Vehicle trajectory reconstruction for intersections: an integrated wavelet transform and Savitzky-Golay filter approach

Abstract

Reconstructing vehicle trajectories is an essential step in obtaining high-precision vehicle trajectory data owing to observation errors. Although considerable efforts have been made to reconstruct trajectories on road segments, it is still challenging to reconstruct two-dimensional vehicle trajectories at intersections. In this study, a three-step vehicle trajectory reconstruction method for intersections was proposed to correct abnormal velocities and direction angles in raw trajectories. First, outliers were identified by wavelet transform. They were then filtered out and interpolated using Gaussian kernel-based locally weighted linear regression. Finally, the trajectories were smoothed using a Savitzky-Golay filter. The proposed method was validated using empirical trajectory data collected at 15 intersections in Shanghai, China. The results demonstrate that the proposed trajectory reconstruction method performs well in terms of trajectory rationality and consistency. The mean outlier rate and root mean square error of the reconstructed trajectories are 3% and 0.821 m, respectively.

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Vehicle trajectory reconstruction for intersections:
an integrated wavelet transform and Savitzky-
Golay filter approach
Jing Zhao, Xiaoliang Yang & Cheng Zhang
To cite this article: Jing Zhao, Xiaoliang Yang & Cheng Zhang (2023): Vehicle trajectory
reconstruction for intersections: an integrated wavelet transform and Savitzky-Golay filter approach,
Transportmetrica A: Transport Science, DOI: 10.1080/23249935.2022.2163207
To link to this article: https://doi.org/10.1080/23249935.2022.2163207
Published online: 04 Jan 2023.
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TRANSPORTMETRICA A: TRANSPORT SCIENCE
https://doi.org/10.1080/23249935.2022.2163207
Vehicle trajectory reconstruction for intersections: an
integrated wavelet transform and Savitzky-Golay filter
approach
Jing Zhao , Xiaoliang Yang and Cheng Zhang
Department of Traffic Engineering, University of Shanghai for Science and Technology, Shanghai, People’s
Republic of China
ABSTRACT
Reconstructing vehicle trajectories is an essential step in obtaining
high-precision vehicle trajectory data owing to observation errors.
Although considerable efforts have been made to reconstruct tra-
jectories on road segments, it is still challenging to reconstruct
two-dimensional vehicle trajectories at intersections. In this study, a
three-step vehicle trajectory reconstruction method for intersections
was proposed to correct abnormal velocities and direction angles in
raw trajectories. First, outliers were identified by wavelet transform.
They were then filtered out and interpolated using Gaussian kernel-
based locally weighted linear regression. Finally, the trajectories were
smoothed using a Savitzky-Golay filter. The proposed method was
validated using empirical trajectory data collected at 15 intersections
in Shanghai, China. The results demonstrate that the proposed tra-
jectory reconstruction method performs well in terms of trajectory
rationality and consistency. The mean outlier rate and root mean
square error of the reconstructed trajectories are 3% and 0.821 m,
respectively.
ARTICLE HISTORY
Received 3 July 2022
Accepted 22 December 2022
KEYWORDS
Two-dimensional trajectory
reconstruction; wavelet
transform; locally weighted
regression; Savitzky-Golay
filter
1. Introduction
Vehicle trajectories with high spatial and temporal resolutions provide rich data for under-
standing driving behaviour and developing microscopic traffic-flow models. Accurate vehi-
cle trajectories can help reproduce traffic flow evolution and analyse the spatio-temporal
characteristics of urban traffic. It has wide application prospects in traffic accident detection
(Wang, Xu, and Dai 2019;Yangetal.2021;NohandYeo2022), signal control optimisation
(Ma, An, and Lo 2016;Wuetal.2019), traffic state estimation (Chen et al. 2019; Ma and Qian
2020;Tangetal.2020), driving behaviour modelling (Park, Kim, and Yeo 2020; Pan et al.
2021; Zhang et al. 2022), and autonomous vehicle control (Yu et al. 2018; Liu and Fan 2021;
Wu et al. 2022).
There are several methods for obtaining vehicle trajectory data, such as an unmanned
aerial vehicle (UAV) video extraction (Xie et al. 2019; Shi et al. 2021), vehicle-mounted
radar, and global positioning systems (Punzo and Simonelli 2005;Fuetal.2020). Among
CONTACT Cheng Zhang chengzhang@usst.edu.cn
© 2023 Hong Kong Society for Transportation Studies Limited

