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Probabilistic Topological Mapping for Mobile Robots using Urn Models

2007 , Ranganathan, Ananth , Dellaert, Frank

We present an application of Bayesian modeling and inference to topological mapping in robotics. This is a potentially difficult problem due to (a) the combinatorial nature of the state space, and (b) perceptual aliasing by which two different landmarks in the environment can appear similar to the robot's sensors. Hence, this presents a challenging approximate inference problem, complicated by the fact that the form of the prior on topologies is far from obvious. We deal with the latter problem by introducing the use of urn models, which very naturally encode prior assumptions in the domain of topological mapping. Secondly, we advance simulated tempering as the basis of two rapidly mixing approximate inference algorithms, based on Markov chain Monte Carlo (MCMC) and Sequential Importance Sampling (SIS), respectively. These algorithms converge quickly even though the posterior being estimated is highly peaked and multimodal. Experiments on real robots and in simulation demonstrate the efficiency and robustness of our technique.

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A Partially Fixed Linearization Approach for Submap-Parametrized Smoothing and Mapping

2005 , Kipp, Alexander , Krauthausen, Peter , Dellaert, Frank

We present an extension of a smoothing approach to Simultaneous Localization and Mapping (SLAM). We have previously introduced Square-Root SAM, a Smoothing and Mapping approach to SLAM based on Levenberg-Marquardt (LM) optimization. It iteratively finds the optimal nonlinear least squares solution (ML), where one iteration comprises of a linearization step, a matrix factorization, and a back-substitution step. We introduce a submap parametrization which enables a rigid transformation of parts relative to each other during the optimization process. This parameterization is used in a multifrontal QR factorization approach, in which we partially fix the linearization point for a subset of the unknowns corresponding to sub-maps. This greatly accelerates the optimization of an entire SAM graph yet yields an exact solution.

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Data Driven MCMC for Appearance-based Topological Mapping

2005 , Dellaert, Frank , Ranganathan, Ananth

Probabilistic techniques have become the mainstay of robotic mapping, particularly for generating metric maps. In previous work, we have presented a hitherto nonexistent general purpose probabilistic framework for dealing with topological mapping. This involves the creation of Probabilistic Topological Maps (PTMs), a sample-based representation that approximates the posterior distribution over topologies given available sensor measurements. The PTM is inferred using Markov Chain Monte Carlo (MCMC) that overcomes the combinatorial nature of the problem. In this paper, we address the problem of integrating appearance measurements into the PTM framework. Specifically, we consider appearance measurements in the form of panoramic images obtained from a camera rig mounted on a robot. We also propose improvements to the efficiency of the MCMC algorithm through the use of an intelligent data-driven proposal distribution. We present experiments t hat illustrate the robustness and wide applicability of our algorithm.

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Grammatical Methods in Computer Vision: An Overview

2004 , Chanda, Gaurav , Dellaert, Frank

We review various methods and applications that have used grammars for solving inference problems in computer vision and pattern recognition. Grammars have been useful because they are intuitively simple to understand, and have very elegant representations. Their ability to model semantic interpretations of patterns, both spatial and temporal, have made them extremely popular in the research community. In this paper, we attempt to give an overview of what syntactic methods exist in the literature, and how they have been used as tools for pattern modeling and recognition. We also describe several practical applications, which have used them with great success.

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Visual SLAM with a Multi-Camera Rig

2006 , Kaess, Michael , Dellaert, Frank

Camera-based simultaneous localization and mapping or visual SLAM has received much attention recently. Typically single cameras, multiple cameras in a stereo setup or omni-directional cameras are used. We propose a different approach, where multiple cameras can be mounted on a robot in an arbitrary configuration. Allowing the cameras to face in different directions yields better constraints than single cameras or stereo setups can provide, simplifying the reconstruction of large-scale environments. And in contrast to omni-directional sensors, the available resolution can be focused on areas of interest depending on the application. We describe a sparse SLAM approach that is suitable for real-time reconstruction from such multi-camera configurations. We have implemented the system and show experimental results in a large-scale environment, using a custom made eight-camera rig.

