In real-world applications, GPS trajectories often suffer from low sampling rates, with large and irregular intervals between consecutive GPS points. This sparse characteristic presents challenges for their direct use in GPS-based systems. This paper addresses the task of map-constrained trajectory recovery, aiming to enhance trajectory sampling rates of GPS trajectories. Previous studies commonly adopt a sequence-to-sequence framework, where an encoder captures the trajectory patterns and a decoder reconstructs the target trajectory. Within this framework, effectively representing the road network and extracting relevant trajectory features are crucial for overall performance. Despite advancements in these models, they fail to fully leverage the complex spatio-temporal dynamics present in both the trajectory and the road network. To overcome these limitations, we categorize the spatio-temporal dynamics of trajectory data into two distinct aspects: spatial-temporal traffic dynamics and trajectory dynamics. Furthermore, We propose TedTrajRec, a novel method for trajectory recovery. To capture spatio-temporal traffic dynamics, we introduce PD-GNN, which models periodic patterns and learns topologically aware dynamics concurrently for each road segment. For spatio-temporal trajectory dynamics, we present TedFormer, a time-aware Transformer that incorporates temporal dynamics for each GPS location by integrating closed-form neural ordinary differential equations into the attention mechanism. This allows TedFormer to effectively handle irregularly sampled data. Extensive experiments on three real-world datasets demonstrate the superior performance of TedTrajRec. The code is publicly available at https://github.com/ysygMhdxw/TEDTrajRec/.
In the real world, trajectory data is often sparse and incomplete due to low collection frequencies or limited device coverage. Trajectory recovery aims to recover these missing trajectory points, making the trajectories denser and more complete. However, this task faces two key challenges: 1) The excessive sparsity of individual trajectories makes it difficult to effectively leverage historical information for recovery; 2) Sparse trajectories make it harder to capture complex individual mobility preferences. To address these challenges, we propose a novel method called DiffMove. Firstly, we harness crowd wisdom for trajectory recovery. Specifically, we construct a group tendency graph using the collective trajectories of all users and then integrate the group mobility trends into the location representations via graph embedding. This solves the challenge of sparse trajectories being unable to rely on individual historical trajectories for recovery. Secondly, we capture individual mobility preferences from both historical and current perspectives. Finally, we integrate group mobility tendencies and individual preferences into the spatiotemporal distribution of the trajectory to recover high-quality trajectories. Extensive experiments on two real-world datasets demonstrate that DiffMove outperforms existing state-of-the-art methods. Further analysis validates the robustness of our method.
We present Deblur-SLAM, a robust RGB SLAM pipeline designed to recover sharp reconstructions from motion-blurred inputs. The proposed method bridges the strengths of both frame-to-frame and frame-to-model approaches to model sub-frame camera trajectories that lead to high-fidelity reconstructions in motion-blurred settings. Moreover, our pipeline incorporates techniques such as online loop closure and global bundle adjustment to achieve a dense and precise global trajectory. We model the physical image formation process of motion-blurred images and minimize the error between the observed blurry images and rendered blurry images obtained by averaging sharp virtual sub-frame images. Additionally, by utilizing a monocular depth estimator alongside the online deformation of Gaussians, we ensure precise mapping and enhanced image deblurring. The proposed SLAM pipeline integrates all these components to improve the results. We achieve state-of-the-art results for sharp map estimation and sub-frame trajectory recovery both on synthetic and real-world blurry input data.
This paper addresses the problem of secure data reconstruction for unknown systems, where data collected from the system are susceptible to malicious manipulation. We aim to recover the real trajectory without prior knowledge of the system model. To achieve this, a behavioral language is used to represent the system, describing it using input/output trajectories instead of state-space models. We consider two attack scenarios. In the first scenario, up to $k$ entries of the collected data are malicious. On the other hand, the second scenario assumes that at most $k$ channels from sensors or actuators can be compromised, implying that any data collected from these channels might be falsified. For both scenarios, we formulate the trajectory recovery problem as an optimization problem and introduce sufficient conditions to ensure successful recovery of the true data. Since finding exact solutions to these problems can be computationally inefficient, we further approximate them using an $\ell_1$-norm and group Least Absolute Shrinkage and Selection Operator (LASSO). We demonstrate that under certain conditions, these approximation problems also find the true trajectory while maintaining low computation complexity. Finally, we extend the proposed algorithms to noisy data. By reconstructing the secure trajectory, this work serves as a safeguard mechanism for subsequent data-driven control methods.
