Pseudo-ground truth trajectory from contaminated data of object
tracking using smoothing algorithms.
ELI G. PALE-RAMON1, YURIY S. SHMALIY1, LUIS J. MORALES-MENDOZA2, MARIO
GONZÁLEZ-LEE2, JORGE A. ORTEGA-CONTRERAS1, RENE FABIÁN VÁZQUEZ-
BAUTISTA2.
1Department of Electronic Engineering
Universidad Veracruzana
Salamanca, Guanajuato, 36885
MEXICO
2 Electronics Engineerig Dept.
Universidad Veracruzana
Poza Rica, Veracruz, 93390
MEXICO
Abstract: - Object tracking is a study area of great interest to various researchers whose main objective is to
improve the trajectory estimation for object tracking. In practical applications, the information available that
allows the application of algorithms to improve the tracking process sometimes is missing. One of the main
obstacles is obtaining ground truth, which takes a long processing time. There are manual methods and
applications of reference algorithms. On the other hand, in most cases, the tracking information obtained using
a camera is contaminated with noise during the acquisition process. In this paper, we applied smoothing
algorithms to compute a pseudo-ground truth achieving lower estimation errors and higher precision than the
measurement data. The test results showed that the proposed algorithms with the highest performance are q-lag
UFIR and q-lag ML FIR. These smoothing algorithms can be useful in practical applications in object-tracking
tasks.
Key-Words: - bounding box, pseudo-ground truth, ground truth, object tracking, smoothing algorithms,
measurement noise, precision.
Received: May 25, 2022. Revised: July 21, 2023. Accepted: August 23, 2023. Published: October 2, 2023.
1 Introduction
Many times, the data about the trajectory of an
object tracking is obtained through a tracking
camera, these data are contaminated by noise in the
tracking process, the factors causing this noise can
be the movement of the camera, lighting, occlusion,
rapid changes of direction, blur, among others.
These factors cause the camera not to follow exactly
the trajectory of the object, there being variations
between the position measured by the camera and
the true position of the object.
When tracking algorithms are implemented, it is
necessary to know the ground truth trajectory to
correctly evaluate the effectiveness of the tracking
process. Estimator algorithms require the
application of a method to eliminate this noise and
compute a pseudo-ground truth that should be a
more accurate estimation of the ground truth.
If a video of the object tracking process is
available, it is possible to manually annotate each
position of the object. This implies a slow process
and human errors are possible as well when the
complete information about the video is unavailable,
such as frame rate and frame size.
Also, in the evaluation of object tracking
algorithms, it is necessary to contrast the estimates
obtained by the tracking algorithms against the
ground truth to evaluate their performance. With an
inadequate ground truth, we will have an evaluation
of the algorithms that is further from the truth.
In this sense, smoothing algorithms are a suitable
tool to remove noise from data. Therefore,
smoothing algorithms are useful for reconstructing
pseudo-ground truth, which can be used in the
estimate process for object tracking [1].
This article shows the application of smoothing
algorithms to reconstruct the ground truth and
derive a pseudo-ground truth that is accurate enough
to the ground truth. The results obtained using
smoothing algorithms prove that they are useful for
the estimation stage in object tracking.
Based on the test results, the smoothing
algorithms provide pseudo-ground truth with lower
estimation errors and higher precision than noise-
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DOI: 10.37394/232014.2023.19.8
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee,
Jorge A. Ortega-Contreras, Rene Fabián Vázquez-Bautista
E-ISSN: 2224-3488
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contaminated measurement data. Being q-lag UFIR
and q-lag ML FIR the algorithms that exhibit the
best results.
Therefore, the pseudo-ground truth
reconstruction through smoothing algorithms will
have a practical application for those researchers
who work in object tracking, where the ground truth
is unavailable or incorrect, i.e., the measured
tracking data is contaminated with noise.
2 Data of video object tracking
In the video object tracking process, image
processing operations seek to identify the target at
each position throughout the entire trajectory, which
implies recognizing the appropriate features to
differentiate the target from the background of the
scene. The target information can be described
through its properties. One of the most common
methods of containing target information during
object tracking is the bounding box [2].
The bounding box (BB) is a rectangular box that
contains the target object information in a sequence
of frames. The target position information is
included in a BB array. The measurements of each
BB represent the coordinates of the upper left and
lower right corners of the box that encloses the
target [3]. The BB matrix consists of the bounding
box measurements, " x" coordinate, " y " coordinate,
width (xw), and height (yh) for each of the object
positions throughout the trajectory.
According to the measurements of each BB, the
object centroid can be obtained in each position.
