Lennart Svensson

CV
h-index68
64papers
3,791citations
Novelty47%
AI Score59

64 Papers

LGMay 11, 2022Code
DoubleMatch: Improving Semi-Supervised Learning with Self-Supervision

Erik Wallin, Lennart Svensson, Fredrik Kahl et al.

Following the success of supervised learning, semi-supervised learning (SSL) is now becoming increasingly popular. SSL is a family of methods, which in addition to a labeled training set, also use a sizable collection of unlabeled data for fitting a model. Most of the recent successful SSL methods are based on pseudo-labeling approaches: letting confident model predictions act as training labels. While these methods have shown impressive results on many benchmark datasets, a drawback of this approach is that not all unlabeled data are used during training. We propose a new SSL algorithm, DoubleMatch, which combines the pseudo-labeling technique with a self-supervised loss, enabling the model to utilize all unlabeled data in the training process. We show that this method achieves state-of-the-art accuracies on multiple benchmark datasets while also reducing training times compared to existing SSL methods. Code is available at https://github.com/walline/doublematch.

LGJan 24, 2023Code
Improving Open-Set Semi-Supervised Learning with Self-Supervision

Erik Wallin, Lennart Svensson, Fredrik Kahl et al.

Open-set semi-supervised learning (OSSL) embodies a practical scenario within semi-supervised learning, wherein the unlabeled training set encompasses classes absent from the labeled set. Many existing OSSL methods assume that these out-of-distribution data are harmful and put effort into excluding data belonging to unknown classes from the training objective. In contrast, we propose an OSSL framework that facilitates learning from all unlabeled data through self-supervision. Additionally, we utilize an energy-based score to accurately recognize data belonging to the known classes, making our method well-suited for handling uncurated data in deployment. We show through extensive experimental evaluations that our method yields state-of-the-art results on many of the evaluated benchmark problems in terms of closed-set accuracy and open-set recognition when compared with existing methods for OSSL. Our code is available at https://github.com/walline/ssl-tf2-sefoss.

CVJul 1, 2022Code
Masked Autoencoder for Self-Supervised Pre-training on Lidar Point Clouds

Georg Hess, Johan Jaxing, Elias Svensson et al.

Masked autoencoding has become a successful pretraining paradigm for Transformer models for text, images, and, recently, point clouds. Raw automotive datasets are suitable candidates for self-supervised pre-training as they generally are cheap to collect compared to annotations for tasks like 3D object detection (OD). However, the development of masked autoencoders for point clouds has focused solely on synthetic and indoor data. Consequently, existing methods have tailored their representations and models toward small and dense point clouds with homogeneous point densities. In this work, we study masked autoencoding for point clouds in an automotive setting, which are sparse and for which the point density can vary drastically among objects in the same scene. To this end, we propose Voxel-MAE, a simple masked autoencoding pre-training scheme designed for voxel representations. We pre-train the backbone of a Transformer-based 3D object detector to reconstruct masked voxels and to distinguish between empty and non-empty voxels. Our method improves the 3D OD performance by 1.75 mAP points and 1.05 NDS on the challenging nuScenes dataset. Further, we show that by pre-training with Voxel-MAE, we require only 40% of the annotated data to outperform a randomly initialized equivalent. Code available at https://github.com/georghess/voxel-mae

CVMar 15, 2022Code
Object Detection as Probabilistic Set Prediction

Georg Hess, Christoffer Petersson, Lennart Svensson

Accurate uncertainty estimates are essential for deploying deep object detectors in safety-critical systems. The development and evaluation of probabilistic object detectors have been hindered by shortcomings in existing performance measures, which tend to involve arbitrary thresholds or limit the detector's choice of distributions. In this work, we propose to view object detection as a set prediction task where detectors predict the distribution over the set of objects. Using the negative log-likelihood for random finite sets, we present a proper scoring rule for evaluating and training probabilistic object detectors. The proposed method can be applied to existing probabilistic detectors, is free from thresholds, and enables fair comparison between architectures. Three different types of detectors are evaluated on the COCO dataset. Our results indicate that the training of existing detectors is optimized toward non-probabilistic metrics. We hope to encourage the development of new object detectors that can accurately estimate their own uncertainty. Code available at https://github.com/georghess/pmb-nll.

50.1CVMay 28Code
SwInception -- Local Attention Meets Convolutions

David Hagerman, Roman Naeem, Jakob Lindqvist et al.

Sparse vision transformers have gained popularity as efficient encoders for medical volumetric segmentation, with Swin emerging as a prominent choice. Swin uses local attention to reduce complexity and yields excellent performance for many tasks but still tends to overfit on small datasets. To mitigate this weakness, we propose a novel architecture that further enhances Swin's inductive bias by introducing Inception blocks in the feed-forward layers. The introduction of these multi-branch convolutions enables more direct reasoning over local, multi-scale features within the transformer block. We have also modified the decoder layers in order to capture finer details using fewer parameters. We demonstrate a performance improvement on eleven different medical datasets through extensive experimentation. We specifically showcase advancements over the previous state-of-the-art backbones on benchmark challenges like the Medical Segmentation Decathlon and Beyond the Cranial Vault. By showing that the existing inductive bias in Swin can be further improved, our work presents a promising avenue for enhancing the capabilities of sparse vision transformers for both medical and natural image segmentation tasks. Code and pre-trained weights can be accessed at https://github.com/Eiphodos/SwInception.

CVDec 13, 2022Code
LidarCLIP or: How I Learned to Talk to Point Clouds

Georg Hess, Adam Tonderski, Christoffer Petersson et al.

Research connecting text and images has recently seen several breakthroughs, with models like CLIP, DALL-E 2, and Stable Diffusion. However, the connection between text and other visual modalities, such as lidar data, has received less attention, prohibited by the lack of text-lidar datasets. In this work, we propose LidarCLIP, a mapping from automotive point clouds to a pre-existing CLIP embedding space. Using image-lidar pairs, we supervise a point cloud encoder with the image CLIP embeddings, effectively relating text and lidar data with the image domain as an intermediary. We show the effectiveness of LidarCLIP by demonstrating that lidar-based retrieval is generally on par with image-based retrieval, but with complementary strengths and weaknesses. By combining image and lidar features, we improve upon both single-modality methods and enable a targeted search for challenging detection scenarios under adverse sensor conditions. We also explore zero-shot classification and show that LidarCLIP outperforms existing attempts to use CLIP for point clouds by a large margin. Finally, we leverage our compatibility with CLIP to explore a range of applications, such as point cloud captioning and lidar-to-image generation, without any additional training. Code and pre-trained models are available at https://github.com/atonderski/lidarclip.