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Square Root SAM: Simultaneous Localization and Mapping via Square Root Information Smoothing

2005 , Dellaert, Frank

Solving the SLAM problem is one way to enable a robot to explore, map, and navigate in a previously unknown environment. We investigate smoothing approaches as a viable alternative to extended Kalman filter-based solutions to the problem. In particular, we look at approaches that factorize either the associated information matrix or the measurement matrix into square root form. Such techniques have several significant advantages over the EKF: they are faster yet exact, they can be used in either batch or incremental mode, are better equipped to deal with non-linear process and measurement models, and yield the entire robot trajectory, at lower cost. In addition, in an indirect but dramatic way, column ordering heuristics automatically exploit the locality inherent in the geographic nature of the SLAM problem. In this paper, we present the theory underlying these methods, an interpretation of factorization in terms of the graphical model associated with the SLAM problem, and simulation results that underscore the potential of these methods for use in practice.

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Dirichlet Process based Bayesian Partition Models for Robot Topological Mapping

2004 , Ranganathan, Ananth , Dellaert, Frank

Robotic mapping involves finding a solution to the correspondence problem. A general purpose solution to this problem is as yet unavailable due to the combinatorial nature of the state space. We present a framework for computing the posterior distribution over the space of topological maps that solves the correspondence problem in the context of topological mapping. Since exact inference in this space is intractable, we present two sampling algorithms that compute sample-based representations of the posterior. Both the algorithms are built on a Bayesian product partition model that is derived from the mixture of Dirichlet processes model. Robot experiments demonstrate the applicability of the algorithms.

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Learning and Inferring Motion Patterns using Parametric Segmental Switching Linear Dynamic Systems

2006 , Oh, Sang Min , Rehg, James M. , Balch, Tucker , Dellaert, Frank

Switching Linear Dynamic System (SLDS) models are a popular technique for modeling complex nonlinear dynamic systems. An SLDS has significantly more descriptive power than an HMM by using continuous hidden states. However, the use of SLDS models in practical applications is challenging for several reasons. First, exact inference in SLDS models is computationally intractable. Second, the geometric duration model induced in standard SLDSs limits their representational power. Third, standard SLDSs do not provide a systematic way to robustly interpret systematic variations governed by higher order parameters. The contributions in this paper address all three challenges above. First, we present a data-driven MCMC sampling method for SLDSs as a robust and efficient approximate inference method. Second, we present segmental switching linear dynamic systems (S-SLDS), where the geometric distributions are replaced with arbitrary duration models. Third, we extend the standard model with a parametric model that can capture systematic temporal and spatial variations. The resulting parametric SLDS model (P-SLDS) uses EM to robustly interpret parametrized motions by incorporating additional global parameters that underly systematic variations of the overall motion. The overall development of the proposed inference methods and extensions for SLDSs provide a robust framework to interpret complex motions. The framework is applied to the honey bee dance interpretation task in the context of the ongoing BioTracking project at Georgia Institute of Technology. The experimental results suggest that the enhanced models provide an effective framework for a wide range of motion analysis applications.

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A Variational inference method for Switching Linear Dynamic Systems

2005 , Oh, Sang Min , Ranganathan, Ananth , Rehg, James M. , Dellaert, Frank

This paper aims to present a structured variational inference algorithm for switching linear dynamical systems (SLDSs) which was initially introduced by Pavlovic and Rehg. Starting with the need for the variational approach, we proceed to the derivation of the generic (model-independent) variational update formulas which are obtained under the mean field assumption. This leads us to the derivation of an approximate variational inference algorithm for an SLDS. The details of deriving the SLDS-specific variational update equations are presented.

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Robust Generative Subspace Modeling: The Subspace t Distribution

2004 , Khan, Zia , Dellaert, Frank

Linear latent variable models such as statistical factor analysis (SFA) and probabilistic principal component analysis (PPCA) assume that the data are distributed according to a multivariate Gaussian. A drawback of this assumption is that parameter learning in these models is sensitive to outliers in the training data. Approaches that rely on M-estimation have been introduced to render principal component analysis (PCA) more robust to outliers. M-estimation approaches assume the data are distributed according to a density with heavier tails than a Gaussian. Yet, these methods are limited in that they fail to define a probability model for the data. Data cannot be generated from these models, and the normalized probability of new data cannot evaluated. To address these limitations, we describe a generative probability model that accounts for outliers. The model is a linear latent variable model in which the marginal density over the data is a multivariate t, a distribution with heavier tails than a Gaussian. We present a computationally efficient expectation maximization (EM) algorithm for estimating the model parameters, and compare our approach with that of PPCA on both synthetic and real data sets.

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