Gauss-Markov noise, and
An∈RK×Kis the state transition matrix, Bn∈RK×P
is the gain matrix model, Cn∈RM×Kis the measurement
matrix. Measurement noise transfer matrix Ψnis chosen such
that the colored noise vnremains stationary. The zero mean
Gaussian noise vectors wn∼ N (0, Qn)∈RPand ξn∼
N(0, Rn)∈RMhave the covariances Qnand Rnand the
property E{wnξT
k}= 0 for all nand k. We will consider the
following estimates: the prior estimate ˆx−
n≜ˆxn|n−1, posterior
estimate ˆxn≜ˆxn|n, prior estimation error ϵ−
n=xn−ˆx−
n,
the posterior estimation error ϵn=xn−ˆxn, the prior error
covariance P−
n≜Pn|n−1=E{ϵ−
nϵ−T
n}, and the posterior
error covariance Pn≜Pn|n=E{ϵnϵnT}.
Considering the motion of a physical system in space, the
next position of the object can be calculated using Newton’s
equation of motion [27]:
cx =cx0+v0τ+1
2act2,(9)
where cx is the position on the x-axis of the object position,
cx0is the object’s initial position, v0is the object’s initial
velocity, ac is the object’s acceleration, and τis the time
interval .
Since the state estimation will be applied for the 4 coordi-
nates of the bounding box, it is necessary to define the equa-
tions of motion for the other 3 bounding box measurements
that make up the bounding box; which are the coordinates on
the y-axis, the width of bounding box and height of bounding
box.
Considering a constant acceleration model [28] , for the
moving object state space model, the state transition (A) is
block diagonal matrix with:
1ττ2
2
0 1 τ
0 0 1
,(10)
where the block is repeated for the ”x”,”y”, ”width”, and
”height” spatial dimensions. τis the sample time.
The gain matrix model (B) and observation matrix (C) are
defined as shown below:
B=
τ2
20 0 0
τ0 0 0
1 0 0 0
0τ2
20 0
0τ0 0
0 0 τ2
20
0 0 τ0
0 0 1 0
0 0 0 τ2
2
0 0 0 τ
0 0 0 1
,(11)
C=
100000000000
000100000000
000000100000
000000000100
.(12)
4. Moving Object State-space Model
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To avoid the CMN vnin ynis necesary the model modifica-
tion using measurement differencing. For which, it
´
s consider
a new observation znas a measurement difference.
zn=yn−Ψnyn−1,
=Cnxn+vn−ΨnHn−1xn−1−Ψnvn−1,(13)
and transform (4) to
zn=Dnxn+ ¯vn,(14)
where Dn=Hn−Γn,Γn= ΨnHn−1F−1
n,¯vn= ΓnBnwn+
ξn, noise ¯vnis now white with the properties
¯
Rn=E{¯vn¯vT
n}= ΓnΦn+Rn,(15)
E{¯vnwT
n}= ΓnBnQn,(16)
where Φn=BnQnBT
nΓT
n, and model (1) and (5) has thus
two white and time-correlated noise sources wnand ¯vn.
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. Where the state is assumed to be dis-
tributed by a white Gaussian noise with zero mean. The KF
consist of two steps, prediction and update and it is a recursive
estimator [29], this means the previous estimated state need
to be combined with new observations to calculate the best
estimate of the current state. The KF requires knowledge of the
system parameters, initial values, and measurement sequences.
The KF can estimate the state dynamics of the system
iteratively [30], as already mentioned, consists of two steps:
predict, where the optimal state ˆx−
nprevious to observing yn
is calculated and update, where after observing ynthe optimal
posterior state ˆxnis calculated. Additionally, it computes
the prior estimation error ϵ−
n=xn−ˆx−
n, the posterior
estimation ϵ−
n=xn−ˆx−
n, a priori estimate error covariance
P−
n=E{ϵ−
nϵ−T
n}, and posterior estimate error covariance
Pn=E{ϵnϵT
n}.
