Factor Graph Framework Research Articles

A Novel Factor Graph Framework for Tightly Coupled GNSS/INS Integration With Carrier-Phase Ambiguity Resolution

A Novel Factor Graph Framework for Tightly Coupled GNSS/INS Integration With Carrier-Phase Ambiguity Resolution

An Enhanced Adaptable Factor Graph for Simultaneous Localization and Calibration in GNSS/IMU/Odometer Integration

The integration of global navigation satellite system (GNSS), inertial measurement unit (IMU) and odometer has been widely utilized in vehicular localization. Due to the advantage in accessing accurate state estimation, factor graph is commonly employed to achieve the integration of vehicular sensors. However, the fusion performance of traditional factor graph reduces when dealing with asynchronous and abnormal measurements, yielding compromised estimation results for localization and calibration. In this paper, an enhanced adaptable factor graph for simultaneous localization and calibration (SLAC) in GNSS/IMU/odometer integration is proposed. First, a novel factor graph framework for GNSS/IMU/odometer integration is designed. In view of the asynchronism among sensors, the variable nodes in factor graph are built according to odometer measurement. IMU pre-integration is then used to establish the relationship between asynchronous GNSS position and corresponding variable nodes to form residuals. To properly deal with the coupling noise in the proposed model, the expectation maximum (EM) algorithm is utilized to estimate the noise parameters based on the proposed model. Simulations and field tests are done to validate the proposed method. It shows that the proposed method has better estimation accuracy for localization and calibration compared with traditional methods. Meanwhile, the computational load of the proposed method is discussed.

A Novel Adaptive Sliding Observation-Based Cooperative Positioning Algorithm Under Factor Graph Framework for Multiple UUVs

A Novel Adaptive Sliding Observation-Based Cooperative Positioning Algorithm Under Factor Graph Framework for Multiple UUVs

Factor Graph Framework for Smartphone Indoor Localization: Integrating Data-Driven PDR and Wi-Fi RTT/RSS Ranging

The classic fusion localization techniques based on the Kalman filter (KF) framework have been a focus of research community in the past decades, due to the limited computing power of the mobile devices. However, with computing-efficient and sensor-rich smartphones now being a commonplace, it is convenient and meaningful to provide more accurate positioning services for smartphones in an indoor environment. In this paper, we design and develop a tightly coupled fusion platform of Wi-Fi RTT, RSS, and data-driven PDR based on factor graph optimization for locating the consumer-grade smartphones in indoor environment. As compared to the existing PDR solutions, including step model-based approaches and data-driven approaches, the proposed PDR solution with magnetic information constraint can track the relative position change of pedestrians at 20 Hz, while supporting multiple smartphone usage poses. A comprehensive comparison between factor graph optimization (FGO), KF frame and its variants is also performed. The experimental results demonstrate that the proposed fusion platform achieves an average positioning accuracy of 0.39 m. In addition, it also improves the accuracy of EFK and ARKF by 45.83% and 27.78%, respectively. The analysis shows that as the smartphone computing performance continues to improve, the FGO-based sensor fusion gradually replaces the KF frame and its variants.

GIVE: A Tightly Coupled RTK-Inertial–Visual State Estimator for Robust and Precise Positioning

Continuous and reliable high-precision positioning in a wide area is an important task for autonomous driving. In this paper, we present GIVE, a nonlinear optimization-based GNSS-inertial-visual (GIV) state estimator that tightly fuses multi-GNSS raw carrier phase and pseudorange observations with inertial and visual information to address the problem of continuous and reliable centimeter-level positioning in urban canyons where GNSS signal may be intermittent or even outage. The primary contribution of this work is the investigation of the high-precision carrier phase observations in terms of continuity and ambiguity resolution (AR) in GIV fusion. In particular, first we propose a carrier phase-enhanced bundle adjustment (CPBA) method for accurate and drift-free GIV state estimation, in which the continuously tracked carrier phase arcs are used to establish the ambiguity constraints of current and historical states which have common-view satellites. The common-view ambiguity constraints are jointly optimized with IMU pre-integration constraints and visual reprojection constraints under the factor graph framework. Furthermore, we develop a marginalization-based carrier phase ambiguity propagation (AP) method to leverage the continuity of the carrier phase and achieve an optimal estimation of the float ambiguity, which is essential for the correct AR of the carrier phase. An integer AR module, in combination with the proposed CPBA, enables the centimeter-level positioning accuracy of the GIV framework. We extensively evaluate the proposed method in both public datasets and challenging datasets collected in deep urban environments. Statistics show that the GIVE can provide smooth centimeter-level positioning results in open-sky conditions. Two challenging urban driving tests demonstrate the GIVE can achieve continuous centimeter to decimeter-level positioning accuracy and obtain 86.5-94.1% 2D positioning availability (<10 cm) and more than 94% 3D positioning availability (<30 cm) in deep urban environments.