This information represents the trajectory followed
by the target object. As previously mentioned, this
information is measured by the tracking camera,
considered as contaminated by measurement noise.
It is necessary to evaluate the performance through
the information provided by the bounding boxes
when using smoothing to reconstruct the pseudo-
ground truth, that is, how accurate the pseudo-
ground truth is compared to the ground truth. The
most common method to evaluate the effectiveness
of smoothing is by estimation error and precision.
3 Ground truth
In the computer vision field in object tracking tasks,
the ground truth (GT) can be interpreted as the set
of true data, that is known to be real or true
positions of the object during the entire trajectory of
the tracking process. These measurements can be
represented through coordinates, bounding box
measurements, camera pose measurements, etc. The
ground truth information can be collected at the
source or can be pre-programmed.
On the other hand, pseudo-ground truth (p-GT)
can be interpreted as an estimation of the ground
truth, which is used as reference data for the
application of tracking algorithms and their
performance evaluation. This data set can be
obtained through hand annotation by a human
operator or using a reference algorithm.
Generally, we can establish that there are two
methods to obtain the ground truth. The first is
through manual annotation of the ground truth data
set and the second is a reference algorithm4].
Two of the most common annotation methods
are:
Bounding annotations. A box is drawn
based on the characteristics of the object
target.
Point annotations. The position of the object
target corresponds to the features extracted
from a single point.
4 Performance evaluation
We used standard metrics for evaluating the
smoothing performance can be done using metrics,
precision, and root mean square error (RMSE).
Precision can be defined as the percentage of the
number of correct predictions over the total number
of predictions [5]-[9].
The RMSE is a measure of the variation between
truth values and estimated values [10]. In the case of
object tracking, it measures the difference between
the truth trajectory and the estimated trajectory. The
equation of RMSE is well known and is shown
below.
(1)
Where is the number of data points, -th
measurement, is the truth value and is the
predicted value.
To calculate the precision, it is necessary to first
calculate another metric, intersection over union
(IoU), which indicates the percentage of overlap 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 computing the precision are
obtained from the comparison of the IoU result with
an established threshold [5]-[7], [11]. The equations
for calculating IoU and precision are (2) and (3),
respectively.
IoU= IA
(TBB- PBB)-IA
(2)
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DOI: 10.37394/232014.2023.19.8
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee,
Jorge A. Ortega-Contreras, Rene Fabián Vázquez-Bautista
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Precision
All detections
(3)
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 overlap 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. Usually, the IoU
threshold is set to 0.5 and 0.75 [12].
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 overlap
in object tracking compared to a given threshold is
shown below [4][6]:
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.
5 State-Space Model
According to the motion of a physical system in
space, the next position of the object can be
calculated using Newton’s equation of motion [13]
as shown below:
(4)
where is the object position, is the
object's initial position, is the object's initial
velocity, is the object's acceleration, and is the
time interval. The state equation is derived from
Newton’s equation of motion. So, we can construct
the dynamic model, the model is represented in
discrete-time state-space using the following state
and observation equations:
(5)
(6)
where is the state vector, is
the observation vector, is the colored
Gauss-Markov noise, and is the state
transition matrix, is the gain matrix
model, is the measurement matrix.
The zero mean Gaussian noise vectors
and have the
covariances and and the property
for all and .
We estimated the state of the 4 coordinates of the
bounding box. So, the state-space model is designed
for the 4 measurements of the bounding box: left
lower corner in x-axis (xc) left upper corner in y-axis
(yc), BB width (xw), and BB height (yh).
For a constant velocity model [14], the state
transition (A) is a block diagonal matrix with:
(7)
where τ is the sample time. This block is
repeated for the xc, yc, xw and yh to build the
complete matrix A.
The gain matrix model (B) and observation
matrix (C) are defined as shown below:
(8)
(9)
6 Smoothing algorithms
6.1 Fixed-lag Kalman smoother
With equations (4) and (5) the Kalman filter (KF)
estimates the state through observation of input and
output. The can estimate the state dynamics of
the system iteratively [15], [16], and 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
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DOI: 10.37394/232014.2023.19.8
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee,
Jorge A. Ortega-Contreras, Rene Fabián Vázquez-Bautista
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estimation error
, the posterior
estimation
, the a priori estimate error
covariance
, and the posterior
estimate error covariance
.
The prior state estimate is computed by (10), and
the prior error covariance matrix is estimated by
(11).
(10)
(11)
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 combination of the
prediction with the current observation is used to
calculate the optimal state estimate and is called the
posterior state estimate. The measurement is
corrupted by colored measurement noise . The
measurement residual is (12).