2J. ZHAO ET AL.
Figure 1. The distribution of the kinetic properties of the trajectory data.
them, with the development of UAV technology and computer technology in vehicle detec-
tion and tracking, UAV traffic data collection technology equipped with a high-resolution
camera has become a popular means of traffic monitoring and data collection due to its
advantages of good flexibility and low cost (Zhang, Jordan, and Livshits 2016;Krajewski
et al. 2018; Barmpounakis and Geroliminis 2020). The locations of all vehicles in each frame
were extracted from the UAV video data, and continuous locations with timestamps from
vehicle trajectories.
However, extracting high-fidelity vehicle trajectories from UAV video is a challenging
task. The UAV video-based vehicle trajectory data usually contain several sources of errors:
(i) instable UAV video camera because of strong wind; (ii) sub-pixel movement of the vehi-
cles (the vehicle movements within the same pixel cannot be detected due to limited
resolutions of videos); and (iii) the jump of the bounding box during vehicle detection and
tracking. Since the time step is small (1/24 s), such error sources cause large errors in calcu-
lating acceleration and curvature with the use of raw vehicle trajectories. As illustrated in
Figure 1, acceleration and curvature distribution demonstrate a large and unrealistic vari-
ance (e.g. the maximum acceleration reaches 300 m/s2). It is thus apparent that the raw
trajectory data from UAV videos contain large errors. The direct use of the raw trajectory
data for deriving the properties of vehicle kinematics and dynamics (e.g. velocity, accelera-
tion, direction angle, and curvature) would lead to unrealistic results and misinterpretation
of traffic dynamics.
Therefore, vehicle trajectory reconstruction in microscopic levels has become a critical
research topic during the last decade. For example, Thiemann, Treiber, and Kesting (2008)
proposed a smoothing kernel based algorithm to deal with the unrealistic situation that
a driver switches between hard acceleration and hard deceleration several times within
one second. Montanino and Punzo (2015) damped out the random noise in the trajectory
by means of traditional low-pass digital filter. Similar efforts to improve trajectory quality
have been made by Zhou, Qu, and Li (2017), Saifuzzaman et al. (2017), Ahn, Vadlamani, and
Laval (2013), Dong et al. (2021), (van Beinum et al. 2018), and Krajewski et al. (2018). Most

TRANSPORTMETRICA A: TRANSPORT SCIENCE 3
of these studies use smoothing methods, including filtering and averaging, as data denois-
ing techniques. Although not the main focus of their studies, different strategies have been
proposed for noise reduction in these papers.
However, the aforementioned studies mainly focused on vehicle following, merging,
overtaking, or lane changing on freeways. Very few studies reconstructed vehicle trajec-
tory data inside intersections. Recently, Venthuruthiyil and Chunchu (2020) proposed a
three-stage trajectory reconstruction framework based on a recursion ensembled low-pass
filter to process the noise in the two-dimensional trajectory of experimental vehicles. An
extended variational solution network was established and Kalman Filtering was applied to
capture and reproduce the stochastic properties of queue boundary curves for signalised
intersections (Chen et al. 2021). Nonetheless, they did not explicitly consider the reconstruc-
tion of two-dimensional direction angles, which are critical to driving behaviours of turning
at intersections.
To address this challenge, a three-step trajectory reconstruction method is proposed
in this study to reduce the noise of vehicle velocity and direction in a two-dimensional
intersection plane. The proposed three-step method includes outlier recognition, data
interpolation, and local smoothing to reconstruct two-dimensional vehicle trajectories at
intersections. Wavelet transforms (WT) and S-G filters are integrated to reduce the noise of
the velocity and direction angle in the raw trajectories. It is worth noting that the proposed
method selects different wavelet bases for each vehicle according to its characteristics
regarding velocity and direction angle, instead of using the same wavelet base as existing
methods did. In addition, the proposed method assigns different smoothing parameters for
acceleration and curvature according to the trajectory rationality of physical kinematics.
The remainder of this paper is organised as follows. Section 2 describes the data sources
used in the study. Section 3introduces the proposed method for reconstructing the vehicle
trajectories at intersections. The effectiveness and reliability of the proposed method are
evaluated in Section 4. The paper is concluded in Section 5.
2. Data source
Raw video data were collected by UAVs at 15 intersections in Shanghai, China, as shown
in Figure 2. The UAV videos had a resolution of 3840 ×2160 pixels and good visibility.
The frame rate of the video was 24 fps. Videos were collected during morning peak hours
(7:00–9:00) and off-peak hours (15:00–17:00) on weekdays in September 2019.
2140 vehicle trajectories in total were extracted from raw video data using a well-
developed video recognition software, in which each vehicle at each frame can be detected
by a bounding box. The centroid of the bounding box is used to represent the detected
vehicle, so that a vehicle is reduced to a pixel and the corresponding X,Y – coordinates rep-
resent the vehicle location (Zhang et al. 2022). As shown in Table 1, each vehicle trajectory
data record contains the following information: timestamp, vehicle ID, X coordinate, and Y
coordinate.
3. Methodology
An example of the motion dynamics (velocity, acceleration, and direction angle) of a raw
vehicle trajectory is shown in Figure 3. It was found that there were unreasonable velocities

4J. ZHAO ET AL.
Figure 2. Real-world video images captured by UAVs.
Tab le 1. Extracted information from video.
Attribute Unit Note
Timestamp second 24 frames per second (1/24 s for each time frame)
Vehicle ID / A random ID assigned to the vehicle
X-coordinate pixel The X-coordinate of the centroid of the vehicle
Y-coordinate pixel The Y-coordinate of the centroid of the vehicle
and accelerations in the trajectory data, as well as abnormalities in the direction angle. For
example, the raw velocity suddenly increases and then decreases, as illustrated by the red
circles in Figure 3(b). The raw direction angle and acceleration also exhibited significant
fluctuations and abnormal ranges.
To improve the quality of trajectory data, in this study, we propose a three-step trajectory
reconstruction framework in the two-dimensional plane of intersections that are influenced
by both the velocity magnitude and direction, as shown in Figure 4. The steps are described
in the following sub-sections.