In the standard Kalman and the KF modified for CMN
filters, in predicted phase is produced the a priori error covari-
ance. Assuming that the process noise wnis white Gaussian
with zero mean, the prior state estimate is computed by (17),
and the prior error covariance matrix is estimated by (18).
ˆx−
n=Aˆxn−1+Bnwn,(17)
P−
n=AnPnA+BnQnBT
n,(18)
Then, in the update phase, the current prior predictions are
combined with the current state observation to redefine the
state estimate and the error covariance matrix. The combina-
tion of the prediction with the current observation is used to
obtain the optimal state estimate, and is called the posterior
state estimate. The measurement ynis corrupted by colored
measurement noise vn. The measurement residual is (34).
yn=Cˆxn−1+ ¯vn(19)
The residual covariance matrix is obtained as follow:
Sn=CnP−
nCT
n+Rn(20)
The optimal Gain for Kalman is given by:
Kn=P−
NCT
nS−1
n(21)
A posteriori state estimate:
ˆxn= ˆx−
n+Kn(zn−cˆx−
n)(22)
A posteriori matrix of error covariance:
Pn= (I−KnC)P−
n(23)
A pseudo code of standard Kalman filter is listed as Algorithm
1.
Algorithm 1: Standard Kalman Filter
Data: yn,ˆx0, P0,Qn, Rn, un
Result: ˆ
xn,Pn
1begin
2for n= 1,2,· · · do
3ˆx−
n=Aˆxn−1+BNun
4¯
P−
n=AnPn−1AT
n+BnQnBT
n
5Sn=CnP−
nCT
n+Rn
6Kn=P−
nCT
nS−1
n
7ˆxn= ˆx−
n+Knzn
8Pn= (I−KnCn)P−
n
9end for
10 end
The design of KF algorithms assuming CMN is deriving a
new bias correction gain and error covariance for correlated
noise. The measurement residual sntransformed to
sn=zn−Dnˆx−
n
=DnAnϵn−1+DnBnwn+ ¯vn,(24)
the innovation covariance Snis given by
Sn=E{snsT
n}
=DnP−
nDT
n+Rn+CnΦn+ ΦT
nDT
n,(25)
and the estimation of KF for CMN is
ˆxn= ˆx−
n+Knsn
=Anˆxn−1+Kn(zn−DnAnˆxn−1),(26)
where Knis the bias correction gain that should be optimized
for correlated wnand ¯vn. the estimation error ϵncan be written
as
ϵn=xn−ˆxn
= (I−KnDn)Anϵn−1
+(I−KnDn)Bnwn−Kn¯vn(27)
4.1. Avoid the CMN
5. Standard and Modified for
CMN Kalman Filter
5.1. Standard Kalman Filter
5.2. Kalman Filter for CMN
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and the error covariance Pn=E{ϵnϵT
n}transformed to
Pn=P−
n−(P−
nDT
n+ Φn)KT
n−Kn(P−
nDT
n+ Φn)T
+KnSnKT
n,(28)
where P−
nis given by (18) and Snby (24). The optimal gain
Knis given by
Kn= (P−
nDT
n+ Φn)S−1
n(29)
and (28) becomes
Pn=P−
n−Kn(DnP−
n+ ΦT
n).(30)
A pseudo code of the KF algorithm for CMN with correlated
wnand ¯vnis listed as Algorithm 2. It can be observed, if the
value Ψn= 0, the algorithm becomes the standard Kalman
Filter algorithm. and it becomes the standard KF by .
Algorithm 2: KF for CMN and Correlated wnand ¯vn
Data: yn,ˆx0,P0,Qn,Rn
Result: ˆ
xn,Pn
1begin
2for n= 1,2,· · · do
3Dn=Hn−ΨnCn−1A−1
n
4zn=yn−Ψnyn−1
5P−
n=AnPn−1AT
n+BnQnBT
n
6Sn=DnP−
nDT
n+Rn+CnΦn+ ΦT
nDT
n
7Kn= (P−
nDT
n+ Φn)S−1
n
8ˆx−
n=Anˆxn−1
9ˆxn= ˆx−
n+Kn(zn−Dnˆx−
n)
10 Pn= (I−KnDn)P−
n−KnΦT
n
11 end for
12 end
Unlike the Kalman Filter, the Unbiased FIR does not require
any information about initial conditions, the initial state, error
covariance matrix P, and statistical noises Q and R, except for
the zero mean assumption [8], [31], [31], [32]. UFIR is also
an option for estimating states with incomplete measurement
Information [33].
