Efficient Simultaneous Localization And Mapping Research Articles

VB-T PHD-SLAM: efficient SLAM under heavy-tailed noise

To address the challenge of simultaneous localization and mapping (SLAM) in the presence of heavy-tailed noise, this paper introduces a robust probability hypothesis density (PHD) SLAM algorithm. This algorithm models measurement noise using the Student's t-distribution, which better captures the heavy-tailed nature of the noise. Since the prior density is assumed to be Gaussian mixture form, the posterior density is no longer Gaussian mixture form after the likelihood update of the t-distribution. A variational Bayesian approach is employed to ensure computable multi-target densities during filtering, minimizing the Kullback-Leibler divergence to obtain an approximate solution for the new marginal likelihood function. Then a new closed-form recursion of PHD-SLAM is derived by using t-distribution. Simulation results and real-world validations demonstrate that the proposed algorithm outperforms PHD-SLAM 1.0 and PHD-SLAM 2.0 in terms of both localization and mapping accuracy while maintaining computational efficiency in SLAM scenarios affected by heavy-tailed noise.

LTA‐OM: Long‐term association LiDAR–IMU odometry and mapping

AbstractThis paper focuses on the Light Detection and Ranging (LiDAR)–Inertial Measurement Unit (IMU) simultaneous localization and mapping (SLAM) problem: How to fuse the sensor measurement from the LiDAR and IMU to online estimate robot's poses and build a consistent map of the environment. This paper presents LTA‐OM: an efficient, robust, and accurate LiDAR SLAM system. Employing fast direct LiDAR‐inertial odometry (FAST‐LIO2) and Stable Triangle Descriptor as LiDAR–IMU odometry and the loop detection method, respectively, LTA‐OM is implemented to be functionally complete, including loop detection and correction, false‐positive loop closure rejection, long‐term association (LTA) mapping, and multisession localization and mapping. One novelty of this paper is the real‐time LTA mapping, which exploits the direct scan‐to‐map registration of FAST‐LIO2 and employs the corrected history map to provide direct global constraints to the LIO mapping process. LTA mapping also has the notable advantage of achieving drift‐free odometry at revisit places. Besides, a multisession mode is designed to allow the user to store the current session's results, including the corrected map points, optimized odometry, and descriptor database for future sessions. The benefits of this mode are additional accuracy improvement and consistent map stitching, which is helpful for life‐long mapping. Furthermore, LTA‐OM has valuable features for robot control and path planning, including high‐frequency and real‐time odometry, driftless odometry at revisit places, and fast loop closing convergence. LTA‐OM is versatile as it is applicable to both multiline spinning and solid‐state LiDARs, mobile robots and handheld platforms. In experiments, we exhaustively benchmark LTA‐OM and other state‐of‐the‐art LiDAR systems with 18 data sequences. The results show that LTA‐OM steadily outperforms other systems regarding trajectory accuracy, map consistency, and time consumption. The robustness of LTA‐OM is validated in a challenging scene—a multilevel building having similar structures at different levels. To demonstrate our system, we created a video which can be found on https://youtu.be/DVwppEKlKps.

Exploiting semantic information in a spiking neural SLAM system.

To navigate in new environments, an animal must be able to keep track of its position while simultaneously creating and updating an internal map of features in the environment, a problem formulated as simultaneous localization and mapping (SLAM) in the field of robotics. This requires integrating information from different domains, including self-motion cues, sensory, and semantic information. Several specialized neuron classes have been identified in the mammalian brain as being involved in solving SLAM. While biology has inspired a whole class of SLAM algorithms, the use of semantic information has not been explored in such work. We present a novel, biologically plausible SLAM model called SSP-SLAM-a spiking neural network designed using tools for large scale cognitive modeling. Our model uses a vector representation of continuous spatial maps, which can be encoded via spiking neural activity and bound with other features (continuous and discrete) to create compressed structures containing semantic information from multiple domains (e.g., spatial, temporal, visual, conceptual). We demonstrate that the dynamics of these representations can be implemented with a hybrid oscillatory-interference and continuous attractor network of head direction cells. The estimated self-position from this network is used to learn an associative memory between semantically encoded landmarks and their positions, i.e., an environment map, which is used for loop closure. Our experiments demonstrate that environment maps can be learned accurately and their use greatly improves self-position estimation. Furthermore, grid cells, place cells, and object vector cells are observed by this model. We also run our path integrator network on the NengoLoihi neuromorphic emulator to demonstrate feasibility for a full neuromorphic implementation for energy efficient SLAM.