Parameter estimation and trajectory reconstruction for data-driven dynamical systems governed by ordinary differential equations (ODEs) are essential tasks in fields such as biology, engineering, and physics. These inverse problems -- estimating ODE parameters from observational data -- are particularly challenging when the data are noisy, sparse, and the dynamics are nonlinear. We propose the Eigen-Fourier Physics-Informed Gaussian Process (EFiGP), an algorithm that integrates Fourier transformation and eigen-decomposition into a physics-informed Gaussian Process framework. This approach eliminates the need for numerical integration, significantly enhancing computational efficiency and accuracy. Built on a principled Bayesian framework, EFiGP incorporates the ODE system through probabilistic conditioning, enforcing governing equations in the Fourier domain while truncating high-frequency terms to achieve denoising and computational savings. The use of eigen-decomposition further simplifies Gaussian Process covariance operations, enabling efficient recovery of trajectories and parameters even in dense-grid settings. We validate the practical effectiveness of EFiGP on three benchmark examples, demonstrating its potential for reliable and interpretable modeling of complex dynamical systems while addressing key challenges in trajectory recovery and computational cost.
Discovering co-movement patterns from urban-scale video data sources has emerged as an attractive topic. This task aims to identify groups of objects that travel together along a common route, which offers effective support for government agencies in enhancing smart city management. However, the previous work has made a strong assumption on the accuracy of recovered trajectories from videos and their co-movement pattern definition requires the group of objects to appear across consecutive cameras along the common route. In practice, this often leads to missing patterns if a vehicle is not correctly identified from a certain camera due to object occlusion or vehicle mis-matching. To address this challenge, we propose a relaxed definition of co-movement patterns from video data, which removes the consecutiveness requirement in the common route and accommodates a certain number of missing captured cameras for objects within the group. Moreover, a novel enumeration framework called MaxGrowth is developed to efficiently retrieve the relaxed patterns. Unlike previous filter-and-refine frameworks comprising both candidate enumeration and subsequent candidate verification procedures, MaxGrowth incurs no verification cost for the candidate patterns. It treats the co-movement pattern as an equivalent sequence of clusters, enumerating candidates with increasing sequence length while avoiding the generation of any false positives. Additionally, we also propose two effective pruning rules to efficiently filter the non-maximal patterns. Extensive experiments are conducted to validate the efficiency of MaxGrowth and the quality of its generated co-movement patterns. Our MaxGrowth runs up to two orders of magnitude faster than the baseline algorithm. It also demonstrates high accuracy in real video dataset when the trajectory recovery algorithm is not perfect.
Spatiotemporal trajectory data is vital for web-of-things services and is extensively collected and analyzed by web-based hardware and platforms. However, issues such as service interruptions and network instability often lead to sparsely recorded trajectories, resulting in a loss of detailed movement data. As a result, recovering these trajectories to restore missing information becomes essential. Despite progress, several challenges remain unresolved. First, the lack of large-scale dense trajectory data hampers the performance of existing deep learning methods, which rely heavily on abundant data for supervised training. Second, current methods struggle to generalize across sparse trajectories with varying sampling intervals, necessitating separate re-training for each interval and increasing computational costs. Third, external factors crucial for the recovery of missing points are not fully incorporated. To address these challenges, we propose a framework called PTR. This framework mitigates the issue of limited dense trajectory data by leveraging the capabilities of pre-trained language models (PLMs). PTR incorporates an explicit trajectory prompt and is trained on datasets with multiple sampling intervals, enabling it to generalize effectively across different intervals in sparse trajectories. To capture external factors, we introduce an implicit trajectory prompt that models road conditions, providing richer information for recovering missing points. Additionally, we present a trajectory embedder that encodes trajectory points and transforms the embeddings of both observed and missing points into a format comprehensible to PLMs. Experimental results on two public trajectory datasets with three sampling intervals demonstrate the efficacy and scalability of PTR.