Instead, it requires an optimal averaging horizon [m, n] to
minimize the mean square error(MSE), that is, the UFIR filter
operates at once with N measurements on a horizon [m, k]
from m=k−N+1. The UFIR filter cannot ignore the CMN
vn, which violates the zero mean assumption on short horizons
[31].
Since the UFIR algorithm does not require noise statistics,
the prediction phase calculates only one value, a priori state6.
In the update step, the state estimate is combined with the
actual observation state to refine the state The estimate is
iteratively updated to the a posteriori state estimate using
generalized noise power gain, measurement residual,and UFIR
gain as shown in a pseudo-code of the UFIR algorithm is listed
as 3. To initialize iterations, the algorithm requires a short
measurement vector y(m.k) = [ym...yk]Tand matrix (31):
Hm,s =
Cm(As...Am+1)−1
Cm+1(As...Am+2)−1
.
.
.
Cs−1A−1
s
Cs
.(31)
Algorithm 3: Standard UFIR Filter
Data: yn,un,N
Result: ˆ
xn
1begin
2for n=N−1, N, · · · do
3m=n−N+ 1,s=m−K+ 1
4Gs=HT
m,sHm,s
5¯xs=GsHT
m,s(ym,s −Lm,sum,s) + Sk
m,sum,s
6for l=s+k, · · · do
7¯x−
l=Aˆxl−1+Bu
Gl= [CT
lCl+ (AlGl−1AT
l)−1]−1
Gainl=GlCT
l
¯xl= ˆx−
l+Gainl(yl−C¯x−
l)
8end for
9ˆxn= ¯xn
10 end for
11 end
Where Sm,s and Lm,s are given by (33) and (32) respec-
tivily, Sk
m,s is the Kth row vector in (33)
Lm,s =diag(Cm,s)Sm,s (32)
Hm,s =
Bm0· · · 0 0
Am+1BmBm+1 · · · 0 0
.
.
..
.
.....
.
..
.
.
Am+1
s−1BmAm+2
s−1Bm+1 · · · Bs−10
Am+1
s−1BmAm+2
s−1Bm· · · AmBs−1Bs
.(33)
Since colored measurement noise is present in the data
needed for the estimation, the following modified UFIR al-
gorithm pseudocode was used to avoid the CMN, as shown
in Algorithm 4. In the same way as with the standard UFIR
algorithm, to initialize the iterations, the algorithm requires a
short measurement vector Ym,s = [ ym. . . ys]Tand matrix
Hm,s =
Dm(Fs...Fm+1)−1
.
.
.
Ds−1F−1
s
Ds
.(34)
It also follows that, by Ψn= 0, the Algorithms 4 and 2
becomes the standard UFIR filter and the standard Kalman
filter, respectively [31].