Low-Complexity Near-Optimum Symbol Detection Based on Neural Enhancement of Factor Graphs

We consider the application of the factor graph framework for symbol detection on linear inter-symbol interference channels. Based on the Ungerboeck observation model, a detection algorithm with appealing complexity properties can be derived. However, since the underlying factor graph contains cycles, the sum-product algorithm (SPA) yields a suboptimal algorithm. In this paper, we develop and evaluate efficient strategies to improve the performance of the factor graph-based symbol detection by means of neural enhancement. In particular, we consider neural belief propagation and generalizations of the factor nodes as an effective way to mitigate the effect of cycles within the factor graph. By applying a generic preprocessor to the channel output, we propose a simple technique to vary the underlying factor graph in every SPA iteration. Using this dynamic factor graph transition, we intend to preserve the extrinsic nature of the SPA messages which is otherwise impaired due to cycles. Simulation results show that the proposed methods can massively improve the detection performance, even approaching the maximum a posteriori performance for various transmission scenarios, while preserving a complexity which is linear in both the block length and the channel memory.

Fault Detection of Resilient Navigation System Based on GNSS Pseudo-Range Measurement

Because of the resilient frame structure, the factor graph is often used in navigation systems to solve the sensor asynchrony problem and realize plug-and-play effectively in the navigation information fusion method. To improve the fault detection performance of resilient integrated navigation systems under complex interference environments, a fault detection method in factor graph navigation framework based on INS measurements and GNSS pseudo-range measurements is proposed in this paper. The proposed method can effectively locate the fault satellite pseudo-range information based on the Chi-square fault detection method. Due to the plug-and-play characteristic of the factor graph framework, this method can quickly isolate faults to improve navigation accuracy. Finally, the effect of the method is verified by simulation and experiment. Compared with the Chi-square fault detection method, positioning accuracy is improved by more than 40%.

A factor graph optimization mapping based on normaldistributions transform

This paper aims to achieve highly accurate mapping results and real time pose estimation of autonomous vehicle by using the normal distribution transform (NDT) algoritm. A factor graph optimization algorithm (FGO-NDT) is proposed to address the poor real-time performance and pose drift errors of the NDT algorithm. Smooth point cloud data are obtained by multisensor calibration and data preprocessing. NDT registration is then used for lidar odometry and feature matching. The global navigation satellite system (GNSS) data and loop detection results are added to the factor graph framework as the pose constraint factors to optimize the pose trajectory and eliminate the pose drift error generated during mapping. In addition, a sliding window method is used for map registration to extract a local map to shorten the map loading time. Thus, the real-time performance of creating point cloud maps of large scenes is significantly improved. Several experiments are conducted in different environments to verify the accuracy and performance of the FGO-NDT. The experimental results demonstrate that the proposed method can eliminate the pose estimation error caused by drift, improve the local structure, and reduce and root mean square error.

Loosely Coupled GNSS/INS Integration Based on Factor Graph and Aided by ARIMA Model

The integration of global navigation satellite system (GNSS) and inertial navigation system (INS) has been widely studied in the past decades. The traditional Kalman filtering method, a procedure that leads to historical states and observations loss, can estimate fast by limiting the update to the latest state. Furthermore, the performance of loosely coupled GNSS/INS navigation integration depends largely on environmental conditions and sensor costs. In cities, GNSS signals may be interrupted due to the influence of obstructions and moving objects, which leads to the discontinuity of GNSS/INS navigation. Therefore, in order to overcome these difficulties an auto regressive integrated moving average (ARIMA) auxiliary model based on time sequence is proposed in this paper, which makes use of the data before interruption to predict the GNSS measurements during the interruption period and makes up for the data gap. In addition, the INS and GNSS measurements are loosely coupled using the most advanced factor graph model. Compared to the filtering algorithm, all previous measurements are used for state estimation through the factor graph framework. In this paper, the experiment is based on the dataset of Tokyo which is a typical challenging city canyon. Experiments showed that, compared with the traditional GNSS/INS integration method, the proposed method can provide a higher precision and a series of continuous navigation results even when the GNSS measurement data was interrupted.