TRANSPORTMETRICA A: TRANSPORT SCIENCE 5
Figure 3. Example of raw vehicle trajectory.
3.1. Step1: identication of outliers using dierent wavelet transforms
In Step 1, the WT is used to identify abnormal data that does not conform to the physical lim-
its of the trajectory data. Considering that the driving characteristics of the raw trajectory
will be lost if filtering or smoothing techniques are directly applied to the raw trajectory,
this step can ensure that the subsequent filtering steps are not influenced by outliers. In
recent years, much research has been conducted on local identification and correction
mechanisms for noise. Generally, low-pass filters based on Fourier transform are applied
to eliminate high – and medium-frequency noise. However, the Fourier transform only pro-
vides information related to the frequency of the trajectory and not the position properties.
In other words, although these methods provide accessibility to frequency information,
they do not control the temporal propagation of the signal (Fard, Mohaymany, and Shahri
2017). This may lead to inefficient noise filtering and unsuitable vehicle trajectory recon-
structions. To overcome this limitation, the WT was used in this study to identify trajectory
outliers.
3.1.1. Construction of wavelet transform
WT enables the multiscale decomposition of signals (in this study, velocity or direction
angle profiles) (Fard, Mohaymany, and Shahri 2017). Such a multi-scale feature of the WT

6J. ZHAO ET AL.
Figure 4. Flowchart of trajectory reconstruction method.

TRANSPORTMETRICA A: TRANSPORT SCIENCE 7
allows the decomposition of a signal into multiple scales, with each scale representing a
particular coarseness of the signal under study. At any level, the signal was decomposed
into approximation and detailed components. The approximation component is further
decomposed into approximation and detail components and this process goes on to level j.
Trajectory longitudinal velocity v(t)and direction angle θ(t)can be described by Equations
(1) and (2), respectively.
v(t)=vAj(t)+
j∈J
vDj(t);(1)
θ(t)=θAj(t)+
j∈J
θDj(t);(2)
where v(t)represents the longitudinal velocity at time t (in m/s); Jis the number of levels of
discrete wavelet; vAj(t)and vDj (t)are the approximation and detail components at level j,
which can be determined by Equations (3) and (4), respectively; θ(t)represents the direction
angle at time t(in rad); θAj(t)andθDj (t)are the approximation and detail components at level
j, which can be determined by Equations (5) and (6), respectively.
vAj(t)=
k∈Z
cvjkϕvjk (t)(3)
vDj(t)=
k∈Z
dvjkψvjk (t)(4)
θAj(t)=
k∈Z
cθjkϕθjk (t)(5)
θDj(t)=
k∈Z
dθjkψθjk (t)(6)
where cvjk denotes the wavelet coefficient of the approximation component, which is
obtained through the dot product of signal v(t)and scale function ϕvjk at j-th level with
translation equal to k, as shown in Equation (7); dvjk is the wavelet coefficient of the detail
component, which is the dot product of signal v(t)and wavelet function ψvjk at j-th level
with translation equal to k, as shown in Equation (8); cθjk denotes the wavelet coefficient
of the approximation component, which is obtained through the dot product of signal
θ(t)and scale function ϕθjk at j-th level with translation equal to k, as shown in Equation (9);
dθjk is the wavelet coefficient of the detail component, which is the dot product of signal
θ(t)and wavelet function ψθjk at j-th level with translation equal to k, as shown in Equation
(10).
cvjk =(v(t),ϕvjk(t)) (7)
dvjk =(v(t),ψvjk(t)) (8)
cθjk =(θ(t),ϕθjk(t)) (9)
dθjk =(θ(t),ψθjk(t)) (10)
To determine the exact location of the wavelet coefficients with highly concentrated sig-
nal energy, the wavelet coefficient of the detail component was compared with a threshold

8J. ZHAO ET AL.
Sz
j. In this study, Sz
jis defined according to Equation (11).
Sz
j=μj±z×σj(11)
where μjand σjare the mean and standard deviation, respectively, of the wavelet coeffi-
cients at level j;zvalue is 1.96 for 95% of the level of confidence (Fard, Mohaymany, and
Shahri 2017).
3.1.2. Selection of wavelet base
The effect of the wavelet transform depends mainly on the choice of wavelet base. There-
fore, for the studied vehicle trajectory signal, an appropriate wavelet transform should
effectively filter out abnormal data from the trajectory and reconstruct trajectory data with
good velocity and acceleration indicators. Since the error patterns for each vehicle in the
longitudinal and lateral directions are different (Venthuruthiyil and Chunchu 2020), the pro-
posed method selects different wavelet bases for each vehicle. From the perspective of
wavelet characteristics, the commonly used methods for selecting an appropriate wavelet
base include maximum energy and minimum entropy criteria, similarity-based criteria, and
their combination (Mallat 1989).
The general characteristics of the signal wave and the physical meaning of the vehicle
trajectory were combined to select the wavelet base. For wavelet base selection for velocity,
we used “the ratio of energy to Shannon entropy and acceleration range” as the criteria. For
wavelet base selection for direction angle, we used “the ratio of energy to Shannon entropy
and curvature range” as the criteria.
The higher the ratio of energy to Shannon entropy the better, which can be calculated
using Equation (12) (Mallat 1989).
radio(j)=E(j)
Sentropy(j)(12)
where radio(j)represents the ratio of energy to Shannon entropy at the j-th level decom-
position; E(j)represents the energy at the j-th level decomposition, which is determined by
Equation (13); Sentropy(j)represents Shannon entropy at the j-th level decomposition, which
is determined by Equation (14).
E(j)=
K