A framework for millimeter-wave multi-user SLAM and its low-cost realization

Millimeter-wave (mmWave) is regarded as a promising technique for accurate simultaneous localization and mapping (SLAM). However, the implementation of a massive antenna array and the extensiveness of mmWave anchor deployment significantly increase the cost of the whole system, and thus mmWave is still not considered in commercial localization scenarios. In this paper, a low-cost and highly efficient mmWave-based SLAM scheme is proposed to address the above limitations. Concretely, mmWave-based sensing is leveraged to obtain the locations of obstacles. Then, multi-user landmark association and fast map merging are performed for accurate global map construction. The simulation and experimental results show that the proposed framework achieves submeter-level mapping accuracy and provides new insight into efficiently designing a SLAM system that meets the stated requirements.

Point–Line-Aware Heterogeneous Graph Attention Network for Visual SLAM System

Simultaneous localization and mapping (SLAM), as an important research topic in robotics, is useful but challenging to estimate robot pose and reconstruct a 3-D map of the surrounding environment. Despite recent success of several deep neural networks for visual SLAM, those methods cannot achieve robust results in complex industrial scenarios for constructing accurate and real-time maps due to the weak texture and complex geometric structure. This paper presents a novel and efficient visual SLAM system based on point–line-aware heterogeneous graph attention network, which combines points and line segments to solve the problem of the insufficient number of reliable features in traditional approaches. Firstly, a simultaneous feature extraction network is constructed based on the geometric relationships between points and points and points and lines. To further improve the efficiency and accuracy of the geometric association features of key regions, we design the point–line-aware attention module to guide the network to pay attention to the trivial features of both points and lines in images. Moreover, the network model is optimized by a transfer-aware knowledge distillation strategy to further improve the system’s real-time performance. Secondly, to improve the accuracy of the point–line matching, we design a point–line heterogeneous graph attention network, which combines an edge aggregation graph attention module and a cross-heterogeneous graph iteration module to conduct learning on the intragraph and intergraph. Finally, the point–line matching process is transformed into an optimal transport problem, and a near-iterative method based on a greedy strategy is presented to solve the optimization problem. The experiments on the KITTI dataset and a self-made dataset demonstrate the better effectiveness, accuracy, and adaptability of our method than those of the state of the art in visual SLAM.

Long-Term Visual Simultaneous Localization and Mapping: Using a Bayesian Persistence Filter-Based Global Map Prediction

With the rapidly growing demand for accurate localization in real-world environments, visual simultaneous localization and mapping (SLAM) has received significant attention in recent years. However, those existing methods still suffer from the degradation of localization accuracy in long-term changing environments. To address these problems, we propose a novel long-term SLAM system with map prediction and dynamics removal. First, a visual point-cloud matching algorithm is designed to efficiently fuse 2D pixel information and 3D voxel information. Second, each map point is classified into three types: static, semistatic, and dynamic based on the Bayesian persistence filter (BPF). Then we remove the dynamic map points to eliminate the influence of those map points. We can obtain a global predicted map by modeling the time series of semistatic map points. Finally, we incorporate the predicted global map into a state-of-the-art SLAM method, achieving an efficient visual SLAM system for long-term, dynamic environments. Extensive experiments are carried out on a wheelchair robot in an indoor environment over several months. The results demonstrate that our method has better map prediction accuracy and achieves more robust localization performance.