Probabilistic model-based fault diagnosis for the cavities of the European XFEL

Abstract The European X-ray Free Electron Laser (EuXFEL) is a complex system with many interconnected components and sensor measurements. We use factor graphs to systematically design a probabilistic fault diagnosis method for its cavity system. This approach is expandable to further subsystems and considers uncertainties from measurements and modeling. After representing a model of the cavity system in the factor graph framework, we infer marginal distributions, e. g., of the fault classes using tabulated message-passing definitions. The emerging fault diagnosis method consists of an unscented Kalman filter-based residual generator and an evaluation of the residuals using a Gaussian mixture model. We include message-passing definitions for the training of the Gaussian Mixture Model from noisy data using the expectation-maximization algorithm.

AOA-Based Three-Dimensional Positioning and Tracking Using the Factor Graph Technique

In this paper, an angle-of-arrival (AOA)-based algorithm is proposed for tracking the position of an anonymous target in three-dimensional (3D) space. Distributed sensors are deployed, which can measure both the azimuth and elevation angles of the AOAs. Assuming the target movement is non-linear, the extended Kalman filter (EKF) is applied, where the observation process is realized by a practical AOA-based position detector, to form a unified factor graph (FG) framework. Moreover, the variance of observation errors, which is needed by EKF, is estimated in real time by using both the AOA measurements and the predicted target state. Such a dynamic estimating approach exhibits higher performance robustness compared to the conventional method, especially when the sensing environment is unstable. Additionally, the predicted target state is also used as the a priori information of the system, in order to reduce the impacts of burst sensing errors. According to the simulations, the proposed system is shown to achieve less root mean squared errors (RMSE) in different evaluation scenarios, with fast convergence behavior.

Iterative Joint Channel Estimation, User Activity Tracking, and Data Detection for FTN-NOMA Systems Supporting Random Access

Given the requirements of increased data rate and massive connectivity in the Internet-of-things (IoT) applications of the fifth-generation communication systems (5G), non-orthogonal multiple access (NOMA) was shown to be capable of supporting more users than OMA. As a further potential enhancement, the faster-than-Nyquist (FTN) signaling is also capable of increasing the symbol rate. Since NOMA and FTN signaling impose non-orthogonalities from different perspectives, it is possible to achieve further increased spectral efficiency by exploiting both. Hence we investigate the FTN-NOMA uplink in the context of random access. Although random access schemes reduce the signaling overheads as well as latency, they require the base station to identify active users before performing data detection. As both inter-symbol and inter-user interferences exist, performing optimal detection requires a prohibitively high complexity. Moreover, in typical mobile communication environments, the channel envelope of users fluctuates violently, which imposes challenges on the receiver design. To tackle this problem, we propose a joint user activity tracking and data detection algorithm based on the factor graph framework, which relies on a sophisticated amalgam of expectation maximization (EM) and hybrid message passing algorithms. The complexity of the algorithm advocated only increases linearly with the number of active users. Our simulation results show that the proposed algorithm is effective in tracking user activity and detecting data symbols in dynamic random access systems.

