Optimal Estimation on the Graph Cycle Space

Russell, Wm. Joshua; Klein, Daniel J.; Hespanha, João P.

doi:10.1109/tsp.2011.2117422

Cited by 24 publications

(5 citation statements)

References 26 publications

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“…Lemma 1 (Orthogonal complements [42]). For a connected graph G, the transpose of the cycle basis matrix C T is an orthogonal complement of the transpose of the reduced incidence matrix A T , i.e., 1) (A T C T ) is a square matrix of full rank; and 2) CA T = 0 �×n .…”

Section: A Computational Graph Theorymentioning

confidence: 99%

From Angular Manifolds to the Integer Lattice: Guaranteed Orientation Estimation With Application to Pose Graph Optimization

Carlone

Censi

2014

IEEE Trans. Robot.

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Estimating the orientations of nodes in a pose graph from relative angular measurements is challenging because the variables live on a manifold product with nontrivial topology and the maximum-likelihood objective function is non-convex and has multiple local minima; these issues prevent iterative solvers to be robust for large amounts of noise. This paper presents an approach that allows working around the problem of multiple minima, and is based on the insight that the original estimation problem on orientations is equivalent to an unconstrained quadratic optimization problem on integer vectors. This equivalence provides a viable way to compute the maximum likelihood estimate and allows guaranteeing that such estimate is almost surely unique. A deeper consequence of the derivation is that the maximum likelihood solution does not necessarily lead to an estimate that is "close" to the actual nodes orientations, hence it is not necessarily the best choice for the problem at hand. To alleviate this issue, our algorithm computes a set of estimates, for which we can derive precise probabilistic guarantees. Experiments show that the method is able to tolerate extreme amounts of noise (e.g., σ = 30 • on each measurement) that are above all noise levels of sensors commonly used in mapping. For most range-finder-based scenarios, the multihypothesis estimator returns only a single hypothesis, because the problem is very well constrained. Finally, using the orientations estimate provided by our method to bootstrap the initial guess of pose graph optimization methods improves their robustness and makes them avoid local minima even for high levels of noise.

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Section: A Computational Graph Theorymentioning

confidence: 99%

From Angular Manifolds to the Integer Lattice: Guaranteed Orientation Estimation With Application to Pose Graph Optimization

Carlone

Censi

2014

IEEE Trans. Robot.

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“…It is based on a convex relaxation of an L1 formulation of the synchronization problem and comes with exact and stable recovery guarantees under a large set of scenarios. Russel et al develop a decentralized algorithm for synchronization on the group of translations R n [33]. Barooah and Hespanha study the covariance of the BLUE estimator for synchronization on R n with anchors.…”

Section: Previous Workmentioning

confidence: 99%

Cramer-Rao bounds for synchronization of rotations

Boumal

Singer²,

Absil

et al. 2013

Information and Inference

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Synchronization of rotations is the problem of estimating a set of rotations Ri ∈ SO(n), i = 1 . . . N based on noisy measurements of relative rotations R i R j . This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cramér-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise.

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“…In this section we will give a brief account of the synchronisation in the group of real numbers R with the sum (see also [18]).…”

Section: B Synchronisation In (R +)mentioning

confidence: 99%

Distributed Interval Synchronization

Fusiello,

Montessoro

2024

IEEE Access

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In this paper we address the problem of using pairwise measures associated with the edges of a graph to obtain absolute consistent measures associated with the nodes. This problem is known in the literature as graph synchronisation. In particular, we rigorously deal with the uncertainty affecting the measures thanks to the interval analysis approach. We propose an asynchronous, distributed algorithm that converges to an interval solution that represents all possible sharp solutions consistent with the measures.INDEX TERMS Uncertainty in measurements, Synchronization, Interval Analysis, Distributed algorithms I. INTRODUCTION

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Optimal Estimation on the Graph Cycle Space

Cited by 24 publications

References 26 publications

From Angular Manifolds to the Integer Lattice: Guaranteed Orientation Estimation With Application to Pose Graph Optimization

From Angular Manifolds to the Integer Lattice: Guaranteed Orientation Estimation With Application to Pose Graph Optimization

Cramer-Rao bounds for synchronization of rotations

Distributed Interval Synchronization

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