Nonparametric Background Model-Based LiDAR SLAM in Highly Dynamic Urban Environments

In urban environments, simultaneous localization and mapping (SLAM) are essential for autonomous driving. Most light detection and ranging (LiDAR) SLAM methodologies have been developed for relatively static environments, despite real-world environments having many dynamic objects such as vehicles, bicycles, and pedestrians. This paper proposes an efficient and robust LiDAR SLAM. Our SLAM framework leverages the estimated background model to achieve robust motion estimation in dynamic urban environments. Based on probabilistic object estimation, the dynamic removal module estimates a nonparametric background model to recognize dynamic objects. This module estimates the probability of the difference of the range values from the accumulated LiDAR frames. Then, dynamic objects are removed by adapting the sensor velocity from the estimated ego-motion. In the local mapping module, our method optimizes the LiDAR motion considering the dynamic characteristics of LiDAR point clouds. Finally, the proposed method results in a global map with static point clouds and accurate LiDAR motion with global pose optimization. We tested the proposed method on the well-known public dataset (KITTI) and the custom dataset with complex environments, including various moving objects. Comparisons with state-of-the-art (SOTA) methods demonstrate that the our approach is more robust and efficient. For example, the proposed method performed an average 0.63% and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.18^{\circ }/100\;m$ </tex-math></inline-formula> errors on the KITTI dataset with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.96ms$ </tex-math></inline-formula> processing time that convinces real-time processing.

Edge Robotics: Edge-Computing-Accelerated Multirobot Simultaneous Localization and Mapping

With the wide penetration of smart robots in multifarious fields, the simultaneous localization and mapping (SLAM) technique in robotics has attracted growing attention in the community. Yet collaborating SLAM over multiple robots still remains challenging due to performance contradiction between the intensive graphics computation of SLAM and the limited computing capability of robots. While traditional solutions resort to the powerful cloud servers acting as an external computation provider, we show by real-world measurements that the significant communication overhead in data offloading prevents its practicability to real deployment. To tackle these challenges, this article promotes the emerging edge-computing paradigm into multirobot SLAM and proposes RecSLAM, a multirobot laser SLAM system that focuses on accelerating the map construction process under the robot–edge–cloud architecture. In contrast to the conventional multirobot SLAM that generates graphic maps on robots and completely merges them on the cloud, RecSLAM develops a hierarchical map fusion technique that directs robots’ raw data to edge servers for real-time fusion and then sends to the cloud for global merging. To optimize the overall pipeline, an efficient multirobot SLAM collaborative processing framework is introduced to adaptively optimize robot-to-edge offloading tailored to heterogeneous edge resource conditions, meanwhile ensuring the workload balancing among the edge servers. Extensive evaluations show RecSLAM can achieve up to 39.31% processing latency reduction over the state of the art. Besides, a proof-of-concept prototype is developed and deployed in real scenes to demonstrate its effectiveness.

Topographic SLAM Using a Single Terrain Altimeter in GNSS-Restricted Environment

In a Global Navigation Satellite System (GNSS)-restricted area, a mobile robot navigation system exploits surrounding environment information. For an aerial or underwater vehicle, undulating terrain of a land or seabed surface is a valuable information resource that leads to the development of terrain-referenced navigation (TRN) algorithms. However, due to the vast amount of a vehicle’s activity area, surveying all the regions to obtain a high-resolution terrain map is impractical and requires simultaneous localization and mapping (SLAM) as a highly desirable capability. This paper presents a topographic SLAM algorithm using only a single terrain altimeter, which is low-cost, computationally efficient, and sufficiently stable for long-term operation. The proposed rectangular panel map structure and update method enable robust and efficient SLAM. As terrain elevation changes are inherently nonlinear, an extended Kalman filter (EKF)-based SLAM filter is adopted. The feasibility and validity of the proposed algorithm are demonstrated through simulations using terrain elevation data from a real-world undersea environment.

Efficient robot localization and SLAM algorithms using Opposition based High Dimensional optimization Algorithm

Particle filter (PF) is introduced to tackle the limitations of the Kalman filter which adopts Gaussian in the state and noise of the system. PFs have the problem of sample impoverishment and one approach to solve this problem is to optimize the proposal distribution shown by particles. This paper introduces a novel evolutionary PF based on Opposition based High Dimensional optimization Algorithm (OHDA) to reposition the particles of PF in high probable regions for estimation. OHDA will preserve the diversity of particles while emphasizing the more informative ones by information sharing and angular movement operators. Opposite particles are introduced in this paper to speed up the convergence of PF. Virtual forward movement by angular movement of OHDA is employed to better guide the search process. The optimized PF can improve the performance of the estimation algorithms in problems such as localization and SLAM. In robot localization problem, particles show the location of the robot in a known environment. For SLAM (Simultaneous Localization And Mapping), particles contain the location of the robot as well as estimated map of the environment. The application of the resulting evolutionary particle filter is tested in both localization and SLAM. Comparing the results of the proposed evolutionary particle filter with other algorithms confirms the efficiency of applying OHDA to PF in terms of improving estimation accuracy in the well-known Victoria park dataset and some other generated test environments. Comparing optimization algorithms on FASTSLAM and UFASTSLAM are PSO, FA, MVO, and MGWO.