(12)
The residual covariance matrix is calculated as
follow:
(13)
The optimal gain for Kalman is given by:
(14)
A posteriori state estimate:
(15)
A posteriori matrix of error covariance:
(16)
For the fixed-lag Kalman smoother, first it is run
the standard Kalman filter and initialize the update
of fixed-lag smoother for ,
as follows:
(17)
(18)
(19)
(20)
6.2 q-lag ML FIR smoother
We used a batch q-lag maximum likelihood (ML)
Finite Impulse Response (FIR) smoother, q-lag ML
FIR, for full covariance matrices. The q-lag ML FIR
smoother can be derived from the ML estimate at
[20]. We use the (4) and (5) and extend them
on in the conventional forms shown below.
(21)
(22)
The state can be defined at for
as
(23)
Where a matrix
can be represented with
(24)
Rearranging the terms in equation (23), we
represent the initial state as
(25)
The q-lag ML FIR estimate and estimation error
in batch forms are calculated in the following.
Substituting equation (25) in (23), which separates
the regular terms and the random terms.
(26)
Where
and the
random term
(27)
The likelihood of can be written as
(28)
We determine the -lag ML FIR estimate
and
as
(29)
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Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee,
Jorge A. Ortega-Contreras, Rene Fabián Vázquez-Bautista
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Setting the derivative equal to zero
(30)
We derive the -lag ML FIR smoothing estimate
in the canonical maximum likelihood form.
(31)
This leads to the q-lag ML FIR smoother gain.
(32)
6.3 q-lag UFIR smoother
The q-lag Unbiased Finite Impulse Response
smoother, q-lag UFIR smoother, can be designed to
satisfy the unbiasedness condition.
(33)
where the -lag estimate can be defined as
(34)
The state model is represented by the
the row vector of the extended state equation (35)
on as
(35)
where
is the th row vector in
.
The batch forms of q-lag UFIR smoother are
given by the following equations. Applying the
condition (33) to (34) and
(35) gives two unbiasedness constraints, the UFIR
smoother gain
is given by.
(36)
Referring to
, then
we transform (36) to
(37)
Where
is the
UFIR filter gain and it gives the homogeneous
smoothed estimate.
(38)
Where is the UFIR filtering estimate. In this
work we use a system without input, , so the
smoothing estimate is obtained by simple projection
of (38) as described in [19], [20].
It is significant to mention that the q-lag UFIR
and q-lag ML FIR are of FIR type, i.e. FIR filtering
structures are bounded input bounded output
(BIBO) stable by design [19]. On the other hand, the
Fixed-lag Kalman is a Kalman filter structure, and it
is known to be asymptotically stable even when the
initial state is unknown [21].
7 Ground truth approximation tests
7.1 Numerical simulation tests
We conducted a computer simulation using the
moving object tracking model. In this case, the
simulation only represents one possible trajectory
followed by an object. The dynamic model
corresponds to a constant velocity. The moving
object model can be described by (4) and (5). A
discrete constant velocity model is simulated where
acceleration is a zero-mean Gaussian white noise
process. The dynamics of simulated movement
correspond to a trajectory in the x and y plane with
the following matrices.
In addition to computing the actual trajectory of
the object tracking, the ground truth, we created a
trajectory that simulates the tracking data by a
camera in the presence of noise that affects the
tracking measurement. We called this trajectory the
measurement data and we used it as input data for
the smoothing algorithms. The purpose of this was
to prove that the pseudo-ground truth obtained by
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the smoothing algorithms is suitable in cases where
the ground truth is unknown.
The numerical stability of the simulation was
verified on the basis that a numerical method is
stable if small changes in the initial data cause small
effects in the final numerical solution. On the other
hand, it is numerically unstable if it produces large
errors in the final solution [22]-[24]. We used the
measurement data, where we perturbed the initial
state x0, and it was compared to the RMSE obtained
between the estimate calculated by q-lag UFIR and
the actual trajectory with the original initial state x0.
The results are shown in Table 1, which
corroborates that small disturbances to the initial
state generate small changes in the solution obtained
by the UFIR q-lag algorithm. So, we can establish
that the simulation method used is stable.
Table 1 Simulation stability.
Initial state x0
RMSE
difference
0.1
0.0103
0.3
0.0600
0.5
0.0020
0.7
0.0111
0.9
0.0469
For the first simulation we consider that an
object target is disturbed by white Gaussian
acceleration noise with a standard deviation of
. The data noise originates from white
Gaussian with . The simulation of the
trajectory was 500 points with sample time
seconds,
.