k=1
|wt(j,k)|2(13)
where, wt(j,k)represents the k-th wavelet coefficient of level j;Kis the total number of
wavelet coefficients.
Sentropy(j)=−
K

k=1
pklog2(pk)(14)
where pkrepresents energy distribution of wavelet coefficient, defined as pk=|wt(j,k)|2
E(j).
For the acceleration range, it has been reported that vehicle accelerations are in the
rangeof–5to5m/s
2and curvature is used as a measure of steering stability and safety
(Bichiou and Rakha 2018).

TRANSPORTMETRICA A: TRANSPORT SCIENCE 9
Tab le 2. The wavelet bases selection for vehicle No. 1105 to identify
outliers (a) velocities and (b) direction angles.
(a)
Wavelet Ratio of energy to Shannon entropy Acceleration range (m/s2)
Haar 13.69 [−4.87, 3.87]
Db2 12.72 [−5.13, 4.11]
Db4 11.74 [−11.02, 36.98]
Db6 11.19 [−51.02, 68.22]
Sym2 12.71 [−6.52, 6.45]
Sym4 12.64 [−7.63, 7.95]
Sym6 12.46 [−7.56, 8.34]
Coif1 12.33 [−7.33, 10.91]
Coif3 12.02 [−8.21, 11.96]
Coif5 12.00 [−14.30, 20.96]
Bior1.3 11.86 [−50.22, 124.68]
Bior2.8 11.43 [−33.09, 50.53]
Bior3.1 11.64 [−25.21, 28.28]
(b)
Wavelet Ratio of energy to Shannon entropy Curvature range (m−1)
Haar 364.17 [−1.07, 1.08]
Db2 333.90 [−0.99, 1.60]
Db4 354.31 [−0.11, 0.58]
Db6 430.14 [−0.08, 0.21]
Sym2 333.90 [−0.99, 1.60]
Sym4 365.77 [−0.09, 0.56]
Sym6 359.20 [−0.08, 0.51]
Coif1 360.86 [−0.24, 0.66]
Coif3 483.76 [−0.08, 0.23]
Coif5 441.12 [−0.17, 0.33]
Bior1.3 343.98 [−1.06, 9.51]
Bior2.8 535.66 [−0.07, 0.19]
Bior3.1 474.94 [−0.88, 1.44]
The curvature of the vehicle path is bounded by the minimum turning radius. Accord-
ing to the road design code (MOHURD 2012), the minimum turning radius is 5 m, and the
corresponding curvature range standard ranges from – 0.2 to 0.2 m−1.
The following five wavelet bases were considered: Haar, Daubechies (db), Symlets (Sym),
Coiflets (Coif), and Biorthogonal (Bior) wavelets. The velocity and direction angle curves
were reconstructed after first-level decomposition and zeroing of the high-frequency com-
ponents. The results of vehicle No.1105, including the ratio of energy to Shannon entropy
and the acceleration/curvature range, are shown in Table 2.
As shown in Table 2(a), the Haar wavelet was selected as the velocity wavelet bases for
vehicle No.1105, that is, the maximum ratio of energy to Shannon entropy and the most
appropriate acceleration range. As shown in Table 2(b), the best results were obtained by
applying the Bior2.8 wavelet, that is, the maximum ratio of energy to Shannon entropy and
the minimum range of curvature.
The selection of velocity and direction angle wavelet bases of all vehicles are shown
in Figure 5. Different vehicle trajectories select different wavelet bases to obtain optimal
identification of outliers in velocity and direction angle curves.

10 J. ZHAO ET AL.
Figure 5. Distributions of wavelet bases selections for all 2140 vehicles.
3.1.3. Results of identification
Figure 6shows the velocity and direction angle of a trajectory processed by the WT. Outliers
in the velocity and direction angle signals can be identified, as shown in Figure 6(a) and 6(b)
based on the WT analysis of the signal detail coefficients in Figure 6(c) and 6(d).
3.2. Step2: data interpolation using locally weighted regression
In Step 2, a Gaussian kernel-based locally weighted regression (LWR) was used to interpo-
late the removed outliers. After identifying outliers through WT, the missing values of the
trajectory data after removing the outliers must be interpolated to facilitate subsequent fil-
tering and smoothing. In this study, the fitted values of the missing points were obtained
using LWR fitting, which is a non-parametric regression technique that fits simple regres-
sion models to local velocities or direction angles. Samples near the predicted sample were
assigned more weights for regression fitting. This method performs better in fitting more
complex and variable data than the traditional linear and polynomial regressions.