Once analysed, location trajectories can provide valuable insights beneficial to various applications. However, such data is also highly sensitive, rendering them susceptible to privacy risks in the event of mismanagement, for example, revealing an individual's identity, home address, or political affiliations. Hence, ensuring that privacy is preserved for this data is a priority. One commonly taken measure to mitigate this concern is aggregation. Previous work by Xu et al. shows that trajectories are still recoverable from anonymised and aggregated datasets. However, the study lacks implementation details, obfuscating the mechanisms of the attack. Additionally, the attack was evaluated on commercial non-public datasets, rendering the results and subsequent claims unverifiable. This study reimplements the trajectory recovery attack from scratch and evaluates it on two open-source datasets, detailing the preprocessing steps and implementation. Results confirm that privacy leakage still exists despite common anonymisation and aggregation methods but also indicate that the initial accuracy claims may have been overly ambitious. We release all code as open-source to ensure the results are entirely reproducible and, therefore, verifiable. Moreover, we propose a stronger attack by designing a series of enhancements to the baseline attack. These enhancements yield higher accuracies by up to 16%, providing an improved benchmark for future research in trajectory recovery methods. Our improvements also enable online execution of the attack, allowing partial attacks on larger datasets previously considered unprocessable, thereby furthering the extent of privacy leakage. The findings emphasise the importance of using strong privacy-preserving mechanisms when releasing aggregated mobility data and not solely relying on aggregation as a means of anonymisation.
With the proliferation of location-aware devices, large amount of trajectories have been generated when agents such as people, vehicles and goods flow around the urban environment. These raw trajectories, typically collected from various sources such as GPS in cars, personal mobile devices, and public transport, are often sparse and fragmented due to limited sampling rates, infrastructure coverage and data loss. In this context, trajectory recovery aims to reconstruct such sparse raw trajectories into their dense and continuous counterparts, so that fine-grained movement of agents across space and time can be captured faithfully. Existing trajectory recovery approaches typically rely on the prior knowledge of travel mode or motion patterns, and often fail in densely populated urban areas where accurate maps are absent. In this paper, we present a new recovery framework called TrajWeaver based on probabilistic diffusion models, which is able to recover dense and refined trajectories from the sparse raw ones, conditioned on various auxiliary features such as Areas of Interest along the way, user identity and waybill information. The core of TrajWeaver is a novel State Propagation Diffusion Model (SPDM), which introduces a new state propagation mechanism on top of the standard diffusion models, so that knowledge computed in earlier diffusion steps can be reused later, improving the recovery performance while reducing the number of steps needed. Extensive experiments show that the proposed TrajWeaver can recover from raw trajectories of various lengths, sparsity levels and heterogeneous travel modes, and outperform the state-of-the-art baselines significantly in recovery accuracy. Our code is available at: https://anonymous.4open.science/r/TrajWeaver/
With the proliferation of GPS-equipped edge devices, huge trajectory data is generated and accumulated in various domains, motivating a variety of urban applications. Due to the limited acquisition capabilities of edge devices, a lot of trajectories are recorded at a low sampling rate, which may lead to the effectiveness drop of urban applications. We aim to recover a high-sampled trajectory based on the low-sampled trajectory in free space, i.e., without road network information, to enhance the usability of trajectory data and support urban applications more effectively. Recent proposals targeting trajectory recovery often assume that trajectories are available at a central location, which fail to handle the decentralized trajectories and hurt privacy. To bridge the gap between decentralized training and trajectory recovery, we propose a lightweight framework, LightTR, for federated trajectory recovery based on a client-server architecture, while keeping the data decentralized and private in each client/platform center (e.g., each data center of a company). Specifically, considering the limited processing capabilities of edge devices, LightTR encompasses a light local trajectory embedding module that offers improved computational efficiency without compromising its feature extraction capabilities. LightTR also features a meta-knowledge enhanced local-global training scheme to reduce communication costs between the server and clients and thus further offer efficiency improvement. Extensive experiments demonstrate the effectiveness and efficiency of the proposed framework.