CVNov 26, 2023Code
NeuRAD: Neural Rendering for Autonomous Driving

Adam Tonderski, Carl Lindström, Georg Hess et al.

Neural radiance fields (NeRFs) have gained popularity in the autonomous driving (AD) community. Recent methods show NeRFs' potential for closed-loop simulation, enabling testing of AD systems, and as an advanced training data augmentation technique. However, existing methods often require long training times, dense semantic supervision, or lack generalizability. This, in turn, hinders the application of NeRFs for AD at scale. In this paper, we propose NeuRAD, a robust novel view synthesis method tailored to dynamic AD data. Our method features simple network design, extensive sensor modeling for both camera and lidar -- including rolling shutter, beam divergence and ray dropping -- and is applicable to multiple datasets out of the box. We verify its performance on five popular AD datasets, achieving state-of-the-art performance across the board. To encourage further development, we will openly release the NeuRAD source code. See https://github.com/georghess/NeuRAD .

LGJul 16, 2024Code
ProSub: Probabilistic Open-Set Semi-Supervised Learning with Subspace-Based Out-of-Distribution Detection

Erik Wallin, Lennart Svensson, Fredrik Kahl et al.

In open-set semi-supervised learning (OSSL), we consider unlabeled datasets that may contain unknown classes. Existing OSSL methods often use the softmax confidence for classifying data as in-distribution (ID) or out-of-distribution (OOD). Additionally, many works for OSSL rely on ad-hoc thresholds for ID/OOD classification, without considering the statistics of the problem. We propose a new score for ID/OOD classification based on angles in feature space between data and an ID subspace. Moreover, we propose an approach to estimate the conditional distributions of scores given ID or OOD data, enabling probabilistic predictions of data being ID or OOD. These components are put together in a framework for OSSL, termed ProSub, that is experimentally shown to reach SOTA performance on several benchmark problems. Our code is available at https://github.com/walline/prosub.

CODec 6, 2019
Poisson multi-Bernoulli conjugate prior for multiple extended object filtering

Karl Granstrom, Maryam Fatemi, Lennart Svensson

This paper presents a Poisson multi-Bernoulli mixture (PMBM) conjugate prior for multiple extended object filtering. A Poisson point process is used to describe the existence of yet undetected targets, while a multi-Bernoulli mixture describes the distribution of the targets that have been detected. The prediction and update equations are presented for the standard transition density and measurement likelihood. Both the prediction and the update preserve the PMBM form of the density, and in this sense the PMBM density is a conjugate prior. However, the unknown data associations lead to an intractably large number of terms in the PMBM density, and approximations are necessary for tractability. A gamma Gaussian inverse Wishart implementation is presented, along with methods to handle the data association problem. A simulation study shows that the extended target PMBM filter performs well in comparison to the extended target d-GLMB and LMB filters. An experiment with Lidar data illustrates the benefit of tracking both detected and undetected targets.

SPDec 12, 2018
Poisson multi-Bernoulli mixture trackers: continuity through random finite sets of trajectories

Karl Granström, Lennart Svensson, Yuxuan Xia et al.

The Poisson multi-Bernoulli mixture (PMBM) is an unlabelled multi-target distribution for which the prediction and update are closed. It has a Poisson birth process, and new Bernoulli components are generated on each new measurement as a part of the Bayesian measurement update. The PMBM filter is similar to the multiple hypothesis tracker (MHT), but seemingly does not provide explicit continuity between time steps. This paper considers a recently developed formulation of the multi-target tracking problem as a random finite set (RFS) of trajectories, and derives two trajectory RFS filters, called PMBM trackers. The PMBM trackers efficiently estimate the set of trajectories, and share hypothesis structure with the PMBM filter. By showing that the prediction and update in the PMBM filter can be viewed as an efficient method for calculating the time marginals of the RFS of trajectories, continuity in the same sense as MHT is established for the PMBM filter.

APJul 13, 2022
A comparison between PMBM Bayesian track initiation and labelled RFS adaptive birth

Ángel F. García-Fernández, Yuxuan Xia, Lennart Svensson

This paper provides a comparative analysis between the adaptive birth model used in the labelled random finite set literature and the track initiation in the Poisson multi-Bernoulli mixture (PMBM) filter, with point-target models. The PMBM track initiation is obtained via Bayes' rule applied on the predicted PMBM density, and creates one Bernoulli component for each received measurement, representing that this measurement may be clutter or a detection from a new target. Adaptive birth mimics this procedure by creating a Bernoulli component for each measurement using a different rule to determine the probability of existence and a user-defined single-target density. This paper first provides an analysis of the differences that arise in track initiation based on isolated measurements. Then, it shows that adaptive birth underestimates the number of objects present in the surveillance area under common modelling assumptions. Finally, we provide numerical simulations to further illustrate the differences.

LGSep 19, 2022
Deep Fusion of Multi-Object Densities Using Transformer

Lechi Li, Chen Dai, Yuxuan Xia et al.

In this paper, we demonstrate that deep learning based method can be used to fuse multi-object densities. Given a scenario with several sensors with possibly different field-of-views, tracking is performed locally in each sensor by a tracker, which produces random finite set multi-object densities. To fuse outputs from different trackers, we adapt a recently proposed transformer-based multi-object tracker, where the fusion result is a global multi-object density, describing the set of all alive objects at the current time. We compare the performance of the transformer-based fusion method with a well-performing model-based Bayesian fusion method in several simulated scenarios with different parameter settings using synthetic data. The simulation results show that the transformer-based fusion method outperforms the model-based Bayesian method in our experimental scenarios.

MLJul 26, 2023
MCMC-Correction of Score-Based Diffusion Models for Model Composition

Anders Sjöberg, Jakob Lindqvist, Magnus Önnheim et al.