TRANSPORTMETRICA A: TRANSPORT SCIENCE 11
Figure 6. Example of outliers identification results using WT.
3.2.1. Locally weighted regression
The general form of LWR is defined in Equation (15). The LWR model must tune two param-
eters: the neighbourhood weights and the number of polynomial functions (Venthuruthiyil
and Chunchu 2018). The weights were generated from the kernel function shown in Section
3.2.2.
f(x0,ψ) =
k

i=1
(ψixi)d(15)
where kis the number of nearest neighbours; ψis the weight vector; ψiis the weight
assigned to each nearest instance which can be obtained from the kernel function K(xi,x0);
xiis the velocity or direction angle of the i-th locus point; x0is the velocity or direction angle
to be fitted; dis the degree of the polynomial, which influences the bias-variance trade-off.
The lower degree polynomials do not always fit the data well. On the other hand, the higher
degree polynomials follow the data more closely but often fit poorly at the extremes points.
During training, the weights of the training examples were updated (ψ∗) using Equation
(16).
ψ∗=
n

i=1
L(yi,f(x0,ψ))K(xi,x0)(16)
where L(yi,f(x0,ψ)) =(yi−xT
iψ)2is the square loss function; yiis the actual output of the
velocity of the direction angle value of the i-th locus point; K(xi,x0)is the kernel function.

12 J. ZHAO ET AL.
Figure 7. Example of data interpolation results using locally weighted regression.
3.2.2. Selection of kernel
As described in the previous section, LWR methods require kernel smoothing functions to
generate the most appropriate weights for neighbouring points of the target data point.
The most used kernel is the Gaussian kernel (radial basis function). Its basic form is given by
Equation (17).
K(xi,x0)=exp −||xi−x0||2
2h2(17)
whereKis a diagonal weight matrix and the value of K(xi,x0)represents the weight of the
i-th velocity or direction angle when fitting the predicted velocity or direction angle; his
the bandwidth of the Gaussian kernel. By adjusting the value of h, the function can be
adjusted to make it more suitable for this method. It is used to control the “local range”
of local weighting. Overfitting will occur if his too small.
According to the study of Alqasrawi, Azzeh, and Elsheikh (2022), LWR polynomial degree
d=3 (in Equation 15) and kernel bandwidth h=0.2 (in Equation 17) are suggested for
most kernels. This set of recommended values is adopted in this study to interpolate using
locally weighted regression.
3.2.3. Results of interpolation
Replace the outliers identified in the example trajectory of Step 1 with interpolation values
obtained by the Gaussian kernel weighted regression of the vehicle trajectory, as shown in
Figure 7.
3.3. Step3: smoothing using Savitzky-Golay lter
In Step 3, the S-G filter for local smoothing was used to remove noise from the trajectory
data, and the velocity and direction angle change curves were processed simultaneously
to ensure that the trajectory was spatially consistent. After filling in the missing values, out-
liers in the trajectory data were removed and interpolated with reasonable values. However,

TRANSPORTMETRICA A: TRANSPORT SCIENCE 13
Figure 8. Example of smoothing results using S-G filter.
there were still some significant unreasonable fluctuations in the vehicle trajectories. There-
fore, in this step, we used an S-G filter to smooth the trajectory data to remove some of the
small-amplitude noise.
Filtering is a data processing technique used to remove noise and restore real data. The
S-G filter was proposed by Savitzky and Golay (1964), and is based on the least square fitting
of the local polynomial in the time domain. The advantage of the S-G filter is that it can keep
the shape and width of the signal unchanged while filtering out the noise. The S-G filtering
is shown in Equation (18).
Yj=i=m
i=−mCiYj+1
N(18)
where Yjis the velocity or direction angle fitting value; Yj+1is the raw velocity or direction
angle value; Ciis the i-th coefficient of the filter, which can be calculated by a polynomial
with order o;mis the half-filter length; Nis the filter length, which is equal to (2m+1).
Two parameters must be determined when the filter is applied to velocity or direction
angle sequence smoothing. The first is the order oof the smoothing polynomial. Smaller
values of oproduce smoother results but may introduce biases. A higher value of oreduces
the bias of the filter, but may overfit the data and produce greater noise. The second is half-
filter length m.Largervaluesofmproduce smoother results but reduce the peak flatness.
The values of oand mcan be determined by prior experiences or trial-and-error methods.
Rahman, Rashid, and Ahmad (2019) suggested order o=3 and half-filter length m=10
(i.e. filter length N=2m+1=21) are the best combination for the SG filter. Here, we use
the recommended values of oand m. Then, taking the sample trajectory used in Step 2,
comparisons of the velocity and direction angle before and after smoothing are shown in
Figure 8.
4. Model validation
We ran the reconstruction programme on an Intel Core i5-10600 CPU processor and
recorded the time required to execute it. It took 403 s to complete the entire reconstruction
process for 2140 trajectories. The reconstruction time for each trajectory was 183 ms, and
the reconstruction time per data point was less than 0.5 ms.