Lightweight 3-D Localization and Mapping for Solid-State LiDAR

The LIght Detection And Ranging (LiDAR) sensor has become one of the most important perceptual devices due to its important role in simultaneous localization and mapping (SLAM). Existing SLAM methods are mainly developed for mechanical LiDAR sensors, which are often adopted by large scale robots. Recently, the solid-state LiDAR is introduced and becomes popular since it provides a cost-effective and lightweight solution for small scale robots. Compared to mechanical LiDAR, solid-state LiDAR sensors have higher update frequency and angular resolution, but also have smaller field of view (FoV), which is very challenging for existing LiDAR SLAM algorithms. Therefore, it is necessary to have a more robust and computationally efficient SLAM method for this new sensing device. To this end, we propose a new SLAM framework for solid-state LiDAR sensors, which involves feature extraction, odometry estimation, and probability map building. The proposed method is evaluated on a warehouse robot and a hand-held device. In the experiments, we demonstrate both the accuracy and efficiency of our method using an Intel L515 solid-state LiDAR. The results show that our method is able to provide precise localization and high quality mapping. We made the source codes public at https://github.com/wh200720041/SSL_SLAM.

Efficient Bathymetric SLAM With Invalid Loop Closure Identification

Bathymetric simultaneous localization and mapping (SLAM) holds high potential for accurate navigation in long endurance autonomous underwater vehicle (AUV) missions. Due to nearly flat seabed topography, the occurrence of invalid loop closures is unavoidable in bathymetric SLAM. Thus, invalid loop closure identification is a crucial component of bathymetric SLAM, a procedure that normally requires a substantial amount of computational resources. An efficient bathymetric SLAM method considering invalid loop closures is proposed in this article, which identifies invalid loop closures via PF-Backend, a robust back end method based on particle filtering theory. Using this set-up, all beliefs of historical loop closures are estimated by a small number of particles and, for each particle, the corresponding map consistency calculated via a global graph optimizer is applied as the measurement update in the particle filter. A loop closure solidification procedure is also discussed to decrease the number of particles in the PF-Backend. The new algorithm is shown to be efficient and capable of providing accurate location and mapping results using play-back experiments.

Robust and Efficient RGB-D SLAM in Dynamic Environments

Simultaneous localization and mapping (SLAM) using an RGB-D camera is a key enabling technique for many augmented reality (AR) applications. However, most existing RGB-D SLAM methods could fail in dynamic scenarios due to non-trivial pose estimation errors arising from moving objects. In this study, we present an accurate and robust RGB-D SLAM system for dynamic scenarios which can run real-time on a single dual-core CPU. The core of our system is a robust and efficient dynamic keypoint exclusion method which consists of three steps: 1) grouping spatially and appearance related pixels of a keyframe into regions; 2) identifying dynamic regions by checking motion consistency of keypoints in every region; 3) excluding keypoints in the identified dynamic regions as well as the matching points in the 3D local map. The dynamic keypoint exclusion method can be easily integrated into any keypoint based RGB-D SLAM system for improving the accuracy and robustness in dynamic scenes with trivial time increase (16.6ms per frame). Experimental results on the TUM dataset demonstrates that our method which runs on an Intel i7-4900 CPU is even 2.3X faster than the state-of-the-art method DS-SLAM [1] which runs parallel on a P4000 GPU and a comparable CPU. In addition, our system outperforms the state-of-the-art methods [1]–[4] in terms of smaller absolute trajectory errors (ATE). We also apply our system to a real AR application and live experiments with a hand-held RGB-D camera demonstrate the robustness and generalizability of our method in practical scenarios. <xref ref-type="fn" rid="fn1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¹</xref> <fn id="fn1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><label>¹</label> A demo video is provided on <uri>https://github.com/cc-qy/Dynamic-RGB-D-SLAM</uri> </fn>