Recovering intermediate missing GPS points in a sparse trajectory, while adhering to the constraints of the road network, could offer deep insights into users' moving behaviors in intelligent transportation systems. Although recent studies have demonstrated the advantages of achieving map-constrained trajectory recovery via an end-to-end manner, they still face two significant challenges. Firstly, existing methods are mostly sequence-based models. It is extremely hard for them to comprehensively capture the micro-semantics of individual trajectory, including the information of each GPS point and the movement between two GPS points. Secondly, existing approaches ignore the impact of the macro-semantics, i.e., the road conditions and the people's shared travel preferences reflected by a group of trajectories. To address the above challenges, we propose a Micro-Macro Spatial-Temporal Graph-based Encoder-Decoder (MM-STGED). Specifically, we model each trajectory as a graph to efficiently describe the micro-semantics of trajectory and design a novel message-passing mechanism to learn trajectory representations. Additionally, we extract the macro-semantics of trajectories and further incorporate them into a well-designed graph-based decoder to guide trajectory recovery. Extensive experiments conducted on sparse trajectories with three different sampling intervals that are respectively constructed from two real-world trajectory datasets demonstrate the superiority of our proposed model.
Vehicle movement is frequently captured in the form of GPS trajectories, i.e., sequences of timestamped GPS locations. Such data is widely used for various tasks such as travel-time estimation, trajectory recovery, and trajectory prediction. A universal vehicle trajectory model could be applied to different tasks, removing the need to maintain multiple specialized models, thereby reducing computational and storage costs. However, creating such a model is challenging when the integrity of trajectory features is compromised, i.e., in scenarios where only partial features are available or the trajectories are sparse. To address these challenges, we propose the Universal Vehicle Trajectory Model (UVTM), which can effectively adapt to different tasks without excessive retraining. UVTM incorporates two specialized designs. First, it divides trajectory features into three distinct domains. Each domain can be masked and generated independently to accommodate tasks with only partially available features. Second, UVTM is pre-trained by reconstructing dense, feature-complete trajectories from sparse, feature-incomplete counterparts, enabling strong performance even when the integrity of trajectory features is compromised. Experiments involving four representative trajectory-related tasks on three real-world vehicle trajectory datasets provide insight into the performance of UVTM and offer evidence that it is capable of meeting its objectives.
As attitude and motion sensing components, inertial sensors are widely used in various portable devices. But the severe errors of inertial sensors restrain their function, especially the trajectory recovery and semantic recognition. As a mainstream signal processing method, wavelet is hailed as the mathematical microscope of signal due to the plentiful and diverse wavelet basis functions. However, complicated noise types and application scenarios of inertial sensors make selecting wavelet basis perplexing. To this end, we propose a wavelet dynamic selection network (WDSNet), which intelligently selects the appropriate wavelet basis for variable inertial signals. In addition, existing deep learning architectures excel at extracting features from input data but neglect to learn the characteristics of target categories, which is essential to enhance the category awareness capability, thereby improving the selection of wavelet basis. Therefore, we propose a category representation mechanism (CRM), which enables the network to extract and represent category features without increasing trainable parameters. Furthermore, CRM transforms the common fully connected network into category representations, which provide closer supervision to the feature extractor than the far and trivial one-hot classification labels. We call this process of imposing interpretability on a network and using it to supervise the feature extractor the feature supervision mechanism, and its effectiveness is demonstrated experimentally and theoretically in this paper. The enhanced inertial signal can perform impracticable tasks with regard to the original signal, such as trajectory reconstruction. Both quantitative and visual results show that WDSNet outperforms the existing methods. Remarkably, WDSNet, as a weakly-supervised method, achieves the state-of-the-art performance of all the compared fully-supervised methods.