14 J. ZHAO ET AL.
The performance of the proposed method is validated in this section based on the
empirical data collected by UAVs. First, we analyse the reconstructed trajectory of a vehi-
cle to illustrate in detail how the proposed method works. Subsequently, all reconstructed
trajectories were analysed to validate the rationality of the proposed method. Finally,
the reconstructed trajectories were compared with the raw trajectories to validate their
consistency.
4.1. Detailed analysis of one vehicle trajectory
Taking one vehicle as an example, the reconstructed vehicle trajectories after applying each
step were analysed first to validate the availability of the method. Subsequently, the ratio-
nality of the reconstructed trajectory and the consistency between the reconstructed and
raw trajectories for the example vehicle are discussed.
4.1.1. Availability validation
Taking one vehicle as an example, the raw and reconstructed trajectories after each step
are shown in Figure 9. The results of each step are presented in each column.
In Figure 9, significant fluctuations can be found in the raw trajectory (More details are
shown in Figure 10). After the wavelet transform of the velocity and direction angle simul-
taneously, the outliers identified by the detail coefficient are shown in the subfigures in the
second and third rows of the first column in Figure 9, respectively. After the interpolation of
the recognised velocity and direction outliers in Step 2, the results are shown in the second
and third row of the second column in Figure 9. The interpolation values are represented
by red and yellow hexagons. In Step 3, the S-G filter was applied to eliminate the residual
high-frequency noise in the data, which was expressed in the velocity and direction angle
curves, as shown in the last column.
The reconstructed velocity and direction angle no longer exhibit rapid oscillations, while
the overall structure of the raw data is preserved. Such processing solves most of the high
velocities and curvatures that are not within the range of human operation. Therefore,
the proposed method can smoothen the vehicle trajectory without altering the overall
trajectory of the observed vehicle.
4.1.2. Rationality validation
We validated the rationality of the reconstructed trajectory of the example vehicle by check-
ing whether the acceleration and curvature follow kinematic laws. The acceleration outlier
rate and curvature outlier rate were used as the evaluation indicators. The acceleration out-
lier rate is defined as the number of trajectory points where the acceleration values are
outside the range of – 5 to 5 m/s2(Bichiou and Rakha 2018) divided by the total number
of trajectory points. The curvature outlier rate is defined as the number of trajectory points
where the curvature values are outside the range from – 0.2 to 0.2 m−1(MOHURD 2012)
divided by the total number of trajectory points.
The acceleration distributions of the raw and reconstructed trajectories are shown in
Figure 11. The raw acceleration was distributed in an unrealistic range from – 196 to
198 m/s2. The acceleration outlier rate is 85%. After applying Steps 1 and 2, the range
of acceleration was reduced by 92%, from – 16 to 14 m/s2. The acceleration outlier rate

TRANSPORTMETRICA A: TRANSPORT SCIENCE 15
Figure 9. Reconstructed trajectories of the example vehicle.
decreased to 78%. After applying Step 3, the range of the acceleration was reduced by 80%,
from – 2 to 4 m/s2. The acceleration outlier rate decreases to 2.8%.
The curvature distributions of the raw and reconstructed trajectories are shown in Figure
12. The raw curvature ranges from – 23 to 8 m−1, and the curvature outlier rate reached
60.4%. The curvature range of the reconstructed trajectory was reduced by 92.1% from –
0.156 to 0.191 m−1. The curvature outlier rate decreased to 0%, implying that the curva-
ture of the reconstructed trajectory fully met the criteria range. This demonstrates that the
proposed trajectory reconstruction method can effectively eliminate noise in the trajectory.

16 J. ZHAO ET AL.
Figure 10. Outliers with anomaly direction and velocity for the example vehicle.
Figure 11. Acceleration rationality analysis of the example vehicle.
Figure 12. Curvature rationality analysis of the example vehicle.

TRANSPORTMETRICA A: TRANSPORT SCIENCE 17
Figure 13. Acceleration rationality comparison of the example vehicle.
Figure 14. Curvature rationality comparison of the example vehicle.
The proposed method was compared with moving average (MA) and S-G methods,
which were widely used in trajectory reconstruction, to demonstrate its advantage. The
comparison results for the acceleration rationality are shown in Figure 13. The accelera-
tion outlier rates were 11% and 5% for MA and S-G, respectively, which are much higher
than those of the proposed method (0%). A comparison of the results of curvature ratio-
nality is shown in Figure 14. The curvature outlier rates were 6% and 13% for MA and S-G,
respectively, which were also much higher than those of the proposed method (0%). The
comparison analysis results indicate that the proposed trajectory reconstruction method
outperforms the MA-only and S-G-only methods in eliminating the noise in the example
vehicle trajectory.
4.1.3. Consistency validation
In this section, we validate the consistency between the reconstructed and raw trajectories
for an example vehicle. To validate the velocity consistency, we compared the velocities of
the raw and reconstructed trajectories, as shown in Table 3. The mean difference between
the reconstructed velocity and raw velocity is 0.18 m/s, which is quite small compared to the

