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Efficient handwriting trajectory reconstruction (TR) requires specific writing surfaces for detecting movements of digital pens. Although several motion-based solutions have been developed to remove the necessity of writing surfaces, most of them are based on classical sensor fusion methods limited, by sensor error accumulation over time, to tracing only single strokes. In this work, we present an approach to map the movements of an IMU-enhanced digital pen to relative displacement data. Training data is collected by means of a tablet. We propose several pre-processing and data-preparation methods to synchronize data between the pen and the tablet, which are of different sampling rates, and train a convolutional neural network (CNN) to reconstruct multiple strokes without the need of writing segmentation or post-processing correction of the predicted trajectory. The proposed system learns the relative displacement of the pen tip over time from the recorded raw sensor data, achieving a normalized error rate of 0.176 relative to unit-scaled tablet ground truth (GT) trajectory. To test the effectiveness of the approach, we train a neural network for character recognition from the reconstructed trajectories, which achieved a character error rate of 19.51%. Finally, a joint model is implemented that makes use of both the IMU data and the generated trajectories, which outperforms the sensor-only-based recognition approach by 0.75%.

This study explores the magnitude of two sources of error that are introduced when extracorporeal bullet trajectories are based on post-mortem computed tomography (PMCT) and/or surface scanning of a body. The first source of error is caused by an altered gravitational pull on soft tissue, which is introduced when a body is scanned in another position than it had when hit. The second source of error is introduced when scanned images are translated into a virtual representation of the victim’s body. To study the combined magnitude of these errors, virtual shooting trajectories with known vertical angles through five “victims” (live test persons) were simulated. The positions of the simulated wounds on the bodies were marked, with the victims in upright positions. Next, the victims were scanned in supine position, using 3D surface scanning, similar to a body’s position when scanned during a PMCT. Seven experts, used to working with 3D data, were asked to determine the bullet trajectories based on the virtual representations of the bodies. The errors between the known and determined trajectories were analysed and discussed. The results of this study give a feel for the magnitude of the introduced errors and can be used to reconstruct actual shooting incidents using PMCT data.

The extraction of vehicle trajectory data using video recognition is often affected by low data matching accuracy due to factors such as video occlusion and noise interference. This paper therefore proposes a two-step vehicle trajectory reconstruction method to improve the accuracy of trajectory data extracted by video recognition. First, the sources of error in video-recognition-based vehicle trajectory data are analyzed, and a two-step reconstruction method integrating interpolation and filtering principles is designed. Anomalous velocity and acceleration are then identified through spatio-temporal thresholding. This process corrects vehicle position data using a backward correction process and replaces outliers in the trajectory using interpolation. At the same time, filtering principles are implemented to optimize the denoising of the trajectory data. Finally, the effectiveness of the proposed method is validated using two metrics: vehicle acceleration standard deviation and jerk value. Case studies based on field-measured data demonstrate that the two-step method can effectively correct raw vehicle trajectory data while preserving its structural features. A comparative analysis of the interpolation effects shows that, compared to Lagrange interpolation, Hermite interpolation preserves the structural features of the original vehicle trajectory data more effectively and reduces interpolation errors more effectively, resulting in higher trajectory data reconstruction accuracy. A comparative analysis of the filtering effects shows that both Kalman filter and the moving average method can effectively remove noise from vehicle trajectory data. However, Kalman filter provides more stable trajectory data after denoising.

Mobile phone data are a popular source of positioning information in many recent studies that have largely improved our understanding of human mobility. These data consist of time-stamped and geo-referenced communication events recorded by network operators, on a per-subscriber basis. They allow for unprecedented tracking of populations of millions of individuals over long periods that span months. Nevertheless, due to the uneven processes that govern mobile communications, the sampling of user locations provided by mobile phone data tends to be sparse and irregular in time, leading to substantial gaps in the resulting trajectory information. In this paper, we illustrate the severity of the problem through an empirical study of a large-scale Call Detail Records (CDR) dataset. We then propose Context-enhanced Trajectory Reconstruction, a new technique that hinges on tensor factorization as a core method to complete individual CDR-based trajectories. The proposed solution infers missing locations with a median displacement within two network cells from the actual position of the user, on an hourly basis and even when as little as 1% of her original mobility is known. Our approach lets us revisit seminal works in the light of complete mobility data, unveiling potential biases that incomplete trajectories obtained from legacy CDR induce on key results about human mobility laws, trajectory uniqueness, and movement predictability.

Modern civil engineering structures, such as footbridges, are increasingly susceptible to vibrations induced by human activities, emphasizing the importance of accurately assessing crowd-induced loading. Developing realistic load models requires detailed insight into the underlying crowd dynamics, which in turn depend on the coordination between individuals and the spatial organization of the group. A deeper understanding of these human–human interactions is therefore essential for capturing the collective behaviour that governs crowd-induced vibrations. This paper presents a vision-based trajectory reconstruction methodology that captures individual movement trajectories in both small groups and large-scale running events. The approach integrates colour-based image segmentation for instrumented participants, deep learning–based object detection for uninstrumented crowds, and a homography-based projection method to map image coordinates to world space. The methodology is applied to empirical data from two urban running events and controlled experiments, including both stationary and dynamic camera perspectives. Results show that the framework reliably reconstructs individual trajectories under varied field conditions, applicable to both walking and running activities. The approach enables scalable monitoring of human activities and provides high-resolution spatio-temporal data for studying human–human interactions and modelling crowd dynamics. In this way, the findings highlight the potential of vision-based methods as practical, non-intrusive tools for analysing human-induced loading in both research and applied engineering contexts.

