Bayesian inference in the space of topological maps
TL;DR: A general framework for modeling measurements is described, and the use of a Markov-chain Monte Carlo algorithm that uses specific instances of these models for odometry and appearance measurements to estimate the posterior distribution is described.
Abstract: While probabilistic techniques have previously been investigated extensively for performing inference over the space of metric maps, no corresponding general-purpose methods exist for topological maps. We present the concept of probabilistic topological maps (PTMs), a sample-based representation that approximates the posterior distribution over topologies, given available sensor measurements. We show that the space of topologies is equivalent to the intractably large space of set partitions on the set of available measurements. The combinatorial nature of the problem is overcome by computing an approximate, sample-based representation of the posterior. The PTM is obtained by performing Bayesian inference over the space of all possible topologies, and provides a systematic solution to the problem of perceptual aliasing in the domain of topological mapping. In this paper, we describe a general framework for modeling measurements, and the use of a Markov-chain Monte Carlo algorithm that uses specific instances of these models for odometry and appearance measurements to estimate the posterior distribution. We present experimental results that validate our technique and generate good maps when using odometry and appearance, derived from panoramic images, as sensor measurements.
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TL;DR: Various issues and problems in multiple-robot SLAM are introduced, current solutions for these problems are reviewed, and their advantages and disadvantages are discussed.
Abstract: Simultaneous localization and mapping SLAM in unknown GPS-denied environments is a major challenge for researchers in the field of mobile robotics. Many solutions for single-robot SLAM exist; however, moving to a platform of multiple robots adds many challenges to the existing problems. This paper reviews state-of-the-art multiple-robot systems, with a major focus on multiple-robot SLAM. Various issues and problems in multiple-robot SLAM are introduced, current solutions for these problems are reviewed, and their advantages and disadvantages are discussed.
269 citations
Cites background from "Bayesian inference in the space of ..."
..., 2004), Bayesian inference (Ranganathan et al., 2006), and the semantic approach with place labelling (Friedman et al....
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TL;DR: An extension of the loop closing technique to a multi-robot mapping problem in which the outputs of several, uncoordinated and SLAM-enabled robots are fused without requiring inter-vehicle observations or a-priori frame alignment.
Abstract: This paper is concerned with "loop closing" for mobile robots. Loop closing is the problem of correctly asserting that a robot has returned to a previously visited area. It is a particularly hard but important component of the Simultaneous Localization and Mapping (SLAM) problem. Here a mobile robot explores an a-priori unknown environment performing on-the-fly mapping while the map is used to localize the vehicle. Many SLAM implementations look to internal map and vehicle estimates (p.d.fs) to make decisions about whether a vehicle is revisiting a previously mapped area or is exploring a new region of workspace. We suggest that one of the reasons loop closing is hard in SLAM is precisely because these internal estimates can, despite best efforts, be in gross error. The "loop closer" we propose, analyze and demonstrate makes no recourse to the metric estimates of the SLAM system it supports and aids---it is entirely independent. At regular intervals the vehicle captures the appearance of the local scene (with camera and laser). We encode the similarity between all possible pairings of scenes in a "similarity matrix". We then pose the loop closing problem as the task of extracting statistically significant sequences of similar scenes from this matrix. We show how suitable analysis (introspection) and decomposition (remediation) of the similarity matrix allows for the reliable detection of loops despite the presence of repetitive and visually ambiguous scenes. We demonstrate the technique supporting a SLAM system driven by scan-matching laser data in a variety of settings. Some of the outdoor settings are beyond the capability of the SLAM system itself in which case GPS was used to provide a ground truth. We further show how the techniques can equally be applied to detect loop closure using spatial images taken with a scanning laser. We conclude with an extension of the loop closing technique to a multi-robot mapping problem in which the outputs of several, uncoordinated and SLAM-enabled robots are fused without requiring inter-vehicle observations or a-priori frame alignment.
244 citations
Cites background from "Bayesian inference in the space of ..."
...The visual appearance aspects of the work in Ranganathan et al. (2006) has a common ground with this paper—visual appearance is used to help close loops....
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01 Jan 2005
TL;DR: A novel system for autonomous mobile robot navigation with only an omnidirectional camera as sensor is presented, able to build automatically and robustly accurate topologically organised environment maps of a complex, natural environment.
Abstract: In this work we present a novel system for autonomous mobile robot navigation. With only an omnidirectional camera as sensor, this system is able to build automatically and robustly accurate topologically organised environment maps of a complex, natural environment. It can localise itself using such a map at each moment, including both at startup (kidnapped robot) or using knowledge of former localisations. The topological nature of the map is similar to the intuitive maps humans use, is memory-efficient and enables fast and simple path planning towards a specified goal. We developed a real-time visual servoing technique to steer the system along the computed path.