Diffusion models can be parameterized in terms of either a score or an energy function. The energy parameterization is attractive as it enables sampling procedures such as Markov Chain Monte Carlo (MCMC) that incorporates a Metropolis-Hastings (MH) correction step based on energy differences between proposed samples. Such corrections can significantly improve sampling quality, particularly in the context of model composition, where pre-trained models are combined to generate samples from novel distributions. Score-based diffusion models, on the other hand, are more widely adopted and come with a rich ecosystem of pre-trained models. However, they do not, in general, define an underlying energy function, making MH-based sampling inapplicable. In this work, we address this limitation by retaining the score parameterization and introducing a novel MH-like acceptance rule based on line integration of the score function. This allows the reuse of existing diffusion models while still combining the reverse process with various MCMC techniques, viewed as an instance of annealed MCMC. Through experiments on synthetic and real-world data, we show that our MH-like samplers offer comparable improvements to those obtained with energy-based models, without requiring explicit energy parameterization.

SINov 10, 2023
Graph GOSPA metric: a metric to measure the discrepancy between graphs of different sizes

Jinhao Gu, Ángel F. García-Fernández, Robert E. Firth et al.

This paper proposes a metric to measure the dissimilarity between graphs that may have a different number of nodes. The proposed metric extends the generalised optimal subpattern assignment (GOSPA) metric, which is a metric for sets, to graphs. The proposed graph GOSPA metric includes costs associated with node attribute errors for properly assigned nodes, missed and false nodes and edge mismatches between graphs. The computation of this metric is based on finding the optimal assignments between nodes in the two graphs, with the possibility of leaving some of the nodes unassigned. We also propose a lower bound for the metric, which is also a metric for graphs and is computable in polynomial time using linear programming. The metric is first derived for undirected unweighted graphs and it is then extended to directed and weighted graphs. The properties of the metric are demonstrated via simulated and empirical datasets.

MLApr 18, 2022
Active Learning with Weak Supervision for Gaussian Processes

Amanda Olmin, Jakob Lindqvist, Lennart Svensson et al.

Annotating data for supervised learning can be costly. When the annotation budget is limited, active learning can be used to select and annotate those observations that are likely to give the most gain in model performance. We propose an active learning algorithm that, in addition to selecting which observation to annotate, selects the precision of the annotation that is acquired. Assuming that annotations with low precision are cheaper to obtain, this allows the model to explore a larger part of the input space, with the same annotation budget. We build our acquisition function on the previously proposed BALD objective for Gaussian Processes, and empirically demonstrate the gains of being able to adjust the annotation precision in the active learning loop.

95.7SYApr 8
Occlusion-Aware Multi-Object Tracking via Expected Probability of Detection

Jan Krejčí, Oliver Kost, Yuxuan Xia et al.

This paper addresses multi-object systems, where objects may occlude one another relative to the sensor. The standard point-object model for detection-based sensors is enhanced so that the probability of detection considers the presence of all objects. A principled tracking method is derived, assigning each object an expected probability of detection, where the expectation is taken over the reduced Palm density, which means conditionally on the object's existence. The assigned probability thus considers the object's visibility relative to the sensor, under the presence of other objects. Unlike existing methods, the proposed method systematically accounts for uncertainties related to all objects in a clear and manageable way. The method is demonstrated through a visual tracking application using the multi-Bernoulli mixture (MBM) filter with marks.

CVSep 20, 2023
You can have your ensemble and run it too -- Deep Ensembles Spread Over Time

Isak Meding, Alexander Bodin, Adam Tonderski et al.

Ensembles of independently trained deep neural networks yield uncertainty estimates that rival Bayesian networks in performance. They also offer sizable improvements in terms of predictive performance over single models. However, deep ensembles are not commonly used in environments with limited computational budget -- such as autonomous driving -- since the complexity grows linearly with the number of ensemble members. An important observation that can be made for robotics applications, such as autonomous driving, is that data is typically sequential. For instance, when an object is to be recognized, an autonomous vehicle typically observes a sequence of images, rather than a single image. This raises the question, could the deep ensemble be spread over time? In this work, we propose and analyze Deep Ensembles Spread Over Time (DESOT). The idea is to apply only a single ensemble member to each data point in the sequence, and fuse the predictions over a sequence of data points. We implement and experiment with DESOT for traffic sign classification, where sequences of tracked image patches are to be classified. We find that DESOT obtains the benefits of deep ensembles, in terms of predictive and uncertainty estimation performance, while avoiding the added computational cost. Moreover, DESOT is simple to implement and does not require sequences during training. Finally, we find that DESOT, like deep ensembles, outperform single models for out-of-distribution detection.

57.9CVMar 27
ARTA: Adaptive Mixed-Resolution Token Allocation for Efficient Dense Feature Extraction

David Hagerman, Roman Naeem, Erik Brorsson et al.

We present ARTA, a mixed-resolution coarse-to-fine vision transformer for efficient dense feature extraction. Unlike models that begin with dense high-resolution (fine) tokens, ARTA starts with low-resolution (coarse) tokens and uses a lightweight allocator to predict which regions require more fine tokens. The allocator iteratively predicts a semantic (class) boundary score and allocates additional tokens to patches above a low threshold, concentrating token density near boundaries while maintaining high sensitivity to weak boundary evidence. This targeted allocation encourages tokens to represent a single semantic class rather than a mixture of classes. Mixed-resolution attention enables interaction between coarse and fine tokens, focusing computation on semantically complex areas while avoiding redundant processing in homogeneous regions. Experiments demonstrate that ARTA achieves state-of-the-art results on ADE20K and COCO-Stuff with substantially fewer FLOPs, and delivers competitive performance on Cityscapes at markedly lower compute. For example, ARTA-Base attains 54.6 mIoU on ADE20K in the ~100M-parameter class while using fewer FLOPs and less memory than comparable backbones.

CVMar 6, 2024Code
ECAP: Extensive Cut-and-Paste Augmentation for Unsupervised Domain Adaptive Semantic Segmentation

Erik Brorsson, Knut Åkesson, Lennart Svensson et al.

We consider unsupervised domain adaptation (UDA) for semantic segmentation in which the model is trained on a labeled source dataset and adapted to an unlabeled target dataset. Unfortunately, current self-training methods are susceptible to misclassified pseudo-labels resulting from erroneous predictions. Since certain classes are typically associated with less reliable predictions in UDA, reducing the impact of such pseudo-labels without skewing the training towards some classes is notoriously difficult. To this end, we propose an extensive cut-and-paste strategy (ECAP) to leverage reliable pseudo-labels through data augmentation. Specifically, ECAP maintains a memory bank of pseudo-labeled target samples throughout training and cut-and-pastes the most confident ones onto the current training batch. We implement ECAP on top of the recent method MIC and boost its performance on two synthetic-to-real domain adaptation benchmarks. Notably, MIC+ECAP reaches an unprecedented performance of 69.1 mIoU on the Synthia->Cityscapes benchmark. Our code is available at https://github.com/ErikBrorsson/ECAP.