Trajectory recovery based on the snapshots from the city-wide multi-camera network facilitates urban mobility sensing and driveway optimization. The state-of-the-art solutions devoted to such a vision-based scheme typically incorporate predefined rules or unsupervised iterative feedback, struggling with multi-fold challenges such as lack of open-source datasets for training the whole pipeline, and the vulnerability to the noises from visual inputs. In response to the dilemma, this paper proposes VisionTraj, the first learning-based model that reconstructs vehicle trajectories from snapshots recorded by road network cameras. Coupled with it, we elaborate on two rational vision-trajectory datasets, which produce extensive trajectory data along with corresponding visual snapshots, enabling supervised vision-trajectory interplay extraction. Following the data creation, based on the results from the off-the-shelf multi-modal vehicle clustering, we first re-formulate the trajectory recovery problem as a generative task and introduce the canonical Transformer as the autoregressive backbone. Then, to identify clustering noises (e.g., false positives) with the bound on the snapshots' spatiotemporal dependencies, a GCN-based soft-denoising module is conducted based on the fine- and coarse-grained Re-ID clusters. Additionally, we harness strong semantic information extracted from the tracklet to provide detailed insights into the vehicle's entry and exit actions during trajectory recovery. The denoising and tracklet components can also act as plug-and-play modules to boost baselines. Experimental results on the two hand-crafted datasets show that the proposed VisionTraj achieves a maximum +11.5% improvement against the sub-best model.
The trajectory on the road traffic is commonly collected at a low sampling rate, and trajectory recovery aims to recover a complete and continuous trajectory from the sparse and discrete inputs. Recently, sequential language models have been innovatively adopted for trajectory recovery in a pre-trained manner: it learns road segment representation vectors, which will be used in the downstream tasks. However, existing methods are incapable of handling complex trajectories: when the trajectory crosses remote road segments or makes several turns, which we call critical nodes, the quality of learned representations deteriorates, and the recovered trajectories skip the critical nodes. This work is dedicated to offering a more robust trajectory recovery for complex trajectories. Firstly, we define the trajectory complexity based on the detour score and entropy score and construct the complexity-aware semantic graphs correspondingly. Then, we propose a Multi-view Graph and Complexity Aware Transformer (MGCAT) model to encode these semantics in trajectory pre-training from two aspects: 1) adaptively aggregate the multi-view graph features considering trajectory pattern, and 2) higher attention to critical nodes in a complex trajectory. Such that, our MGCAT is perceptual when handling the critical scenario of complex trajectories. Extensive experiments are conducted on large-scale datasets. The results prove that our method learns better representations for trajectory recovery, with 5.22% higher F1-score overall and 8.16% higher F1-score for complex trajectories particularly. The code is available at https://github.com/bonaldli/ComplexTraj.
GPS trajectories are the essential foundations for many trajectory-based applications, such as travel time estimation, traffic prediction and trajectory similarity measurement. Most applications require a large amount of high sample rate trajectories to achieve a good performance. However, many real-life trajectories are collected with low sample rate due to energy concern or other constraints.We study the task of trajectory recovery in this paper as a means for increasing the sample rate of low sample trajectories. Currently, most existing works on trajectory recovery follow a sequence-to-sequence diagram, with an encoder to encode a trajectory and a decoder to recover real GPS points in the trajectory. However, these works ignore the topology of road network and only use grid information or raw GPS points as input. Therefore, the encoder model is not able to capture rich spatial information of the GPS points along the trajectory, making the prediction less accurate and lack spatial consistency. In this paper, we propose a road network enhanced transformer-based framework, namely RNTrajRec, for trajectory recovery. RNTrajRec first uses a graph model, namely GridGNN, to learn the embedding features of each road segment. It next develops a spatial-temporal transformer model, namely GPSFormer, to learn rich spatial and temporal features along with a Sub-Graph Generation module to capture the spatial features for each GPS point in the trajectory. It finally forwards the outputs of encoder model into a multi-task decoder model to recover the missing GPS points. Extensive experiments based on three large-scale real-life trajectory datasets confirm the effectiveness of our approach.