A key technology making this all possible is the novel fast wide baseline feature matching, which yields an efficient description of the scene, with a focus on man-made environments.
198 citations
Cites methods from "Bayesian inference in the space of ..."
...(Ranganathan et al., 2005) for instance use Bayesian inference to find the topological structure that explains best a set of panoramic observations, while (Shatkay & Kaelbling, 1997) fit hidden Markov models to the data....
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TL;DR: In this paper, the authors present a system for autonomous mobile robot navigation with only an omnidirectional camera as sensor, which is able to build automatically and robustly accurate topologically organized environment maps of a complex, natural environment.
Abstract: In this work we present a novel system for autonomous mobile robot navigation. With only an omnidirectional camera as sensor, this system is able to build automatically and robustly accurate topologically organised environment maps of a complex, natural environment. It can localise itself using such a map at each moment, including both at startup (kidnapped robot) or using knowledge of former localisations. The topological nature of the map is similar to the intuitive maps humans use, is memory-efficient and enables fast and simple path planning towards a specified goal. We developed a real-time visual servoing technique to steer the system along the computed path.
A key technology making this all possible is the novel fast wide baseline feature matching, which yields an efficient description of the scene, with a focus on man-made environments.
189 citations
TL;DR: A new relative bundle adjustment is derived which, instead of optimizing in a single Euclidean space, works in a metric space defined by a manifold, and it is shown experimentally that it is possible to solve for the full maximum-likelihood solution incrementally in constant time, even at loop closure.
Abstract: In this paper we describe a relative approach to simultaneous localization and mapping, based on the insight that a continuous relative representation can make the problem tractable at large scales. First, it is well known that bundle adjustment is the optimal non-linear least-squares formulation for this problem, in that its maximum-likelihood form matches the definition of the CramerâRao lower bound. Unfortunately, computing the maximum-likelihood solution is often prohibitively expensive: this is especially true during loop closures, which often necessitate adjusting all parameters in a loop. In this paper we note that it is precisely the choice of a single privileged coordinate frame that makes bundle adjustment costly, and that this expense can be avoided by adopting a completely relative approach. We derive a new relative bundle adjustment which, instead of optimizing in a single Euclidean space, works in a metric space defined by a manifold. Using an adaptive optimization strategy, we show experimentally that it is possible to solve for the full maximum-likelihood solution incrementally in constant time, even at loop closure. Our approach is, by definition, everywhere locally Euclidean, and we show that the local Euclidean estimate matches that of traditional bundle adjustment. Our system operates online in realtime using stereo data, with fast appearance-based loop closure detection. We show results on over 850,000 images that indicate the accuracy and scalability of the approach, and process over 330 GB of image data into a relative map covering 142 km of Southern England. To demonstrate a baseline sufficiency for navigation, we show that it is possible to find shortest paths in the relative maps we build, in terms of both time and distance. Query images from the web of popular landmarks around London, such as the London Eye or Trafalgar Square, are matched to the relative map to provide route planning goals.
183 citations
Cites methods from "Bayesian inference in the space of ..."
...Topological mapping, in conjunction with bag-of-words place recognition, has also recently been placed on firm probabilistic ground (Ranganathan et al. 2006 Ranganathan 2008)....
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References
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Book•
01 May 1986TL;DR: In this article, the authors present a graphical representation of data using Principal Component Analysis (PCA) for time series and other non-independent data, as well as a generalization and adaptation of principal component analysis.
Abstract: Introduction * Properties of Population Principal Components * Properties of Sample Principal Components * Interpreting Principal Components: Examples * Graphical Representation of Data Using Principal Components * Choosing a Subset of Principal Components or Variables * Principal Component Analysis and Factor Analysis * Principal Components in Regression Analysis * Principal Components Used with Other Multivariate Techniques * Outlier Detection, Influential Observations and Robust Estimation * Rotation and Interpretation of Principal Components * Principal Component Analysis for Time Series and Other Non-Independent Data * Principal Component Analysis for Special Types of Data * Generalizations and Adaptations of Principal Component Analysis
17,446 citations
Book•
01 Jan 1995TL;DR: Detailed notes on Bayesian Computation Basics of Markov Chain Simulation, Regression Models, and Asymptotic Theorems are provided.