CVJul 15, 2025Code
Trexplorer Super: Topologically Correct Centerline Tree Tracking of Tubular Objects in CT Volumes

Roman Naeem, David Hagerman, Jennifer Alvén et al.

Tubular tree structures, such as blood vessels and airways, are essential in human anatomy and accurately tracking them while preserving their topology is crucial for various downstream tasks. Trexplorer is a recurrent model designed for centerline tracking in 3D medical images but it struggles with predicting duplicate branches and terminating tracking prematurely. To address these issues, we present Trexplorer Super, an enhanced version that notably improves performance through novel advancements. However, evaluating centerline tracking models is challenging due to the lack of public datasets. To enable thorough evaluation, we develop three centerline datasets, one synthetic and two real, each with increasing difficulty. Using these datasets, we conduct a comprehensive evaluation of existing state-of-the-art (SOTA) models and compare them with our approach. Trexplorer Super outperforms previous SOTA models on every dataset. Our results also highlight that strong performance on synthetic data does not necessarily translate to real datasets. The code and datasets are available at https://github.com/RomStriker/Trexplorer-Super.

LGApr 1, 2021Code
Next Generation Multitarget Trackers: Random Finite Set Methods vs Transformer-based Deep Learning

Juliano Pinto, Georg Hess, William Ljungbergh et al.

Multitarget Tracking (MTT) is the problem of tracking the states of an unknown number of objects using noisy measurements, with important applications to autonomous driving, surveillance, robotics, and others. In the model-based Bayesian setting, there are conjugate priors that enable us to express the multi-object posterior in closed form, which could theoretically provide Bayes-optimal estimates. However, the posterior involves a super-exponential growth of the number of hypotheses over time, forcing state-of-the-art methods to resort to approximations for remaining tractable, which can impact their performance in complex scenarios. Model-free methods based on deep-learning provide an attractive alternative, as they can, in principle, learn the optimal filter from data, but to the best of our knowledge were never compared to current state-of-the-art Bayesian filters, specially not in contexts where accurate models are available. In this paper, we propose a high-performing deep-learning method for MTT based on the Transformer architecture and compare it to two state-of-the-art Bayesian filters, in a setting where we assume the correct model is provided. Although this gives an edge to the model-based filters, it also allows us to generate unlimited training data. We show that the proposed model outperforms state-of-the-art Bayesian filters in complex scenarios, while matching their performance in simpler cases, which validates the applicability of deep-learning also in the model-based regime. The code for all our implementations is made available at https://github.com/JulianoLagana/MT3 .

CVNov 25, 2024
SplatAD: Real-Time Lidar and Camera Rendering with 3D Gaussian Splatting for Autonomous Driving

Georg Hess, Carl Lindström, Maryam Fatemi et al.

Ensuring the safety of autonomous robots, such as self-driving vehicles, requires extensive testing across diverse driving scenarios. Simulation is a key ingredient for conducting such testing in a cost-effective and scalable way. Neural rendering methods have gained popularity, as they can build simulation environments from collected logs in a data-driven manner. However, existing neural radiance field (NeRF) methods for sensor-realistic rendering of camera and lidar data suffer from low rendering speeds, limiting their applicability for large-scale testing. While 3D Gaussian Splatting (3DGS) enables real-time rendering, current methods are limited to camera data and are unable to render lidar data essential for autonomous driving. To address these limitations, we propose SplatAD, the first 3DGS-based method for realistic, real-time rendering of dynamic scenes for both camera and lidar data. SplatAD accurately models key sensor-specific phenomena such as rolling shutter effects, lidar intensity, and lidar ray dropouts, using purpose-built algorithms to optimize rendering efficiency. Evaluation across three autonomous driving datasets demonstrates that SplatAD achieves state-of-the-art rendering quality with up to +2 PSNR for NVS and +3 PSNR for reconstruction while increasing rendering speed over NeRF-based methods by an order of magnitude. See https://research.zenseact.com/publications/splatad/ for our project page.

CVMar 24, 2024
Are NeRFs ready for autonomous driving? Towards closing the real-to-simulation gap

Carl Lindström, Georg Hess, Adam Lilja et al.

Neural Radiance Fields (NeRFs) have emerged as promising tools for advancing autonomous driving (AD) research, offering scalable closed-loop simulation and data augmentation capabilities. However, to trust the results achieved in simulation, one needs to ensure that AD systems perceive real and rendered data in the same way. Although the performance of rendering methods is increasing, many scenarios will remain inherently challenging to reconstruct faithfully. To this end, we propose a novel perspective for addressing the real-to-simulated data gap. Rather than solely focusing on improving rendering fidelity, we explore simple yet effective methods to enhance perception model robustness to NeRF artifacts without compromising performance on real data. Moreover, we conduct the first large-scale investigation into the real-to-simulated data gap in an AD setting using a state-of-the-art neural rendering technique. Specifically, we evaluate object detectors and an online mapping model on real and simulated data, and study the effects of different fine-tuning strategies.Our results show notable improvements in model robustness to simulated data, even improving real-world performance in some cases. Last, we delve into the correlation between the real-to-simulated gap and image reconstruction metrics, identifying FID and LPIPS as strong indicators. See https://research.zenseact.com/publications/closing-real2sim-gap for our project page.

CVDec 22, 2023
Transformer-Based Multi-Object Smoothing with Decoupled Data Association and Smoothing

Juliano Pinto, Georg Hess, Yuxuan Xia et al.

Multi-object tracking (MOT) is the task of estimating the state trajectories of an unknown and time-varying number of objects over a certain time window. Several algorithms have been proposed to tackle the multi-object smoothing task, where object detections can be conditioned on all the measurements in the time window. However, the best-performing methods suffer from intractable computational complexity and require approximations, performing suboptimally in complex settings. Deep learning based algorithms are a possible venue for tackling this issue but have not been applied extensively in settings where accurate multi-object models are available and measurements are low-dimensional. We propose a novel DL architecture specifically tailored for this setting that decouples the data association task from the smoothing task. We compare the performance of the proposed smoother to the state-of-the-art in different tasks of varying difficulty and provide, to the best of our knowledge, the first comparison between traditional Bayesian trackers and DL trackers in the smoothing problem setting.