6. Standard and Modified for CMN
Unbiased)LQLWH,PSXOVH5HVSRQVH)LOWHU
6.1. Standard Unbiased Finite Impulse
Response Filter
6.2. Unbiased Finite Impulse Response
Filter Modified for CMN
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Algorithm 4: UFIR Filter for CMN
Data: N,yn
Result: ˆxn
1begin
2for n=N−1, N, · · · do
3m=n−N+ 1 ,s=n−N+K
4Gs= (CT
m,sCm,s)−1
5¯xs=GsCT
m,sYm,s
6for l=s+ 1 : ndo
7Dl=Hl−ΨlHl−1F−1
l
8zl=yl−Ψlyl−1
9Gl= [DT
lDl+ (FlGl−1FT
l)−1]−1
10 Kl=GlDT
l
11 ¯x−
l=Fl¯xl−1
12 ¯xl= ¯x−
l+Kl(zl−Dl¯x−
l)
13 end for
14 ˆxn= ¯xn
15 end for
16 end
The simulation was performed using the object tracking
model and matrices proposed in section IV. Where, the first
state is the distance, and the second state is the velocity and
the third state is the acceleration, which corresponds to a
constant acceleration state model. The moving object model
can be described by (1) and (2). For the first simulation
test we consider that an object target is disturbed by white
Gaussian acceleration noise with a standard deviation of
σw= 1m/s2. The data noise originates from white Gaussian
with σξ= 20 m. The simulation of the trajectory was 2000
points with sample time T= 0.05 seconds, P0= 0,Q=σ2
w,
R=σ2
V, on a short horizon Nopt = 100.
The RMSE results obtained by the standard and modified
filters for CMN, KF and UFIR, from the color factor Ψ 0 to
0.95 are shown in Fig. 1. In the calculation of RMSE with the
standard and modified UFIR algorithm for CMN, a Nopt of
the same size was used for the entire range of the coloration
factor Ψ.
Results of standard UFIR are shown with magenta line,
standard KF with black line, and modified UFIR for CMN
with red line. For the KF modified for CMN, the object
tracking model states were established under ideal conditions,
p=q= 1, and with errors in the noise statistics, p= 1,
represented with blue line, and q= 1 with represented with
blue dotted line.
Assuming that a complete information about noise is in-
complete, in Fig. 1 we showed the relevant filtering errors
produced by substituting in the algorithms Qwith p2Qand
Rwith q2Rfor {p, q}>0[8], [31]. As can be seen, even
slight modifications in error factors (p= 2, q = 0.5) make
the KF CMN less accurate than the UFIR CMN. It even has
a poorer performance than standard filters. It is important to
note that the assumption of incomplete noise information did
not apply to the standard KF.Since both the standard and
modified UFIR filter for CMN do not require knowing the
noise conditions Qand Rit shows that the UFIR filter is more
preferable for object target tracking. To corroborate the effect
of a model with incomplete or unknown noise information.
We performed another test considering only the highest noise
factor Ψ = 0.95. For this test, we propose a model with
standard deviation of σw= 5m/s2, The data noise with
σv= 10m.The simulated trajectory with 2000 points with
a sample time T= 0.05 seconds, P0= 0 on a short horizon
Nopt = 100.
Derived from the results obtained in the previous test,
where the filters modified for CMN showed better results,
it was decided to carry out the second test only with the
filters modified for CMN. The results obtained are shown in
Fig. 2,we showed the relevant filtering errors produced by
substituting in the algorithms Qwith p2Qand Rwith q2R
for {p, q}>0, (p= 2, q = 0.5).
As can be seen, that before a slight modification of the error
factor, a significant change is generated in the performance of
the KF CMN algorithm. On the other hand, since UFIR does
not require the initial noise conditions, it is more robust to state
models in which the information of noise is not available. This
gives another proof that the UFIR CMN is more suitable for
object tracking process.
In this section we work with benchmark data available
on [11], the data with which the tests were performed are
called ”Car4” and ”SUV”. Taking into account the tests carried
out with the simulated data, in the previous subsection, we
decided to carry out the state estimation test with the Nopt
corresponding to the standard and modified UFIR. The tests
Fig. 1. RMSE results obtained b,y the KF and UFIR standard and modified
filters for CMN with Ψ 0 to 0.95.
7. Object Tracking Test
7.1. Results of Simulation Test
7.2. Results of Benchmark Data Test
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Eli G. Pale-Ramon, Yuriy S. Shmaliy,
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were performed with the highest noise factor Ψ=0.95 in
order to evaluate the robustness of the algorithms.
Fig. 2. RMSE results obtained by modified filters for CMN, KF and UFIR
(Nopt = 100), with the colored factor Ψ=0.95.