The RMSE results assessed from smoothing
algorithms, fixed lag Kalman, q-lag UFIR, and q-lag
ML FIR, and measurement data are presented in
Table 2. For the computed RMSE with the q-lag
UFIR and q-lag ML FIR, the was 4. The
results show that q-lag UFIR smoother presented a
higher performance since the value is lower than the
other algorithms, followed by ML FIR, which was
only surpassed by a small value of . Fixed-lag
Kalman presents a higher RMSE value compared to
the other smoothing algorithms. However, it reduces
the estimation error of the measurement data.
Fig 1. presents the trajectories reconstructed
through the smoothing algorithms, as well as the
measurement data. It describes the resulting
smoothing of measurement data as observed,
reducing the noise and calculating a pseudo-ground
truth that closely follows the ground truth. Results
of the q-lag UFIR and the q-lag ML FIR smoothers
are similar, as mentioned above UFIR is slightly
better than ML FIR, which is consistent with the
RMSE results.
Table 2 RMSE results of simulated data 1.
Data
RMSE results of
algorithms
RMSE
Value
Performance
ranking
Measurement data
3.0353
4
Fixed-lag Kalman
2.6938
3
q-lag UFIR
2.1054
1
q-lag ML FIR
2.1054
2
Fig.2 presents separately ground truth,
measurement data, and the smoothed estimates to
clarify the smoothing algorithms' performance. The
tracking measurement data, which represents the
measurements obtained by a tracking camera under
noise conditions, presents large estimation errors.
Fixed-lag Kalman smooths the estimates, which is
similar to the ground truth; However, the behaviour
pattern differs. On the other hand, q-lag UFIR and
q-lag ML FIR perform higher at smoothing the
estimates, computing a pseudo-ground truth close to
the ground truth.
To corroborate the effect of smooth we
performed another simulation test. For this test, the
moving object model is the same as the example
above with the same matrices. The model was
developed with a standard deviation of
, and the data noise with . The
simulated trajectory with 500 points with a sample
time seconds.
Fig. 1 Smoothing and measurement trajectories of
simulated data 1.
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Eli G. Pale-Ramon, Yuriy S. Shmaliy,
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Fig. 2 Smoothing and measurement trajectories
(separated) of simulated data 1.
The RMSE results obtained by smoothing
algorithms, fixed-lag Kalman, q-lag UFIR, and q-
lag ML FIR, and measurement data are shown in
Table 3. For the computed RMSE with the q-lag
UFIR and q-lag ML FIR, the was 6. In the
same way, as in the previous simulation test, q-lag
UFIR smoother presented a higher performance
with a lower value than the other algorithms,
followed by ML FIR, which was only surpassed by
a small value of . The UFIR and ML FIR
obtain an estimation error equivalent to half that
generated by measurement data. In this case fixed-
lag Kalman with a higher RMSE value compared to
the other smoothing algorithms. However, it reduces
the estimation error of the measurement data.
Table 3 RMSE results of simulated data 2.
Data
RMSE results of
algorithms
RMSE
Value
Performance
ranking
Measurement data
9.7477
4
Fixed-lag Kalman
7.1000
3
q-lag UFIR
4.8168
1
q-lag ML FIR
4.8168
2
Fig 3 shows the trajectories reconstructed using
the smoothing algorithms and the measurement
data. As in the previous test, the measurement data
represents data measured by a tracking camera,
which explain the observed high estimation errors.
The smoothing algorithms reduce estimation errors
providing a pseudo-ground truth more similar to the
ground truth. Results of the q-lag UFIR smoother
and the q-lag ML FIR smoother are similar, as
mentioned above UFIR slightly performs higher
than ML FIR. In this case, fixed-lag Kalman shows
poor performance, reducing the estimation errors to
a lesser extent.
For a broader visualization of the smoothing
algorithms' performance, in Fig. 4 ground truth,
measurement data, and the smoothed estimates are
shown separately. Fixed-lag Kalman smoothing with
poor performance, although pseudo-ground truth has
similarities with ground truth, the pattern of
behaviour is different. While q-lag UFIR and q-lag
ML FIR perform better in obtaining a pseudo-ground
truth whose behaviour is more similar to the GT.
Fig. 3 Smoothing and measurement trajectories of
simulated data 2.
Fig. 4 Smoothing and measurement trajectories
(separated) of simulated data 2.