SYDec 11, 2024
TGOSPA Metric Parameters Selection and Evaluation for Visual Multi-object Tracking

Jan Krejčí, Oliver Kost, Ondřej Straka et al.

Multi-object tracking algorithms are deployed in various applications, each with different performance requirements. For example, track switches pose significant challenges for offline scene understanding, as they hinder the accuracy of data interpretation. Conversely, in online surveillance applications, their impact is often minimal. This disparity underscores the need for application-specific performance evaluations that are both simple and mathematically sound. The trajectory generalized optimal sub-pattern assignment (TGOSPA) metric offers a principled approach to evaluate multi-object tracking performance. It accounts for localization errors, the number of missed and false objects, and the number of track switches, providing a comprehensive assessment framework. This paper illustrates the effective use of the TGOSPA metric in computer vision tasks, addressing challenges posed by the need for application-specific scoring methodologies. By exploring the TGOSPA parameter selection, we enable users to compare, comprehend, and optimize the performance of algorithms tailored for specific tasks, such as target tracking and training of detector or re-ID modules.

ROMay 1, 2025
Future-Oriented Navigation: Dynamic Obstacle Avoidance with One-Shot Energy-Based Multimodal Motion Prediction

Ze Zhang, Georg Hess, Junjie Hu et al.

This paper proposes an integrated approach for the safe and efficient control of mobile robots in dynamic and uncertain environments. The approach consists of two key steps: one-shot multimodal motion prediction to anticipate motions of dynamic obstacles and model predictive control to incorporate these predictions into the motion planning process. Motion prediction is driven by an energy-based neural network that generates high-resolution, multi-step predictions in a single operation. The prediction outcomes are further utilized to create geometric shapes formulated as mathematical constraints. Instead of treating each dynamic obstacle individually, predicted obstacles are grouped by proximity in an unsupervised way to improve performance and efficiency. The overall collision-free navigation is handled by model predictive control with a specific design for proactive dynamic obstacle avoidance. The proposed approach allows mobile robots to navigate effectively in dynamic environments. Its performance is accessed across various scenarios that represent typical warehouse settings. The results demonstrate that the proposed approach outperforms other existing dynamic obstacle avoidance methods.

CVApr 1, 2025
NeuRadar: Neural Radiance Fields for Automotive Radar Point Clouds

Mahan Rafidashti, Ji Lan, Maryam Fatemi et al.

Radar is an important sensor for autonomous driving (AD) systems due to its robustness to adverse weather and different lighting conditions. Novel view synthesis using neural radiance fields (NeRFs) has recently received considerable attention in AD due to its potential to enable efficient testing and validation but remains unexplored for radar point clouds. In this paper, we present NeuRadar, a NeRF-based model that jointly generates radar point clouds, camera images, and lidar point clouds. We explore set-based object detection methods such as DETR, and propose an encoder-based solution grounded in the NeRF geometry for improved generalizability. We propose both a deterministic and a probabilistic point cloud representation to accurately model the radar behavior, with the latter being able to capture radar's stochastic behavior. We achieve realistic reconstruction results for two automotive datasets, establishing a baseline for NeRF-based radar point cloud simulation models. In addition, we release radar data for ZOD's Sequences and Drives to enable further research in this field. To encourage further development of radar NeRFs, we release the source code for NeuRadar.

QMFeb 19, 2025
Optimizing Gene-Based Testing for Antibiotic Resistance Prediction

David Hagerman, Anna Johnning, Roman Naeem et al.

Antibiotic Resistance (AR) is a critical global health challenge that necessitates the development of cost-effective, efficient, and accurate diagnostic tools. Given the genetic basis of AR, techniques such as Polymerase Chain Reaction (PCR) that target specific resistance genes offer a promising approach for predictive diagnostics using a limited set of key genes. This study introduces GenoARM, a novel framework that integrates reinforcement learning (RL) with transformer-based models to optimize the selection of PCR gene tests and improve AR predictions, leveraging observed metadata for improved accuracy. In our evaluation, we developed several high-performing baselines and compared them using publicly available datasets derived from real-world bacterial samples representing multiple clinically relevant pathogens. The results show that all evaluated methods achieve strong and reliable performance when metadata is not utilized. When metadata is introduced and the number of selected genes increases, GenoARM demonstrates superior performance due to its capacity to approximate rewards for unseen and sparse combinations. Overall, our framework represents a major advancement in optimizing diagnostic tools for AR in clinical settings.

CVDec 5, 2024
MVUDA: Unsupervised Domain Adaptation for Multi-view Pedestrian Detection

Erik Brorsson, Lennart Svensson, Kristofer Bengtsson et al.

We address multi-view pedestrian detection in a setting where labeled data is collected using a multi-camera setup different from the one used for testing. While recent multi-view pedestrian detectors perform well on the camera rig used for training, their performance declines when applied to a different setup. To facilitate seamless deployment across varied camera rigs, we propose an unsupervised domain adaptation (UDA) method that adapts the model to new rigs without requiring additional labeled data. Specifically, we leverage the mean teacher self-training framework with a novel pseudo-labeling technique tailored to multi-view pedestrian detection. This method achieves state-of-the-art performance on multiple benchmarks, including MultiviewX$\rightarrow$Wildtrack. Unlike previous methods, our approach eliminates the need for external labeled monocular datasets, thereby reducing reliance on labeled data. Extensive evaluations demonstrate the effectiveness of our method and validate key design choices. By enabling robust adaptation across camera setups, our work enhances the practicality of multi-view pedestrian detectors and establishes a strong UDA baseline for future research.

CVNov 24, 2025
IDSplat: Instance-Decomposed 3D Gaussian Splatting for Driving Scenes

Carl Lindström, Mahan Rafidashti, Maryam Fatemi et al.

Reconstructing dynamic driving scenes is essential for developing autonomous systems through sensor-realistic simulation. Although recent methods achieve high-fidelity reconstructions, they either rely on costly human annotations for object trajectories or use time-varying representations without explicit object-level decomposition, leading to intertwined static and dynamic elements that hinder scene separation. We present IDSplat, a self-supervised 3D Gaussian Splatting framework that reconstructs dynamic scenes with explicit instance decomposition and learnable motion trajectories, without requiring human annotations. Our key insight is to model dynamic objects as coherent instances undergoing rigid transformations, rather than unstructured time-varying primitives. For instance decomposition, we employ zero-shot, language-grounded video tracking anchored to 3D using lidar, and estimate consistent poses via feature correspondences. We introduce a coordinated-turn smoothing scheme to obtain temporally and physically consistent motion trajectories, mitigating pose misalignments and tracking failures, followed by joint optimization of object poses and Gaussian parameters. Experiments on the Waymo Open Dataset demonstrate that our method achieves competitive reconstruction quality while maintaining instance-level decomposition and generalizes across diverse sequences and view densities without retraining, making it practical for large-scale autonomous driving applications. Code will be released.