18 J. ZHAO ET AL.
Tab le 3. Velocity consistency analysis of the example vehicle.
Trajectory velocity Maximum (m/s) Minimum (m/s) Mean (m/s) Standard deviation (m/s)
Raw trajectory 21.84 1.68 5.95 2.60
Reconstructed trajectory 7.01 4.53 5.77 0.76
Tab le 4. Paired-sample T-test for path consistency analysis of the example vehicle.
Difference 95% confidence interval
Indicator Mean Standard deviation T statistics Pvalues Upper limit Lower limit
x coordinate difference (m) −0.24 0.32 −11.12 0.001 −0.28 −0.20
y coordinate difference (m) 0.18 0.21 13.00 0.001 0.16 0.21
mean velocity of 5.95 m/s. The standard deviation decreased by 70.8% because the outliers
were filtered out by the proposed method.
A paired-sample T-test of the coordinates between the reconstructed and raw trajecto-
ries was applied for path consistency validation. The results are presented in Table 4.This
shows that there is no significant difference in the coordinates between the reconstructed
and raw trajectories for the example vehicle, which indicates that the proposed method can
maintain the overall trajectory of the raw data.
4.2. Overall analysis of all vehicle trajectories
We validated the effectiveness of the proposed method by applying it to all the extracted
trajectory data (2140 vehicle trajectories). The validation was conducted in two parts:
rationality and consistency validations.
4.2.1. Rationality validation
The rationality of all the reconstructed trajectories was validated by calculating the accel-
eration and curvature outlier rates. The acceleration distributions of the reconstructed
trajectories are shown in Figure 15. The raw acceleration was distributed in an unreal-
istic range from – 400 to 400 m/s2. The acceleration outlier rate for all the trajectories
reached 77.2%. Using the proposed method, the acceleration of all reconstructed trajec-
tories ranged from – 6 to 5 m/s2. The acceleration outlier rate decreased to 3.5%. In total,
96.5% of the acceleration outliers was eliminated. In comparison, the acceleration outlier
rates of the MA – and S-G-only methods were 11.2% and 16.8%, respectively. Therefore,
the proposed method outperforms the MA-only and S-G-only methods in reducing the
acceleration outlier rate, and can effectively eliminate velocity noise.
The curvature distributions of the reconstructed trajectories are shown in Figure 16.
The raw curvature ranged from – 25 to 30 m−1, and the curvature outlier rate was 80.6%.
The curvature range of the trajectories reconstructed by the proposed method was – 0.19
to 0.30 m−1. The curvature outlier rate was 2.7%, which indicates that the reconstructed
trajectories of 97.3% of the vehicles can fully satisfy the criterion range of the curvature.
In comparison, the curvature outlier rates of the MA – and S-G-only methods were 6.6%
and 11.4%, respectively. Therefore, the proposed method outperforms the MA – and S-G
– methods in reducing the curvature outlier rate, and can effectively eliminate curvature
noise.

TRANSPORTMETRICA A: TRANSPORT SCIENCE 19
Figure 15. Acceleration rationality analysis of all vehicles.
Figure 16. Curvature rationality analysis of all vehicles.
4.2.2. Consistency validation
To validate the consistency between the reconstructed and raw trajectories, the root mean
square error (RMSE) and the mean absolute percentage error (MAPE) of the trajectory at
all time frames was used as indicators. The trajectory error at each time frame (1/24 s)
was defined as the Euclidean distance between the reconstructed and raw trajectories
(Zhao, Knoop, and Wang 2020). Subsequently, the RMSE and MAPE of one vehicle can be
calculated using Equations (19) and (20), respectively.
RMSEi=



Ti

t=1
((xi(t)−x
i(t))2+(yi(t)−y
i(t))2)(19)
MAPEi=1
Ti
Ti

t=1
(



xi(t)−x
i(t)
x
i(t)



+



yi(t)−y
i(t)
y
i(t)



)(20)
where RMSEiand MAPEirepresent the RMSE and MAPE values of vehicle irespectively; xi(t)
and yi(t)are the coordinates of the reconstructed trajectory of vehicle iat time t, in meters;

20 J. ZHAO ET AL.
Tab le 5. RMSE and MAPE comparisons of all vehicle trajectories.
Maximum Minimum Mean Standard deviation
Method RMSE(m) MAPE(%) RMSE(m) MAPE(%) RMSE(m) MAPE(%) RMSE(m) MAPE(%)
Proposed method 0.921 20.678 0.127 1.627 0.821 5.872 1.257 3.152
MA method 2.132 48.888 1.153 4.900 1.775 8.154 2.396 4.350
S−G method 2.055 31.320 1.241 2.646 1.826 9.444 2.878 6.807
x
i(t)and y
i(t)represent the coordinates of the raw trajectory of vehicle iat time t, in meters;
Tirepresents the number of sampling points for vehicle i.
The RMSE and MAPE of all vehicles can be calculated using Equations (21) and (22),
respectively.
RMSEA=1
N
N