Integrated Factor Graph Algorithm for DOA-Based Geolocation and Tracking

This paper proposes a new position tracking algorithm by integrating extended Kalman filter (EKF) and direction-of-arrival (DOA)-based geolocation into one factor graph (FG) framework. A distributed sensor network is assumed for detecting an anonymous target, where the process and observation equations in the state space model (SSM) are unknown. Importantly, the predicted state information can be utilized not only for filtering, but also for enhancing the observation process. To be specific, by taking the prediction into account as the a priori, a new FG scheme is proposed for GEolocation, denoted by FG-GE. The benefits are two-fold, compared with the conventional geolocation scheme which does not rely on the a priori information. First of all, significant performance improvement can be observed, in terms of the root mean square error (RMSE), when severe sensing errors are suddenly encountered. Furthermore, the proposed FG-GE can achieve dramatic reduction of computational complexity. In addition, this paper also proposes the use of a predicted Cramer-Rao lower bound (P-CRLB) to dynamically estimate the observation error variance, which demonstrates more robust tracking performance than that with only fixed average variance approximation.

Iterative Receiver Design for FTN Signaling Aided Sparse Code Multiple Access

The sparse code multiple access (SCMA) is a promising candidate for bandwidth-efficient next generation wireless communications, since it can support more users than the number of resource elements. On the same note, faster-than-Nyquist (FTN) signaling can also be used to improve the spectral efficiency. Hence in this paper, we consider a combined uplink FTN-SCMA system in which the data symbols corresponding to a user are further packed using FTN signaling. As a result, a higher spectral efficiency is achieved at the cost of introducing intentional inter-symbol interference (ISI). To perform joint channel estimation and detection, we design a low complexity iterative receiver based on the factor graph framework. In addition, to reduce the signaling overhead and transmission latency of our SCMA system, we intrinsically amalgamate it with grant-free scheme. Consequently, the active and inactive users should be distinguished. To address this problem, we extend the aforementioned receiver and develop a new algorithm for jointly estimating the channel state information, detecting the user activity and for performs data detection. In order to further reduce the complexity, an energy minimization based approximation is employed for restricting the user state to Gaussian. Finally, a hybrid message passing algorithm is conceived. Our Simulation results show that the FTN-SCMA system relying on the proposed receiver design has a higher throughput than conventional SCMA scheme at a negligible performance loss.

Iterative Approximate Nonlinear Inference via Gaussian Message Passing on Factor Graphs

Factor graphs are graphical models able to represent the factorization of probability density functions. By visualizing conditional independence statements, they provide an intuitive and versatile interface to sparsity exploiting message passing algorithms as a unified framework for constructing algorithms in signal processing, estimation, and control in a mix-and-match style. Especially, when assuming Gaussian distributed variables, tabulated message passing rules allow for easy automated derivations of algorithms. This letter’s contribution consists in the combination of statistical or Jacobian-based linearization approaches to handling nonlinear factors with efficient message parametrizations in a Gaussian message passing setting. Tabulated message passing rules for a multivariate nonlinear factor node are presented that implement a re-linearization about the most current belief (marginal) of each adjacent variable. When utilized in a nonlinear Kalman smoothing setting, the iterated nonlinear modified Bryson–Frazier smoother is recovered, while retaining the flexibility of the factor graph framework. This application is illustrated by deriving an input estimation algorithm for a nonlinear system.

Indoor 3-D Localization Based on Received Signal Strength Difference and Factor Graph for Unknown Radio Transmitter.

Accurate localization of the radio transmitter is an important work in radio management. Previous research is more focused on two-dimensional (2-D) scenarios, but the localization of an unknown radio transmitter under three-dimensional (3-D) scenarios has more practical significance. In this paper, we propose a novel 3-D localization algorithm with received signal strength difference (RSSD) information and factor graph (FG), which is suitable for both line-of-sight (LOS) and non-line-of-sight (NLOS) condition. Considering the stochastic properties of measurement errors caused by the indoor environment, RSSD measurements are processed with mean and variance in the form of Gaussian distribution in the FG framework. A new 3-D RSSD-based FG model is constructed with the relationship between RSSD and location coordinates by local linearization technique. The soft-information computation and iterative process of the proposed model are derived by using the sum-product algorithm. In addition, the impacts of different grid distances and number of signal receivers on positioning accuracy are explored. Finally, the performance of our proposed approach is experimentally evaluated in a real scenario. The results show that the positioning performance of the proposed algorithm is not only superior to the k-nearest neighbors (kNN) algorithm and least square (LS) algorithm, but also it can achieve a mean localization error as low as 1.15 m. Our proposed scheme provides a good solution for the accurate detection of an unknown radio transmitter under indoor 3-D space and has a good application prospect.