SYAug 19, 2025
Model-based Multi-object Visual Tracking: Identification and Standard Model Limitations

Jan Krejčí, Oliver Kost, Yuxuan Xia et al.

This paper uses multi-object tracking methods known from the radar tracking community to address the problem of pedestrian tracking using 2D bounding box detections. The standard point-object (SPO) model is adopted, and the posterior density is computed using the Poisson multi-Bernoulli mixture (PMBM) filter. The selection of the model parameters rooted in continuous time is discussed, including the birth and survival probabilities. Some parameters are selected from the first principles, while others are identified from the data, which is, in this case, the publicly available MOT-17 dataset. Although the resulting PMBM algorithm yields promising results, a mismatch between the SPO model and the data is revealed. The model-based approach assumes that modifying the problematic components causing the SPO model-data mismatch will lead to better model-based algorithms in future developments.

CVJul 18, 2025
GOSPA and T-GOSPA quasi-metrics for evaluation of multi-object tracking algorithms

Ángel F. García-Fernández, Jinhao Gu, Lennart Svensson et al.

This paper introduces two quasi-metrics for performance assessment of multi-object tracking (MOT) algorithms. In particular, one quasi-metric is an extension of the generalised optimal subpattern assignment (GOSPA) metric and measures the discrepancy between sets of objects. The other quasi-metric is an extension of the trajectory GOSPA (T-GOSPA) metric and measures the discrepancy between sets of trajectories. Similar to the GOSPA-based metrics, these quasi-metrics include costs for localisation error for properly detected objects, the number of false objects and the number of missed objects. The T-GOSPA quasi-metric also includes a track switching cost. Differently from the GOSPA and T-GOSPA metrics, the proposed quasi-metrics have the flexibility of penalising missed and false objects with different costs, and the localisation costs are not required to be symmetric. These properties can be useful in MOT evaluation in certain applications. The performance of several Bayesian MOT algorithms is assessed with the T-GOSPA quasi-metric via simulations.

SIJun 18, 2025
A family of graph GOSPA metrics for graphs with different sizes

Jinhao Gu, Ángel F. García-Fernández, Robert E. Firth et al.

This paper proposes a family of graph metrics for measuring distances between graphs of different sizes. The proposed metric family defines a general form of the graph generalised optimal sub-pattern assignment (GOSPA) metric and is also proved to satisfy the metric properties. Similarly to the graph GOSPA metric, the proposed graph GOSPA metric family also penalises the node attribute costs for assigned nodes between the two graphs, and the number of unassigned nodes. However, the proposed family of metrics provides more general penalties for edge mismatches than the graph GOSPA metric. This paper also shows that the graph GOSPA metric family can be approximately computed using linear programming. Simulation experiments are performed to illustrate the characteristics of the proposed graph GOSPA metric family with different choices of hyperparameters. The benefits of the proposed graph GOSPA metric family for classification tasks are also shown on real-world datasets.

MLFeb 26, 2024
On the connection between Noise-Contrastive Estimation and Contrastive Divergence

Amanda Olmin, Jakob Lindqvist, Lennart Svensson et al.

Noise-contrastive estimation (NCE) is a popular method for estimating unnormalised probabilistic models, such as energy-based models, which are effective for modelling complex data distributions. Unlike classical maximum likelihood (ML) estimation that relies on importance sampling (resulting in ML-IS) or MCMC (resulting in contrastive divergence, CD), NCE uses a proxy criterion to avoid the need for evaluating an often intractable normalisation constant. Despite apparent conceptual differences, we show that two NCE criteria, ranking NCE (RNCE) and conditional NCE (CNCE), can be viewed as ML estimation methods. Specifically, RNCE is equivalent to ML estimation combined with conditional importance sampling, and both RNCE and CNCE are special cases of CD. These findings bridge the gap between the two method classes and allow us to apply techniques from the ML-IS and CD literature to NCE, offering several advantageous extensions.

AIMay 5, 2023
Set-Type Belief Propagation with Applications to Poisson Multi-Bernoulli SLAM

Hyowon Kim, Angel F. García-Fernández, Yu Ge et al.

Belief propagation (BP) is a useful probabilistic inference algorithm for efficiently computing approximate marginal probability densities of random variables. However, in its standard form, BP is only applicable to the vector-type random variables with a fixed and known number of vector elements, while certain applications rely on RFSs with an unknown number of vector elements. In this paper, we develop BP rules for factor graphs defined on sequences of RFSs where each RFS has an unknown number of elements, with the intention of deriving novel inference methods for RFSs. Furthermore, we show that vector-type BP is a special case of set-type BP, where each RFS follows the Bernoulli process. To demonstrate the validity of developed set-type BP, we apply it to the PMB filter for SLAM, which naturally leads to new set-type BP-mapping, SLAM, multi-target tracking, and simultaneous localization and tracking filters. Finally, we explore the relationships between the vector-type BP and the proposed set-type BP PMB-SLAM implementations and show a performance gain of the proposed set-type BP PMB-SLAM filter in comparison with the vector-type BP-SLAM filter.

LGFeb 16, 2022
Can Deep Learning be Applied to Model-Based Multi-Object Tracking?

Juliano Pinto, Georg Hess, William Ljungbergh et al.

Multi-object tracking (MOT) is the problem of tracking the state of an unknown and time-varying number of objects using noisy measurements, with important applications such as autonomous driving, tracking animal behavior, defense systems, and others. In recent years, deep learning (DL) has been increasingly used in MOT for improving tracking performance, but mostly in settings where the measurements are high-dimensional and there are no available models of the measurement likelihood and the object dynamics. The model-based setting instead has not attracted as much attention, and it is still unclear if DL methods can outperform traditional model-based Bayesian methods, which are the state of the art (SOTA) in this context. In this paper, we propose a Transformer-based DL tracker and evaluate its performance in the model-based setting, comparing it to SOTA model-based Bayesian methods in a variety of different tasks. Our results show that the proposed DL method can match the performance of the model-based methods in simple tasks, while outperforming them when the task gets more complicated, either due to an increase in the data association complexity, or to stronger nonlinearities of the models of the environment.