For the benchmark data test,”Car4”, for the object tracking
model we considered that the car target is disturbed by white
Gaussian acceleration noise with the standard deviation of
σw= 3m/s2. The data noise (CMN) originates from white
Gaussian σv= 2m. The sample time T= 0.05 seconds,
P0= 0,, Q=σ2
w,R=σ2
V, on a short horizon Nopt = 110.
The model of a moving target is completed according to what
is established in the section IV.
The precision values of each of the filters in the entire
intersection over union (IoU) threshold range are shown in
Fig.3 and the F-score values in 4. The UFIR CMN and KF
CMN filters presented the best performance, with the UFIR
CMN being slightly better in the full range of IoU threshold.
With this test it can be seen that the standard filters had a poor
performance in the estimation of states under colored noise
with a high color factor Ψ=0.95. The UFIR CMN and KF
CMN filters presented the best performance, where with UFIR
CMN beign slightly better than KF CMN in the full range of
IoU treshold. Analyzing the F-score metric,the performance
is similar to that given by the precision metric. Both metrics
serve us to measure the performance of the tracking algorithm.
As opposed to precision which focuses on what percentage of
the target bounding box was detected; with F-score we can see
if there were target losses. In this case, it is observed that the
standard Kalman filter presented the lowest value of F-score.
Fig. 3. Precision of benchmark ”Car4”(N opt = 110),with Ψ=0.95.
Fig. 4. F-score of benchmark ”Car4”(N opt = 110),with Ψ=0.95.
For the benchmark data test,”SUV”, for the object tracking
model we considered that the car target is disturbed by white
Gaussian acceleration noise with the standard deviation of
σw= 10m/s2. The data noise (CMN) originates from white
Gaussian σv= 5m. The sample time T= 0.05 seconds,
P0= 0,, Q=σ2
w,R=σ2
V, on a short horizon Nopt = 70.
The model of a moving target is completed according to what
is established in the section IV.
The precision values of each of the filters over the entire
Intersection over Union (IoU) threshold range are shown in
Fig.5 and the F-score values in 6. In this case, both values
are similar, the F-score is slightly lower. The UFIR CMN
presented the best performance at a higher value of IoU
threshold, being the UFIR CMN slightly better after the
threshold of IoU 0.4 for precision and F-score. Overall, the
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KF CMN filter was the second best estimator. However, its
accuracy and F-score were lower at high IoU values, being
outperformed by the standard UFIR filter. This may be because
the initial conditions of the motion model are not exactly
known. As with the ”Car4” test, it can be seen that the standard
Kalman filter had a performance poorly in the estimation of
states under colored noise with a high color factor Ψ=0.95.
Fig. 5. Precision of benchmark ”SUV”(N opt = 70),with Ψ=0.95.
Fig. 6. F-score of benchmark ”SUV”(N opt = 110),with Ψ=0.95.
Based on the results of this work, we find that, in general,
the standard filters, UFIR and KF, presented a low performance
in the estimation of states with data in the presence of colored
measurement noise and with noise factor Ψpresenting low
precision and accuracy values (F-score). The KF CMN filter
performs well when used simulated data with ideal conditions
and known data and process noise. In the same way, it was
shown to be a good estimator with real data under non-ideal
conditions, but it is highly dependent on a correct approach to
the movement model, as was demonstrated in the estimation
with simulated data when using incomplete or unknown noise.
The algorithm UFIR CMN generally obtained better results
and proved to be more robust, since it does not require
knowing the initial conditions of process noise and data.
Its performance was good under non-ideal conditions and
with a high noise factor value. These characteristics make it
an estimation algorithm with great application in movement
models where the information is not well known or the
complete information is not available.
Therefore, we conclude that the incorporation of UFIR
CMN state estimation algorithms would contribute to the
development and improvement of applications and research
in the field of object tracking research.
We are currently working on modifications of the UFIR
CMN algorithm, since although we consider good results were
obtained, we are focused to the improvement in the estimates
when the values of Ψand IoU threshold are the highest. This
in order to develop a more robust and efficient algorithm.