A Mobile Robot Visual SLAM System With Enhanced Semantics Segmentation

Traditional visual simultaneous localization and mapping (SLAM) systems mostly based on small-area static environments. In recent years, some studies focused on combining semantic information with visual SLAM. However, most of them are hard to obtain better performance in the large-scale dynamic environment. And the accuracy, rapidity of the system still needs to strengthen. In this paper, we develop a more efficient semantic SLAM system in the two-wheeled mobile robot by using semantic segmentation to recognize people, chairs, and other objects in every keyframe. With a preliminary understanding of the environment, fusing the RGB-D camera and encoders information, to localization and creating a dense colored octree map without dynamic objects. Besides, for the incomplete identification of movable objects, we used image processing algorithms to enhance the semantic segmentation effect. In the proposed method, enhanced semantic segmentation in keyframes dramatically increases the efficiency of the system. Moreover, fusing the different sensors can highly raise localization accuracy. We conducted experiments on various datasets and in some real environments and compared them with DRE-SLAM, DS-SLAM, to evaluate the performance of the proposed approach. The results suggest we significantly improve the processing efficiency, robustness, and quality of the map.

A Computationally Efficient Semantic SLAM Solution for Dynamic Scenes

In various dynamic scenes, there are moveable objects such as pedestrians, which may challenge simultaneous localization and mapping (SLAM) algorithms. Consequently, the localization accuracy may be degraded, and a moving object may negatively impact the constructed maps. Maps that contain semantic information of dynamic objects impart humans or robots with the ability to semantically understand the environment, and they are critical for various intelligent systems and location-based services. In this study, we developed a computationally efficient SLAM solution that is able to accomplish three tasks in real time: (1) complete localization without accuracy loss due to the existence of dynamic objects and generate a static map that does not contain moving objects, (2) extract semantic information of dynamic objects through a computionally efficient approach, and (3) eventually generate semantic maps, which overlay semantic objects on static maps. The proposed semantic SLAM solution was evaluated through four different experiments on two data sets, respectively verifying the tracking accuracy, computational efficiency, and the quality of the generated static maps and semantic maps. The results show that the proposed SLAM solution is computationally efficient by reducing the time consumption for building maps by 2/3; moreover, the relative localization accuracy is improved, with a translational error of only 0.028 m, and is not degraded by dynamic objects. Finally, the proposed solution generates static maps of a dynamic scene without moving objects and semantic maps with high-precision semantic information of specific objects.

Vision and laser fused SLAM in indoor environments with multi-robot system

Purpose Multi-robot laser-based simultaneous localization and mapping (SLAM) in large-scale environments is an essential but challenging issue in mobile robotics, especially in situations wherein no prior knowledge is available between robots. Moreover, the cumulative errors of every individual robot exert a serious negative effect on loop detection and map fusion. To address these problems, this paper aims to propose an efficient approach that combines laser and vision measurements. Design/methodology/approach A multi-robot visual laser-SLAM is developed to realize robust and efficient SLAM in large-scale environments; both vision and laser loop detections are integrated to detect robust loops. A method based on oriented brief (ORB) feature detection and bag of words (BoW) is developed, to ensure the robustness and computational effectiveness of the multi-robot SLAM system. A robust and efficient graph fusion algorithm is proposed to merge pose graphs from different robots. Findings The proposed method can detect loops more quickly and accurately than the laser-only SLAM, and it can fuse the submaps of each single robot to promote the efficiency, accuracy and robustness of the system. Originality/value Compared with the state of art of multi-robot SLAM approaches, the paper proposed a novel and more sophisticated approach. The vision-based and laser-based loops are integrated to realize a robust loop detection. The ORB features and BoW technologies are further utilized to gain real-time performance. Finally, random sample consensus and least-square methodologies are used to remove the outlier loops among robots.