SPDec 5, 2021
Iterated Posterior Linearization PMB Filter for 5G SLAM

Yu Ge, Yibo Wu, Fan Jiang et al.

5G millimeter wave (mmWave) signals have inherent geometric connections to the propagation channel and the propagation environment. Thus, they can be used to jointly localize the receiver and map the propagation environment, which is termed as simultaneous localization and mapping (SLAM). One of the most important tasks in the 5G SLAM is to deal with the nonlinearity of the measurement model. To solve this problem, existing 5G SLAM approaches rely on sigma-point or extended Kalman filters, linearizing the measurement function with respect to the prior probability density function (PDF). In this paper, we study the linearization of the measurement function with respect to the posterior PDF, and implement the iterated posterior linearization filter into the Poisson multi-Bernoulli SLAM filter. Simulation results demonstrate the accuracy and precision improvements of the resulting SLAM filter.

CVOct 26, 2021
A time-weighted metric for sets of trajectories to assess multi-object tracking algorithms

Ángel F. García-Fernández, Abu Sajana Rahmathullah, Lennart Svensson

This paper proposes a metric for sets of trajectories to evaluate multi-object tracking algorithms that includes time-weighted costs for localisation errors of properly detected targets, for false targets, missed targets and track switches. The proposed metric extends the metric in [1] by including weights to the costs associated to different time steps. The time-weighted costs increase the flexibility of the metric [1] to fit more applications and user preferences. We first introduce a metric based on multi-dimensional assignments, and then its linear programming relaxation, which is computable in polynomial time and is also a metric. The metrics can also be extended to metrics on random finite sets of trajectories to evaluate and rank algorithms across different scenarios, each with a ground truth set of trajectories.

CVSep 2, 2021
Extended Object Tracking Using Sets Of Trajectories with a PHD Filter

Jakob Sjudin, Martin Marcusson, Lennart Svensson et al.

PHD filtering is a common and effective multiple object tracking (MOT) algorithm used in scenarios where the number of objects and their states are unknown. In scenarios where each object can generate multiple measurements per scan, some PHD filters can estimate the extent of the objects as well as their kinematic properties. Most of these approaches are, however, not able to inherently estimate trajectories and rely on ad-hoc methods, such as different labeling schemes, to build trajectories from the state estimates. This paper presents a Gamma Gaussian inverse Wishart mixture PHD filter that can directly estimate sets of trajectories of extended targets by expanding previous research on tracking sets of trajectories for point source objects to handle extended objects. The new filter is compared to an existing extended PHD filter that uses a labeling scheme to build trajectories, and it is shown that the new filter can estimate object trajectories more reliably.

MENov 9, 2020
A Poisson multi-Bernoulli mixture filter for coexisting point and extended targets

Ángel F. García-Fernández, Jason L. Williams, Lennart Svensson et al.

This paper proposes a Poisson multi-Bernoulli mixture (PMBM) filter for coexisting point and extended targets, i.e., for scenarios where there may be simultaneous point and extended targets. The PMBM filter provides a recursion to compute the multi-target filtering posterior based on probabilistic information on data associations, and single-target predictions and updates. In this paper, we first derive the PMBM filter update for a generalised measurement model, which can include measurements originated from point and extended targets. Second, we propose a single-target space that accommodates both point and extended targets and derive the filtering recursion that propagates Gaussian densities for point targets and gamma Gaussian inverse Wishart densities for extended targets. As a computationally efficient approximation of the PMBM filter, we also develop a Poisson multi-Bernoulli (PMB) filter for coexisting point and extended targets. The resulting filters are analysed via numerical simulations.

CVJul 17, 2020
DACS: Domain Adaptation via Cross-domain Mixed Sampling

Wilhelm Tranheden, Viktor Olsson, Juliano Pinto et al.

Semantic segmentation models based on convolutional neural networks have recently displayed remarkable performance for a multitude of applications. However, these models typically do not generalize well when applied on new domains, especially when going from synthetic to real data. In this paper we address the problem of unsupervised domain adaptation (UDA), which attempts to train on labelled data from one domain (source domain), and simultaneously learn from unlabelled data in the domain of interest (target domain). Existing methods have seen success by training on pseudo-labels for these unlabelled images. Multiple techniques have been proposed to mitigate low-quality pseudo-labels arising from the domain shift, with varying degrees of success. We propose DACS: Domain Adaptation via Cross-domain mixed Sampling, which mixes images from the two domains along with the corresponding labels and pseudo-labels. These mixed samples are then trained on, in addition to the labelled data itself. We demonstrate the effectiveness of our solution by achieving state-of-the-art results for GTA5 to Cityscapes, a common synthetic-to-real semantic segmentation benchmark for UDA.

CVJul 15, 2020
ClassMix: Segmentation-Based Data Augmentation for Semi-Supervised Learning

Viktor Olsson, Wilhelm Tranheden, Juliano Pinto et al.

The state of the art in semantic segmentation is steadily increasing in performance, resulting in more precise and reliable segmentations in many different applications. However, progress is limited by the cost of generating labels for training, which sometimes requires hours of manual labor for a single image. Because of this, semi-supervised methods have been applied to this task, with varying degrees of success. A key challenge is that common augmentations used in semi-supervised classification are less effective for semantic segmentation. We propose a novel data augmentation mechanism called ClassMix, which generates augmentations by mixing unlabelled samples, by leveraging on the network's predictions for respecting object boundaries. We evaluate this augmentation technique on two common semi-supervised semantic segmentation benchmarks, showing that it attains state-of-the-art results. Lastly, we also provide extensive ablation studies comparing different design decisions and training regimes.

CVMar 28, 2020
Trajectory Poisson multi-Bernoulli filters

Ángel F. García-Fernández, Lennart Svensson, Jason L. Williams et al.

This paper presents two trajectory Poisson multi-Bernoulli (TPMB) filters for multi-target tracking: one to estimate the set of alive trajectories at each time step and another to estimate the set of all trajectories, which includes alive and dead trajectories, at each time step. The filters are based on propagating a Poisson multi-Bernoulli (PMB) density on the corresponding set of trajectories through the filtering recursion. After the update step, the posterior is a PMB mixture (PMBM) so, in order to obtain a PMB density, a Kullback-Leibler divergence minimisation on an augmented space is performed. The developed filters are computationally lighter alternatives to the trajectory PMBM filters, which provide the closed-form recursion for sets of trajectories with Poisson birth model, and are shown to outperform previous multi-target tracking algorithms.

