GVU Technical Report Series

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GVU Technical Report Series

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Now showing 1 - 9 of 9

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

(Georgia Institute of Technology, 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.
Square Root SAM: Simultaneous Localization and Mapping via Square Root Information Smoothing

(Georgia Institute of Technology, 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.
A Variational inference method for Switching Linear Dynamic Systems

(Georgia Institute of Technology, 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.
Data Driven MCMC for Appearance-based Topological Mapping

(Georgia Institute of Technology, 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.
Dirichlet Process based Bayesian Partition Models for Robot Topological Mapping

(Georgia Institute of Technology, 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.
Robust Generative Subspace Modeling: The Subspace t Distribution

(Georgia Institute of Technology, 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.
Grammatical Methods in Computer Vision: An Overview

(Georgia Institute of Technology, 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.
An MCMC-based Particle Filter for Tracking Multiple Interacting Targets

(Georgia Institute of Technology, 2003) Khan, Zia ; Balch, Tucker ; Dellaert, Frank

We describe a Markov chain Monte Carlo based particle filter that effectively deals with interacting targets, i.e., targets that are influenced by the proximity and/or behavior of other targets. Such interactions cause problems for traditional approaches to the data association problem. In response, we developed a joint tracker that includes a more sophisticated motion model to maintain the identity of targets throughout an interaction, drastically reducing tracker failures. The paper presents two main contributions: (1) we show how a Markov random field (MRF) motion prior, built on the fly at each time step, can substantially improve tracking when targets interact, and (2) we show how this can be done efficiently using Markov chain Monte Carlo (MCMC) sampling. We prove that incorporating an MRF to model interactions is equivalent to adding an additional interaction factor to the importance weights in a joint particle filter. Since a joint particle filter suffers from exponential complexity in the number of tracked targets, we replace the traditional importance sampling step in the particle filter with an MCMC sampling step. The resulting filter deals efficiently and effectively with complicated interactions when targets approach each other. We present both qualitative and quantitative results to substantiate the claims made in the paper, including a large scale experiment on a video-sequence of over 10,000 frames in length.
The Expectation Maximization Algorithm

(Georgia Institute of Technology, 2002) Dellaert, Frank

This note represents my attempt at explaining the EM algorithm. This is just a slight variation on Tom Minka's tutorial, perhaps a little easier (or perhaps not). It includes a graphical example to provide some intuition.