Abstract: FUNDAMENTALS OF BAYESIAN INFERENCE Probability and Inference Single-Parameter Models Introduction to Multiparameter Models Asymptotics and Connections to Non-Bayesian Approaches Hierarchical Models FUNDAMENTALS OF BAYESIAN DATA ANALYSIS Model Checking Evaluating, Comparing, and Expanding Models Modeling Accounting for Data Collection Decision Analysis ADVANCED COMPUTATION Introduction to Bayesian Computation Basics of Markov Chain Simulation Computationally Efficient Markov Chain Simulation Modal and Distributional Approximations REGRESSION MODELS Introduction to Regression Models Hierarchical Linear Models Generalized Linear Models Models for Robust Inference Models for Missing Data NONLINEAR AND NONPARAMETRIC MODELS Parametric Nonlinear Models Basic Function Models Gaussian Process Models Finite Mixture Models Dirichlet Process Models APPENDICES A: Standard Probability Distributions B: Outline of Proofs of Asymptotic Theorems C: Computation in R and Stan Bibliographic Notes and Exercises appear at the end of each chapter.
16,079 citations
TL;DR: A generalization of the sampling method introduced by Metropolis et al. as mentioned in this paper is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates.
Abstract: SUMMARY A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates. Examples of the methods, including the generation of random orthogonal matrices and potential applications of the methods to numerical problems arising in statistics, are discussed. For numerical problems in a large number of dimensions, Monte Carlo methods are often more efficient than conventional numerical methods. However, implementation of the Monte Carlo methods requires sampling from high dimensional probability distributions and this may be very difficult and expensive in analysis and computer time. General methods for sampling from, or estimating expectations with respect to, such distributions are as follows. (i) If possible, factorize the distribution into the product of one-dimensional conditional distributions from which samples may be obtained. (ii) Use importance sampling, which may also be used for variance reduction. That is, in order to evaluate the integral J = X) p(x)dx = Ev(f), where p(x) is a probability density function, instead of obtaining independent samples XI, ..., Xv from p(x) and using the estimate J, = Zf(xi)/N, we instead obtain the sample from a distribution with density q(x) and use the estimate J2 = Y{f(xj)p(x1)}/{q(xj)N}. This may be advantageous if it is easier to sample from q(x) thanp(x), but it is a difficult method to use in a large number of dimensions, since the values of the weights w(xi) = p(x1)/q(xj) for reasonable values of N may all be extremely small, or a few may be extremely large. In estimating the probability of an event A, however, these difficulties may not be as serious since the only values of w(x) which are important are those for which x -A. Since the methods proposed by Trotter & Tukey (1956) for the estimation of conditional expectations require the use of importance sampling, the same difficulties may be encountered in their use. (iii) Use a simulation technique; that is, if it is difficult to sample directly from p(x) or if p(x) is unknown, sample from some distribution q(y) and obtain the sample x values as some function of the corresponding y values. If we want samples from the conditional dis
14,965 citations
"Bayesian inference in the space of ..." refers methods in this paper
...We now present a concrete implementation of the theory that uses the Metropolis–Hastings algorithm [19], a very general MCMC method, for performing the inference....
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TL;DR: The Markov Chain Monte Carlo Implementation Results Summary and Discussion MEDICAL MONITORING Introduction Modelling Medical Monitoring Computing Posterior Distributions Forecasting Model Criticism Illustrative Application Discussion MCMC for NONLINEAR HIERARCHICAL MODELS.