i=1
RMSEi(21)
MAPEA=1
N
N

i=1
MAPEi(22)
where RMSEAis the average RMSE value for all trajectories; MAPEAis the average MAPE value
for all trajectories; Nindicates the number of vehicles.
The maximum, minimum, mean, and standard deviation of the RMSE and the MAPE of
the trajectories reconstructed using the proposed method are listed in Table 5. Compared
with the MA and S-G methods, the RMSE of the proposed method ranged from 0.127 to
0.921 m, with a mean value of 0.821 m and a standard deviation of 1.257 m, which were
smaller than those of the MA and S-G methods. The MAPE of the proposed method ranged
from 1.627% to 20.678%, with a mean value of 5.872% and a standard deviation of 3.152%.
The average RMSE values were reduced by 53.7% and 55.1%, respectively, compared to
those of the MA and S-G methods. The standard deviations of the RMSE were reduced by
47.5% and 56.3%, respectively. The average MAPE values were reduced by 30% and 37.8%,
respectively, compared to those of the MA and S-G methods. The standard deviations of the
MAPE were reduced by 27.5% and 53.7%, respectively. This indicates that the trajectories
reconstructed by the proposed method are more consistent with the raw data than those
reconstructed by the MA and S-G methods.
A paired-sample T-test was used to compare whether there were significant differences
in the paths before and after reconstruction, and the results are shown in Table 6. A total of
95.9% (2053 out of 2140) of the reconstructed trajectories of the proposed method showed
no significant difference from the raw trajectories at a confidence level of 95%. This pro-
portion is 39.6% and 29.8% higher than that of the MA and S-G methods, respectively.
This indicates that the proposed method performs better in maintaining the overall raw
trajectory.
5. Conclusions
In this study, a three-step trajectory reconstruction method for the two-dimensional move-
ment of vehicles inside intersections is proposed. The wavelet transform, locally weighted
regression, and S-G filter were used to deal with outlier identification, data interpolation,

TRANSPORTMETRICA A: TRANSPORT SCIENCE 21
Tab le 6. Paired-sample T-test for path consistency analysis of all vehicle trajectories.
Method Sample size
Number of vehicle trajectories
with no significant difference
Proportion of vehicle trajectories
with no significant difference
Proposed method 2140 2053 95.90%
MA method 2140 1204 56.30%
S-G method 2140 1415 66.10%
and trajectory smoothing, respectively. Different wavelet bases were selected for outlier
identification of the velocity and direction angle for each vehicle according to the trajectory
rationality of physical kinematics. The rationality and consistency analysis were conducted
using empirical trajectory data by comparing it with the MA-only and S-G-only methods.
The quality of the reconstructed trajectory, including its rationality and consistency, was
confirmed. The following findings were drawn from this study:
(1) The proposed method is promising for reconstructing two-dimensional vehicle trajec-
tories. It can eliminate acceleration and curvature noises while maintaining consistency
with the raw trajectory. The large variance and range of errors in the trajectory data can
be effectively addressed by the proposed method.
(2) In terms of rationality, 96.5% and 97.3% of the trajectory acceleration and curvature
outliers, respectively, can be eliminated. The acceleration and curvature outlier rates of
the reconstructed trajectories are 3.5% and 2.7%, respectively.
(3) In terms of consistency, 95.9% of the reconstructed trajectories were not significantly
different from the raw trajectories. The reconstructed trajectory resulted from the
proposed method only have 0.82 m of RMSE (5.9% of MAPE) compared to the raw
trajectories.
This study provided a promising method for reconstructing video-based vehicle trajecto-
ries at intersections with the aim of correcting abnormal velocities and direction angles in
raw trajectories. The reconstructed vehicle trajectories resulted from the proposed method
outperforms those from existing approaches regarding rationality of accelerations and cur-
vatures. However, trajectories of each vehicle are processed individually. Further studies
are required to add the constraint conditions of the interaction between vehicles to make
the reconstructed vehicle trajectories more consistent with the reality (Zhao, Knoop, and
Wang 2022). The driving behaviour of various vehicle types and interactions between vehi-
cles at intersections are worth to be further investigated (Zhang et al. 2022). Additionally,
in this study, videos were collected by stable UAV cameras under low wind speed condi-
tions. Further research should be conducted to address instable UAV videos due to strong
wind. Moreover, some intersections are partly occluded by trees and highways from the
top views; therefore, videos with different views should be fused for vehicle trajectory
reconstruction.
Acknowledgements
Thank Dr. Jan-Dirk Schmoecker from Kyoto University for providing us with insightful suggestions
during the study.

22 J. ZHAO ET AL.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
This work was supported by the National Natural Science Foundation of China under Grants
52122215, 71971140 and 52202401; the Natural Science Foundation of Shanghai under Grant
20ZR1439300; and China Postdoctoral Science Foundation under Grant 2022M712139.
ORCID
Jing Zhao http://orcid.org/0000-0003-0741-4911
Xiaoliang Yang http://orcid.org/0000-0002-2461-8222
Cheng Zhang http://orcid.org/0000-0002-6529-2468
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