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According to the results of the simulation data,
the performance of the q-lag UFIR and q-lag ML
FIR show great capacity to reduce disturbances,
providing a smoothed pseudo-ground truth with
fewer estimation errors. This gives another proof
that these smoothing algorithms are suitable for
obtaining a pseudo-ground truth when the ground
truth is unavailable.
7.2 Results of experimental test
The smoothing algorithms have shown a high
performance capable of reducing estimation errors
by close to . Therefore, we decided to test these
algorithms with true tracking data. For this purpose,
we use the data called "Remotecar" available in
[25]. In this case, we used the bounding box data to
estimate and assess the performance of the
smoothing algorithms.
The experimental test was performed using the
moving object tracking model and matrices
proposed in section . In the object tracking model,
we considered that the car target is disturbed by
white Gaussian acceleration noise with a standard
deviation of . The data noise
originates from white Gaussian . The
sample time seconds,
, on a short horizon . The
model of a moving target is completed according to
what is established in section .
The estimated smoothing trajectories and
measurement data are shown in Fig. 5. In this case,
a more complex trajectory with greater variation
between states is observed. As in the tests with
simulated data, the q-lag UFIR and q-lag ML FIR
present a higher performance with similar results, an
overview of these results indicates that these
algorithms manage to reduce the noise present in the
measurement to a greater degree. Likewise, fixed-
lag Kalman performed with lower performance, but
managed to reduce the noise of the measurement
data.
In Fig. 6 ground truth, measurement data, and the
smoothed estimates are shown separately. It can be
observed that the measurement data contains a high
variation concerning to the ground truth,
representing the measurements obtained by a
tracking camera in noise conditions. Fixed-lag
Kalman had a lower performance. While q-lag
UFIR and q-lag ML FIR smooth the estimates in a
better way. The performance of the smoothing
algorithms will be better analyzed by the precision
metric.
We have the complete information on the object
tracking available for assessing the precision,
therefore we have the measurements of the
bounding box at each point of the trajectory so we
can evaluate the performance of the smoothing
algorithms.
The precision values of each of the smoothing
filters in the entire intersection over union (IoU)
threshold range are shown in Fig.7. The q-lag UFIR
and q-lag ML FIR smoothers presented the best
performance.
Setting the IoU threshold equal to 0.5, the
precision of all smoothing algorithms is close to
, obtaining a higher precision than the
measurement data which is below . The
average precision over the full range of the IoU
threshold of the smoothing algorithms and the
measurement data are shown in Table 4. With these
Fig. 5 Smoothing and measurement trajectories of
Remotecar
Fig. 6 Smoothing and measurement trajectories
(separated) of Remotecar
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results it is confirmed that q-lag UFIR and q-lag ML
FIR generate a pseudo-ground truth reducing the
noise of the measurement data, providing adequate
information to use as a reference in the development
of object tracking tasks when the ground truth does
not is available.
Table 4 Precision results of Remotecar
Data
Precision results of
algorithms
Precision
Performance
ranking
Measurement data
27%
4
Fixed-lag Kalman
31%
3
q-lag UFIR
37%
1
q-lag ML FIR
37%
2
4 Conclusion
Smoothing algorithms for deriving the ground truth
from the measurement data proved to be able to
reduce noise, producing pseudo-ground truth with
less estimation error than the measured data.
With both simulated data and truth object
tracking data, the q-lag UFIR and q-lag ML FIR
algorithms exhibited the highest performance, being
able to provide a pseudo-ground truth with higher
precision and lower estimation error.
Since the q-lag UFIR and q-lag ML FIR
algorithms are more robust against measurement
data under noise, they provide a reliable pseudo-
ground truth for use as a reference in object-tracking
research in the field of computer vision. Being
practical and robust methods against the lack of
ground truth information, data noise, and when
complete information on video object tracking is
unavailable, they can be useful for state estimation
applied with different artificial intelligence
methodologies, neural networks, and machine
learning, among others, to improve the tracking
object process.
Due to the higher robustness of the q-lag UFIR
and q-lag ML FIR smoothing algorithms, we are
currently working on efficient algorithms that use
smoothing and state estimator algorithms for object
tracking and plan to report the results in the near
future.
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Contribution of Individual Authors to the
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research, at all stages from the formulation of the
problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
The funding was received from Universidad
Veracruzana for conducting this study.
Conflict of Interest
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WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.8
Eli G. Pale-Ramon, Yuriy S. Shmaliy,
Luis J. Morales-Mendoza, Mario González-Lee,
Jorge A. Ortega-Contreras, Rene Fabián Vázquez-Bautista
E-ISSN: 2224-3488
76
Volume 19, 2023