Selective image registration for efficient visual SLAM on planar surface structures in underwater environment

This paper presents a computationally efficient approach that can be applied to visual simultaneous localization and mapping (SLAM) for the autonomous inspection of underwater structures using monocular vision. A selective image registration scheme consisting of key-frame selection and key-pair selection is proposed to effectively use visual features that may not be evenly distributed on the surface of underwater structures. The computational cost of the visual SLAM algorithm can be substantially reduced using only potentially effective images and image pairs by applying the proposed image registration scheme. The performance of the proposed approach is demonstrated on two different experimental datasets obtained using autonomous underwater vehicles.

Computationally efficient algorithm for vision-based simultaneous localization and mapping of mobile robots

PurposeFastSLAM is a popular method to solve the problem of simultaneous localization and mapping (SLAM). However, when the number of landmarks present in real environments increases, there are excessive comparisons of the measurement with all the existing landmarks in each particle. As a result, the execution speed will be too slow to achieve the objective of real-time navigation. Thus, this paper aims to improve the computational efficiency and estimation accuracy of conventional SLAM algorithms.Design/methodology/approachAs an attempt to solve this problem, this paper presents a computationally efficient SLAM (CESLAM) algorithm, where odometer information is considered for updating the robot’s pose in particles. When a measurement has a maximum likelihood with the known landmark in the particle, the particle state is updated before updating the landmark estimates.FindingsSimulation results show that the proposed CESLAM can overcome the problem of heavy computational burden while improving the accuracy of localization and mapping building. To practically evaluate the performance of the proposed method, a Pioneer 3-DX robot with a Kinect sensor is used to develop an RGB-D-based computationally efficient visual SLAM (CEVSLAM) based on Speeded-Up Robust Features (SURF). Experimental results confirm that the proposed CEVSLAM system is capable of successfully estimating the robot pose and building the map with satisfactory accuracy.Originality/valueThe proposed CESLAM algorithm overcomes the problem of the time-consuming process because of unnecessary comparisons in existing FastSLAM algorithms. Simulations show that accuracy of robot pose and landmark estimation is greatly improved by the CESLAM. Combining CESLAM and SURF, the authors establish a CEVSLAM to significantly improve the estimation accuracy and computational efficiency. Practical experiments by using a Kinect visual sensor show that the variance and average error by using the proposed CEVSLAM are smaller than those by using the other visual SLAM algorithms.

Accurate Mobile Urban Mapping via Digital Map-Based SLAM.

This paper presents accurate urban map generation using digital map-based Simultaneous Localization and Mapping (SLAM). Throughout this work, our main objective is generating a 3D and lane map aiming for sub-meter accuracy. In conventional mapping approaches, achieving extremely high accuracy was performed by either (i) exploiting costly airborne sensors or (ii) surveying with a static mapping system in a stationary platform. Mobile scanning systems recently have gathered popularity but are mostly limited by the availability of the Global Positioning System (GPS). We focus on the fact that the availability of GPS and urban structures are both sporadic but complementary. By modeling both GPS and digital map data as measurements and integrating them with other sensor measurements, we leverage SLAM for an accurate mobile mapping system. Our proposed algorithm generates an efficient graph SLAM and achieves a framework running in real-time and targeting sub-meter accuracy with a mobile platform. Integrated with the SLAM framework, we implement a motion-adaptive model for the Inverse Perspective Mapping (IPM). Using motion estimation derived from SLAM, the experimental results show that the proposed approaches provide stable bird’s-eye view images, even with significant motion during the drive. Our real-time map generation framework is validated via a long-distance urban test and evaluated at randomly sampled points using Real-Time Kinematic (RTK)-GPS.

A Review: Simultaneous Localization and Mapping Algorithms

Simultaneous Localization and Mapping (SLAM) involves creating an environmental map based on sensor data, while concurrently keeping track of the robot’s current position. Efficient and accurate SLAM is crucial for any mobile robot to perform robust navigation. It is also the keystone for higher-level tasks such as path planning and autonomous navigation. The past two decades have seen rapid and exciting progress in solving the SLAM problem together with many compelling implementations of SLAM methods. In this paper, we will review the two common families of SLAM algorithms: Kalman filter with its variations and particle filters. This article complements other surveys in this field by reviewing the representative algorithms and the state-of-the-art in each family. It clearly identifies the inherent relationship between the state estimation via the KF versus PF techniques, all of which are derivations of Bayes rule.