[1] A. N. Bishop, A. V. Savkin, and P. N. Pathirana, “Vision-based target
tracking and surveillance with robust set-valued state estimation,” IEEE
Signal Processing Letters, vol. 17, no. 3, pp. 289–292, 2010.
[2] A. Yilmaz, O. Javed, and M. Shah, “Object tracking: A survey,” Acm
computing surveys (CSUR), vol. 38, no. 4, pp. 13–es, 2006.
[3] “Robust visual tracking framework in the presence of blurring by
author = Kang, TaeKoo and Mo, YungHak and Pae, DongSung and
Ahn, ChoonKi and Lim, MyoTaeg, journal=Measurement, volume=95,
pages=50–69, year=2017, publisher=Elsevier.”
[4] Y. S. Shmaliy, S. Zhao, and C. K. Ahn, “Optimal and unbiased
filtering with colored process noise using state differencing,” IEEE
Signal Processing Letters, vol. 26, no. 4, pp. 548–551, 2019.
[5] Y. Yoon, A. Kosaka, and A. C. Kak, “A new kalman-filter-based
framework for fast and accurate visual tracking of rigid objects,” IEEE
Transactions on Robotics, vol. 24, no. 5, pp. 1238–1251, 2008.
[6] D. Simon, Optimal state estimation: Kalman, H∞, and nonlinear
approaches. Hoboken,NJ: John Wiley & Sons, 2006.
[7] P. Liang, E. Blasch, and H. Ling, “Encoding color information for visual
tracking: Algorithms and benchmark,” IEEE transactions on image
processing, vol. 24, no. 12, pp. 5630–5644, 2015.
[8] Y. S. Shmaliy, S. Zhao, and C. K. Ahn, “Kalman and ufir state estimation
with coloured measurement noise using backward euler method,” IET
Signal Processing, vol. 14, no. 2, pp. 64–71, 2020.
[9] Y. S. Shmaliy, “Linear optimal fir estimation of discrete time-invariant
state-space models,” IEEE Transactions on Signal Processing, vol. 58,
no. 6, pp. 3086–3096, 2010.
[10] S. Zhao, Y. S. Shmaliy, and C. K. Ahn, “Bias-constrained optimal
fusion filtering for decentralized wsn with correlated noise sources,”
IEEE Transactions on Signal and Information Processing over Networks,
vol. 4, no. 4, pp. 727–735, 2018.
[11] (2015) Datasets-visual tracker benchmark. [Online]. Available: http:
//www.visual-tracking.net
[12] W. Burger, M. J. Burge, M. J. Burge, and M. J. Burge, Principles of
digital image processing. Springer, 2009, vol. 54.
[13] A. Dave, T. Khurana, P. Tokmakov, C. Schmid, and D. Ramanan,
“Tao: A large-scale benchmark for tracking any object,” in European
conference on computer vision. Springer, 2020, pp. 436–454.
[14] P. Voigtlaender, L. Luo, C. Yuan, Y. Jiang, and B. Leibe, “Reducing the
annotation effort for video object segmentation datasets,” in Proceedings
of the IEEE/CVF Winter Conference on Applications of Computer Vision,
2021, pp. 3060–3069.
8. Conclusions
References
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.40
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario Gonzalez-lee,
Jorge A. Ortega-Contreras, Karen Uribe-Murcia
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Volume 17, 2022
[15] Y. Hu, H. Liu, M. Pfeiffer, and T. Delbruck, “Dvs benchmark datasets
for object tracking, action recognition, and object recognition,” Frontiers
in neuroscience, vol. 10, p. 405, 2016.
[16] O. S. Google. (2020) Open images dataset v6. [Online]. Available:
https://storage.googleapis.com/openimages/web/download.html
[17] K. Choeychuen, P. Kumhom, and K. Chamnongthai, “An efficient
implementation of the nearest neighbor based visual objects tracking,”
in 2006 International Symposium on Intelligent Signal Processing and
Communications. IEEE, 2006, pp. 574–577.