The advance in neuroscience and computer technology over the past decades have made brain-computer interface (BCI) a most promising area of neurorehabilitation and neurophysiology research. Limb motion decoding has gradually become a hot topic in the field of BCI. Decoding neural activity related to limb movement trajectory is considered to be of great help to the development of assistive and rehabilitation strategies for motor-impaired users. Although a variety of decoding methods have been proposed for limb trajectory reconstruction, there does not yet exist a review that covers the performance evaluation of these decoding methods. To alleviate this vacancy, in this paper, we evaluate EEG-based limb trajectory decoding methods regarding their advantages and disadvantages from a variety of perspectives. Specifically, we first introduce the differences in motor execution and motor imagery in limb trajectory reconstruction with different spaces (2D and 3D). Then, we discuss the limb motion trajectory reconstruction methods including experiment paradigm, EEG pre-processing, feature extraction and selection, decoding methods, and result evaluation. Finally, we expound on the open problem and future outlooks.

This paper presents the results of a study on bullet trajectory deflection, for 9 mm Luger Full Metal Jacket Round Nose (FMJ-RN) bullets fired through 23–24 cm of ballistic gelatine. The bullets were fired at different velocities. Impact velocity, energy transfer and bullet trajectory deflection after gelatine perforation were measured and calculated. As was expected, energy transfer to the gelatine blocks generally increased with increasing impact velocity, indicating an altering bullet/gelatine interaction with altering velocity. This alteration did not result in a discernible alteration of bullet trajectory deflection. Deflection angles fell between 5.7° and 7.4° for 136 of the 140 fired shots, with four outliers below 5.7°. •Bullets transfer kinetic energy when passing through gelatine.•Energy transfer increases with increasing impact velocity.•Round nose bullets deflect when passing through gelatine.•Trajectory deflection does not increase with impact velocity.

Vehicle trajectory data contain extensive spatiotemporal information and are of great significance for analyzing the operational patterns of urban traffic networks, optimizing traffic signal control and achieving refined traffic management. However, due to the low penetration rate of probe vehicles and the limited coverage of fixed sensors, it remains challenging to obtain comprehensive full-sample vehicle trajectory data. To address this issue, this paper proposes a multi-source data fusion-based vehicle trajectory reconstruction method, which comprises vehicle trajectory state estimation and a self-optimization algorithm. First, the trajectory states of undetected vehicles are categorized into four types based on the trajectory states of adjacent probe vehicles. Four corresponding trajectory estimation models are then established using an extended Intelligent Driver Model to reconstruct the initial trajectories of undetected vehicles. Second, a particle filter-based trajectory self-optimization algorithm is proposed, integrating upstream and downstream fixed sensor data to iteratively correct and optimize the initial trajectories by minimizing errors, thereby enhancing trajectory accuracy and smoothness. Case studies demonstrate that the proposed method achieves outstanding performance under low PV penetration rates and in complex traffic environments. Compared to baseline methods, MAE, MAPE, and RMSE are reduced by 14.31%, 22.87%, and 13.36%, respectively. Furthermore, the reconstruction errors of the proposed method gradually decrease as traffic density and PV penetration rates increase. Notably, PV penetration has a more significant impact on model accuracy. These findings confirm the robustness and effectiveness of the proposed method in complex traffic scenarios, providing critical technical support for refined urban traffic management and optimized decision-making.

A greater variety of technologies are being applied in sports and health with the advancement of technology, but most optoelectronic systems have strict environmental restrictions and are usually costly. To visualize and perform quantitative analysis on the football kick, we introduce a 3D motion analysis system based on a six-axis inertial measurement unit (IMU) to reconstruct the motion trajectory, in the meantime analyzing the velocity and the highest point of the foot during the backswing. We build a signal processing system in MATLAB and standardize the experimental process, allowing users to reconstruct the foot trajectory and obtain information about the motion within a short time. This paper presents a system that directly analyzes the instep kicking motion rather than recognizing different motions or obtaining biomechanical parameters. For the instep kicking motion of path length around 3.63 m, the root mean square error (RMSE) is about 0.07 m. The RMSE of the foot velocity is 0.034 m/s, which is around 0.45% of the maximum velocity. For the maximum velocity of the foot and the highest point of the backswing, the error is approximately 4% and 2.8%, respectively. With less complex hardware, our experimental results achieve excellent velocity accuracy.

•Ballistic gelatin 10% is used in wound ballistics tests to reproduce wound characteristics.•During shooting incidents reconstruction bullet deflection caused by soft tissues perforation is oft underestimated.•Shots with rifle bullets through ballistic gelatin having different depth have been performed.•Bullet deflection after perforation of soft tissue simulant has been observed.•The deflection increase with the increasing of soft tissue simulant depth. Trajectory deflections of 5.56 NATO and 7.62×39mm rifle bullets, fired through 7.5, 15 and 22.5cm of gelatine, were studied. The magnitude of the deflections from the bullets’ original trajectories after perforation are related to the length and the profile of the (wound) channels through gelatine. After 7.5cm of penetration depth, deflection was less than 1°. With the longer channel lengths, bullet instability set in and subsequently, deflection was much larger. Deflection was highest with fragmented 5.56 NATO bullets after perforating 22.5cm of gelatine. The data from this study can be used to assess the degree of bullet deflection in trajectory reconstructions after incidents where human bodies were perforated with rifle bullets of the respective calibres and cartridge types.