MLFeb 26, 2020
A general framework for ensemble distribution distillation

Jakob Lindqvist, Amanda Olmin, Fredrik Lindsten et al.

Ensembles of neural networks have been shown to give better performance than single networks, both in terms of predictions and uncertainty estimation. Additionally, ensembles allow the uncertainty to be decomposed into aleatoric (data) and epistemic (model) components, giving a more complete picture of the predictive uncertainty. Ensemble distillation is the process of compressing an ensemble into a single model, often resulting in a leaner model that still outperforms the individual ensemble members. Unfortunately, standard distillation erases the natural uncertainty decomposition of the ensemble. We present a general framework for distilling both regression and classification ensembles in a way that preserves the decomposition. We demonstrate the desired behaviour of our framework and show that its predictive performance is on par with standard distillation.

SPDec 17, 2019
Poisson Multi-Bernoulli Mixtures for Sets of Trajectories

Karl Granström, Lennart Svensson, Yuxuan Xia et al.

The Poisson Multi-Bernoulli Mixture (PMBM) density is a conjugate multi-target density for the standard point target model with Poisson point process birth. This means that both the filtering and predicted densities for the set of targets are PMBM. In this paper, we first show that the PMBM density is also conjugate for sets of trajectories with the standard point target measurement model. Second, based on this theoretical foundation, we develop two trajectory PMBM filters that provide recursions to calculate the posterior density for the set of all trajectories that have ever been present in the surveillance area, and the posterior density of the set of trajectories present at the current time step in the surveillance area. These two filters therefore provide complete probabilistic information on the considered trajectories enabling optimal trajectory estimation. Third, we establish that the density of the set of trajectories in any time window, given the measurements in a possibly different time window, is also a PMBM. Finally, the trajectory PMBM filters are evaluated via simulations, and are shown to yield state-of-the-art performance compared to other multi-target tracking algorithms based on random finite sets and multiple hypothesis tracking.

LGDec 14, 2019
Bayesian Linear Regression on Deep Representations

John Moberg, Lennart Svensson, Juliano Pinto et al.

A simple approach to obtaining uncertainty-aware neural networks for regression is to do Bayesian linear regression (BLR) on the representation from the last hidden layer. Recent work [Riquelme et al., 2018, Azizzadenesheli et al., 2018] indicates that the method is promising, though it has been limited to homoscedastic noise. In this paper, we propose a novel variation that enables the method to flexibly model heteroscedastic noise. The method is benchmarked against two prominent alternative methods on a set of standard datasets, and finally evaluated as an uncertainty-aware model in model-based reinforcement learning. Our experiments indicate that the method is competitive with standard ensembling, and ensembles of BLR outperforms the methods we compared to.

CVNov 28, 2019
Lidar-Camera Co-Training for Semi-Supervised Road Detection

Luca Caltagirone, Lennart Svensson, Mattias Wahde et al.

Recent advances in the field of machine learning and computer vision have enabled the development of fast and accurate road detectors. Commonly such systems are trained within a supervised learning paradigm where both an input sensor's data and the corresponding ground truth label must be provided. The task of generating labels is commonly carried out by human annotators and it is notoriously time consuming and expensive. In this work, it is shown that a semi-supervised approach known as co-training can provide significant F1-score average improvements compared to supervised learning. In co-training, two classifiers acting on different views of the data cooperatively improve each other's performance by leveraging unlabeled examples. Depending on the amount of labeled data used, the improvements ranged from 1.12 to 6.10 percentage points for a camera-based road detector and from 1.04 to 8.14 percentage points for a lidar-based road detector. Lastly, the co-training algorithm is validated on the KITTI road benchmark, achieving high performance using only 36 labeled training examples together with several thousands unlabeled ones.

SPAug 23, 2019
Gaussian implementation of the multi-Bernoulli mixture filter

Ángel F. García-Fernández, Yuxuan Xia, Karl Granström et al.

This paper presents the Gaussian implementation of the multi-Bernoulli mixture (MBM) filter. The MBM filter provides the filtering (multi-target) density for the standard dynamic and radar measurement models when the birth model is multi-Bernoulli or multi-Bernoulli mixture. Under linear/Gaussian models, the single target densities of the MBM mixture admit Gaussian closed-form expressions. Murty's algorithm is used to select the global hypotheses with highest weights. The MBM filter is compared with other algorithms in the literature via numerical simulations.

SPAug 23, 2019
Spooky effect in optimal OSPA estimation and how GOSPA solves it

Ángel F. García-Fernández, Lennart Svensson

In this paper, we show the spooky effect at a distance that arises in optimal estimation of multiple targets with the optimal sub-pattern assignment (OSPA) metric. This effect refers to the fact that if we have several independent potential targets at distant locations, a change in the probability of existence of one of them can completely change the optimal estimation of the rest of the potential targets. As opposed to OSPA, the generalised OSPA (GOSPA) metric ($α=2$) penalises localisation errors for properly detected targets, false targets and missed targets. As a consequence, optimal GOSPA estimation aims to lower the number of false and missed targets, as well as the localisation error for properly detected targets, and avoids the spooky effect.

CVNov 21, 2018
Trajectory PHD and CPHD filters

Ángel F. García-Fernández, Lennart Svensson

This paper presents the probability hypothesis density filter (PHD) and the cardinality PHD (CPHD) filter for sets of trajectories, which are referred to as the trajectory PHD (TPHD) and trajectory CPHD (TCPHD) filters. Contrary to the PHD/CPHD filters, the TPHD/TCPHD filters are able to produce trajectory estimates from first principles. The TPHD filter is derived by recursively obtaining the best Poisson multitrajectory density approximation to the posterior density over the alive trajectories by minimising the Kullback-Leibler divergence. The TCPHD is derived in the same way but propagating an independent identically distributed (IID) cluster multitrajectory density approximation. We also propose the Gaussian mixture implementations of the TPHD and TCPHD recursions, the Gaussian mixture TPHD (GMTPHD) and the Gaussian mixture TCPHD (GMTCPHD), and the L-scan computationally efficient implementations, which only update the density of the trajectory states of the last L time steps.