Cooperative Simultaneous Localization and Synchronization in Mobile Agent Networks

Cooperative localization in agent networks based on interagent time-of-flight measurements is closely related to synchronization. To leverage this relation, we propose a Bayesian factor graph framework for cooperative simultaneous localization and synchronization (CoSLAS). This framework is suited to mobile agents and time-varying local clock parameters. Building on the CoSLAS factor graph, we develop a distributed (decentralized) belief propagation algorithm for CoSLAS in the practically important case of an affine clock model and asymmetric time stamping. Our algorithm allows for real-time operation and is suitable for a time-varying network connectivity. To achieve high accuracy at reduced complexity and communication cost, the algorithm combines particle implementations with parametric message representations and takes advantage of a conditional independence property. Simulation results demonstrate the good performance of the proposed algorithm in a challenging scenario with time-varying network connectivity.

Linear Optimal Control on Factor Graphs — A Message Passing Perspective —

Factor graphs form a class of probabilistic graphical models representing the factorization of probability density functions as bipartite graphs. They can be used to exploit the conditional independence structure of the underlying model to efficiently solve inference problems by message passing. The present paper advocates the use of factor graphs in control and highlights similarities to, e. g., signal processing and communications where this class of models is widely used. By applying the factor graph framework to a probabilistic interpretation of optimal control, several classical results are recovered. The dynamic programming approach to linear quadratic Gaussian control is described as a message passing algorithm on factor graph on which possible extensions are exemplified. A factor graph-based iterative learning control scheme is outlined and an expectation maximization-based estimation of normal unknown variance priors is adapted for the derivation of sparse control signals, highlighting the benefits of using a unified framework across disciplines by mixing and matching corresponding graphical algorithms.

Significance evaluation in factor graphs

BackgroundFactor graphs provide a flexible and general framework for specifying probability distributions. They can capture a range of popular and recent models for analysis of both genomics data as well as data from other scientific fields. Owing to the ever larger data sets encountered in genomics and the multiple-testing issues accompanying them, accurate significance evaluation is of great importance. We here address the problem of evaluating statistical significance of observations from factor graph models.ResultsTwo novel numerical approximations for evaluation of statistical significance are presented. First a method using importance sampling. Second a saddlepoint approximation based method. We develop algorithms to efficiently compute the approximations and compare them to naive sampling and the normal approximation. The individual merits of the methods are analysed both from a theoretical viewpoint and with simulations. A guideline for choosing between the normal approximation, saddle-point approximation and importance sampling is also provided. Finally, the applicability of the methods is demonstrated with examples from cancer genomics, motif-analysis and phylogenetics.ConclusionsThe applicability of saddlepoint approximation and importance sampling is demonstrated on known models in the factor graph framework. Using the two methods we can substantially improve computational cost without compromising accuracy. This contribution allows analyses of large datasets in the general factor graph framework.

On-Manifold Preintegration for Real-Time Visual--Inertial Odometry

Current approaches for visual-inertial odometry (VIO) are able to attain highly accurate state estimation via nonlinear optimization. However, real-time optimization quickly becomes infeasible as the trajectory grows over time, this problem is further emphasized by the fact that inertial measurements come at high rate, hence leading to fast growth of the number of variables in the optimization. In this paper, we address this issue by preintegrating inertial measurements between selected keyframes into single relative motion constraints. Our first contribution is a \emph{preintegration theory} that properly addresses the manifold structure of the rotation group. We formally discuss the generative measurement model as well as the nature of the rotation noise and derive the expression for the \emph{maximum a posteriori} state estimator. Our theoretical development enables the computation of all necessary Jacobians for the optimization and a-posteriori bias correction in analytic form. The second contribution is to show that the preintegrated IMU model can be seamlessly integrated into a visual-inertial pipeline under the unifying framework of factor graphs. This enables the application of incremental-smoothing algorithms and the use of a \emph{structureless} model for visual measurements, which avoids optimizing over the 3D points, further accelerating the computation. We perform an extensive evaluation of our monocular \VIO pipeline on real and simulated datasets. The results confirm that our modelling effort leads to accurate state estimation in real-time, outperforming state-of-the-art approaches.