[18] P. Deepak and S. Suresh, “Design and utilization of bounding box
in human detection and activity identification,” in Emerging ICT for
Bridging the Future-Proceedings of the 49th Annual Convention of the
Computer Society of India CSI Volume 2. Springer, 2015, pp. 59–70.
[19] M. S. Grewal and A. P. Andrews, Kalman filtering: Theory and Practice
with MATLAB. Hoboken,NJ: John Wiley & Sons, 2014.
[20] Y. Zhou and S. Suri, “Analysis of a bounding box heuristic for object
intersection,” Journal of the ACM (JACM), vol. 46, no. 6, pp. 833–857,
1999.
[21] B. Karasulu and S. Korukoglu, “A software for performance evaluation
and comparison of people detection and tracking methods in video
processing,” Multimedia Tools and Applications, vol. 55, no. 3, pp. 677–
723, 2011.
[22] D. L. Olson and D. Delen, Advanced data mining techniques. Springer
Science & Business Media, 2008.
[23] R. Padilla, W. L. Passos, T. L. Dias, S. L. Netto, and E. A. da Silva,
“A comparative analysis of object detection metrics with a companion
open-source toolkit,” Electronics, vol. 10, no. 3, p. 279, 2021.
[24] E. Ranguelova, B. Weel, D. Roy, M. Kuffer, K. Pfeffer, and M. Lees,
“Image based classification of slums, built-up and non-built-up areas
in kalyan and bangalore, india,” European journal of remote sensing,
vol. 52, no. sup1, pp. 40–61, 2019.
[25] F. Sun, H. Li, Z. Liu, X. Li, and Z. Wu, “Arbitrary-angle bounding box
based location for object detection in remote sensing image,” European
Journal of Remote Sensing, vol. 54, no. 1, pp. 102–116, 2021.
[26] A. W. Smeulders, D. M. Chu, R. Cucchiara, S. Calderara, A. Dehghan,
and M. Shah, “Visual tracking: An experimental survey,” IEEE transac-
tions on pattern analysis and machine intelligence, vol. 36, no. 7, pp.
1442–1468, 2013.
[27] C. Xiu, X. Su, and X. Pan, “Improved target tracking algorithm based
on camshift,” in 2018 Chinese control and decision conference (CCDC).
IEEE, 2018, pp. 4449–4454.
[28] X. R. Li and V. P. Jilkov, “Survey of maneuvering target tracking. part
i. dynamic models,” IEEE Transactions on aerospace and electronic
systems, vol. 39, no. 4, pp. 1333–1364, 2003.
[29] Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with applica-
tions to tracking and navigation: theory algorithms and software. John
Wiley & Sons, 2001.
[30] R. G. Brown and P. Y. Hwang, “Introduction to random signals and
applied kalman filtering: with matlab exercises and solutions,” Intro-
duction to random signals and applied Kalman filtering: with MATLAB
exercises and solutions, 1997.
[31] Y. S. Shmaliy, S. Zhao, and C. K. Ahn, “Unbiased finite impluse
response filtering: An iterative alternative to kalman filtering ignoring
noise and initial conditions,” IEEE Control Systems Magazine, vol. 37,
no. 5, pp. 70–89, 2017.
[32] S. Zhao, Y. S. Shmaliy, and F. Liu, “Fast kalman-like optimal unbiased
fir filtering with applications,” IEEE Transactions on Signal Processing,
vol. 64, no. 9, pp. 2284–2297, 2016.
[33] D. K. Ryu, C. J. Lee, S. K. Park, and M. T. Lim, “Unbiased fir filtering
with incomplete measurement information,” International Journal of
Control, Automation and Systems, vol. 18, no. 2, pp. 330–338, 2020.
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.40
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario Gonzalez-lee,
Jorge A. Ortega-Contreras, Karen Uribe-Murcia
E-ISSN: 2224-285
367
Volume 17, 2022