Trajectory Representation Learning (TRL) is a powerful tool for spatial-temporal data analysis and management. TRL aims to convert complicated raw trajectories into low-dimensional representation vectors, which can be applied to various downstream tasks, such as trajectory classification, clustering, and similarity computation. Existing TRL works usually treat trajectories as ordinary sequence data, while some important spatial-temporal characteristics, such as temporal regularities and travel semantics, are not fully exploited. To fill this gap, we propose a novel Self-supervised trajectory representation learning framework with TemporAl Regularities and Travel semantics, namely START. The proposed method consists of two stages. The first stage is a Trajectory Pattern-Enhanced Graph Attention Network (TPE-GAT), which converts the road network features and travel semantics into representation vectors of road segments. The second stage is a Time-Aware Trajectory Encoder (TAT-Enc), which encodes representation vectors of road segments in the same trajectory as a trajectory representation vector, meanwhile incorporating temporal regularities with the trajectory representation. Moreover, we also design two self-supervised tasks, i.e., span-masked trajectory recovery and trajectory contrastive learning, to introduce spatial-temporal characteristics of trajectories into the training process of our START framework. The effectiveness of the proposed method is verified by extensive experiments on two large-scale real-world datasets for three downstream tasks. The experiments also demonstrate that our method can be transferred across different cities to adapt heterogeneous trajectory datasets.
Many important tasks such as forensic signature verification, calligraphy synthesis, etc, rely on handwriting trajectory recovery of which, however, even an appropriate evaluation metric is still missing. Indeed, existing metrics only focus on the writing orders but overlook the fidelity of glyphs. Taking both facets into account, we come up with two new metrics, the adaptive intersection on union (AIoU) which eliminates the influence of various stroke widths, and the length-independent dynamic time warping (LDTW) which solves the trajectory-point alignment problem. After that, we then propose a novel handwriting trajectory recovery model named Parsing-and-tracing ENcoder-decoder Network (PEN-Net), in particular for characters with both complex glyph and long trajectory, which was believed very challenging. In the PEN-Net, a carefully designed double-stream parsing encoder parses the glyph structure, and a global tracing decoder overcomes the memory difficulty of long trajectory prediction. Our experiments demonstrate that the two new metrics AIoU and LDTW together can truly assess the quality of handwriting trajectory recovery and the proposed PEN-Net exhibits satisfactory performance in various complex-glyph languages including Chinese, Japanese and Indic.
Stroke order and velocity are helpful features in the fields of signature verification, handwriting recognition, and handwriting synthesis. Recovering these features from offline handwritten text is a challenging and well-studied problem. We propose a new model called TRACE (Trajectory Recovery by an Adaptively-trained Convolutional Encoder). TRACE is a differentiable approach that uses a convolutional recurrent neural network (CRNN) to infer temporal stroke information from long lines of offline handwritten text with many characters and dynamic time warping (DTW) to align predictions and ground truth points. TRACE is perhaps the first system to be trained end-to-end on entire lines of text of arbitrary width and does not require the use of dynamic exemplars. Moreover, the system does not require images to undergo any pre-processing, nor do the predictions require any post-processing. Consequently, the recovered trajectory is differentiable and can be used as a loss function for other tasks, including synthesizing offline handwritten text. We demonstrate that temporal stroke information recovered by TRACE from offline data can be used for handwriting synthesis and establish the first benchmarks for a stroke trajectory recovery system trained on the IAM online handwriting dataset.
A considerable amount of mobility data has been accumulated due to the proliferation of location-based service. Nevertheless, compared with mobility data from transportation systems like the GPS module in taxis, this kind of data is commonly sparse in terms of individual trajectories in the sense that users do not access mobile services and contribute their data all the time. Consequently, the sparsity inevitably weakens the practical value of the data even it has a high user penetration rate. To solve this problem, we propose a novel attentional neural network-based model, named AttnMove, to densify individual trajectories by recovering unobserved locations at a fine-grained spatial-temporal resolution. To tackle the challenges posed by sparsity, we design various intra- and inter- trajectory attention mechanisms to better model the mobility regularity of users and fully exploit the periodical pattern from long-term history. We evaluate our model on two real-world datasets, and extensive results demonstrate the performance gain compared with the state-of-the-art methods. This also shows that, by providing high-quality mobility data, our model can benefit a variety of mobility-oriented down-stream applications.