Abstract: INTRODUCING MARKOV CHAIN MONTE CARLO Introduction The Problem Markov Chain Monte Carlo Implementation Discussion HEPATITIS B: A CASE STUDY IN MCMC METHODS Introduction Hepatitis B Immunization Modelling Fitting a Model Using Gibbs Sampling Model Elaboration Conclusion MARKOV CHAIN CONCEPTS RELATED TO SAMPLING ALGORITHMS Markov Chains Rates of Convergence Estimation The Gibbs Sampler and Metropolis-Hastings Algorithm INTRODUCTION TO GENERAL STATE-SPACE MARKOV CHAIN THEORY Introduction Notation and Definitions Irreducibility, Recurrence, and Convergence Harris Recurrence Mixing Rates and Central Limit Theorems Regeneration Discussion FULL CONDITIONAL DISTRIBUTIONS Introduction Deriving Full Conditional Distributions Sampling from Full Conditional Distributions Discussion STRATEGIES FOR IMPROVING MCMC Introduction Reparameterization Random and Adaptive Direction Sampling Modifying the Stationary Distribution Methods Based on Continuous-Time Processes Discussion IMPLEMENTING MCMC Introduction Determining the Number of Iterations Software and Implementation Output Analysis Generic Metropolis Algorithms Discussion INFERENCE AND MONITORING CONVERGENCE Difficulties in Inference from Markov Chain Simulation The Risk of Undiagnosed Slow Convergence Multiple Sequences and Overdispersed Starting Points Monitoring Convergence Using Simulation Output Output Analysis for Inference Output Analysis for Improving Efficiency MODEL DETERMINATION USING SAMPLING-BASED METHODS Introduction Classical Approaches The Bayesian Perspective and the Bayes Factor Alternative Predictive Distributions How to Use Predictive Distributions Computational Issues An Example Discussion HYPOTHESIS TESTING AND MODEL SELECTION Introduction Uses of Bayes Factors Marginal Likelihood Estimation by Importance Sampling Marginal Likelihood Estimation Using Maximum Likelihood Application: How Many Components in a Mixture? Discussion Appendix: S-PLUS Code for the Laplace-Metropolis Estimator MODEL CHECKING AND MODEL IMPROVEMENT Introduction Model Checking Using Posterior Predictive Simulation Model Improvement via Expansion Example: Hierarchical Mixture Modelling of Reaction Times STOCHASTIC SEARCH VARIABLE SELECTION Introduction A Hierarchical Bayesian Model for Variable Selection Searching the Posterior by Gibbs Sampling Extensions Constructing Stock Portfolios With SSVS Discussion BAYESIAN MODEL COMPARISON VIA JUMP DIFFUSIONS Introduction Model Choice Jump-Diffusion Sampling Mixture Deconvolution Object Recognition Variable Selection Change-Point Identification Conclusions ESTIMATION AND OPTIMIZATION OF FUNCTIONS Non-Bayesian Applications of MCMC Monte Carlo Optimization Monte Carlo Likelihood Analysis Normalizing-Constant Families Missing Data Decision Theory Which Sampling Distribution? Importance Sampling Discussion STOCHASTIC EM: METHOD AND APPLICATION Introduction The EM Algorithm The Stochastic EM Algorithm Examples GENERALIZED LINEAR MIXED MODELS Introduction Generalized Linear Models (GLMs) Bayesian Estimation of GLMs Gibbs Sampling for GLMs Generalized Linear Mixed Models (GLMMs) Specification of Random-Effect Distributions Hyperpriors and the Estimation of Hyperparameters Some Examples Discussion HIERARCHICAL LONGITUDINAL MODELLING Introduction Clinical Background Model Detail and MCMC Implementation Results Summary and Discussion MEDICAL MONITORING Introduction Modelling Medical Monitoring Computing Posterior Distributions Forecasting Model Criticism Illustrative Application Discussion MCMC FOR NONLINEAR HIERARCHICAL MODELS Introduction Implementing MCMC Comparison of Strategies A Case Study from Pharmacokinetics-Pharmacodynamics Extensions and Discussion BAYESIAN MAPPING OF DISEASE Introduction Hypotheses and Notation Maximum Likelihood Estimation of Relative Risks Hierarchical Bayesian Model of Relative Risks Empirical Bayes Estimation of Relative Risks Fully Bayesian Estimation of Relative Risks Discussion MCMC IN IMAGE ANALYSIS Introduction The Relevance of MCMC to Image Analysis Image Models at Different Levels Methodological Innovations in MCMC Stimulated by Imaging Discussion MEASUREMENT ERROR Introduction Conditional-Independence Modelling Illustrative examples Discussion GIBBS SAMPLING METHODS IN GENETICS Introduction Standard Methods in Genetics Gibbs Sampling Approaches MCMC Maximum Likelihood Application to a Family Study of Breast Cancer Conclusions MIXTURES OF DISTRIBUTIONS: INFERENCE AND ESTIMATION Introduction The Missing Data Structure Gibbs Sampling Implementation Convergence of the Algorithm Testing for Mixtures Infinite Mixtures and Other Extensions AN ARCHAEOLOGICAL EXAMPLE: RADIOCARBON DATING Introduction Background to Radiocarbon Dating Archaeological Problems and Questions Illustrative Examples Discussion Index
7,399 citations
01 Jan 1996
3,908 citations
"Bayesian inference in the space of ..." refers background in this paper
...Below we review relevant prior research in these areas....
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...We simply state the expression for the prior, given the total number of landmarks in the environment (including those not visited by the robot) (8) where is the number of measurements, is the number of distinct landmarks in the topology , and is a normalization constant....
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