Planar Pose Graph Optimization: Duality, Optimal Solutions, and Verification

Carlone, Luca; Calafiore, Giuseppe Carlo; Tommolillo, Carlo; Dellaert, Frank

doi:10.1109/tro.2016.2544304

Cited by 82 publications

(118 citation statements)

References 55 publications

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“…Results: First we will focus on the results under increasing rotation noise. Previous works has extensively pointed [1], [8], [16], [18] that the tightness of the Lagrangian relaxation (6) is sensitive to rotation noise above certain threshold. This trend appears in Fig.…”

Section: Methodsmentioning

confidence: 99%

“…In the present work we present an appropriate recovery procedure that allows us to get a feasible estimate for the original problem from the solution of the dual (SDP) problem 1 When the graph underlying the problem is balanced or a tree [16].…”

Section: Related Workmentioning

confidence: 99%

“…In [16], a similar heuristic was exploited in the context of 2D PGO, outperforming all current 2D alternatives in convergence success. Our main contribution involves the non-trivial development of the necessary framework for exploiting this same heuristic in the case of 3D PGO.…”

Section: Related Workmentioning

confidence: 99%

“…If the solution S R for this linear relaxation (17) has 3 < rank(S R ) ≤ k, S R is not feasible for the lifted metric upgrade problem [UpgL]. In this case, we compute the rank-3 matrix S 0 closest to S R applying the matrix approximation lemma 9 and use this as an initial estimate to perform a local iterative search on the constrained problem (16). This operation can be readily performed using optimization approaches tailored to this kind of problem, e.g.…”

Section: A Metric Upgrade Of the Nullspacementioning

confidence: 99%

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Initialization of 3D pose graph optimization using Lagrangian duality

Briales

González-Jiménez

2017

2017 IEEE International Conference on Robotics and Automation (ICRA)

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Abstract-Pose Graph Optimization (PGO) is the de facto choice to solve the trajectory of an agent in Simultaneous Localization and Mapping (SLAM). The Maximum Likelihood Estimation (MLE) for PGO is a non-convex problem for which no known technique is able to guarantee a globally optimal solution under general conditions. In recent years, Lagrangian duality has proved suitable to provide good, frequently tight relaxations of the hard PGO problem through convex Semidefinite Programming (SDP). In this work, we build from the state-ofthe-art Lagrangian relaxation [1] and contribute a complete recovery procedure that, given the (tractable) optimal solution of the relaxation, provides either the optimal MLE solution if the relaxation is tight, or a remarkably good feasible guess if the relaxation is non-tight, which occurs in specially challenging PGO problems (very noisy observations, low graph connectivity, etc.). In the latter case, when used for initialization of local iterative methods, our approach outperforms other state-ofthe-art approaches converging to better solutions. We support our claims with extensive experiments.

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Section: Methodsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: A Metric Upgrade Of the Nullspacementioning

confidence: 99%

See 2 more Smart Citations

Initialization of 3D pose graph optimization using Lagrangian duality

Briales

González-Jiménez

2017

2017 IEEE International Conference on Robotics and Automation (ICRA)

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“…This provides particularly good approximations for many problems that can be reformulated as a Quadratically Constrained Quadratic Program (QCQP), where the relaxed problem becomes a Semidefinite Program (SDP) [4,15]. Some problems involving rotations can be characterized as QCQPs, such as Pose Graph Optimization, for which recent literature applying the Lagrangian dual relaxation has shown impressive results finding globally optimal solutions based solely on convex relaxations [11,10,7,40,8].…”

Section: Global Optimizationmentioning

confidence: 99%

Convex Global 3D Registration with Lagrangian Duality

Briales

González-Jiménez

2017

2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)

138

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The registration of 3D models by a Euclidean transformation is a fundamental task at the core of many application in computer vision. This problem is non-convex due to the presence of rotational constraints, making traditional local optimization methods prone to getting stuck in local minima. This paper addresses finding the globally optimal transformation in various 3D registration problems by a unified formulation that integrates common geometric registration modalities (namely point-to-point, point-to-line and point-to-plane). This formulation renders the optimization problem independent of both the number and nature of the correspondences.The main novelty of our proposal is the introduction of a strengthened Lagrangian dual relaxation for this problem, which surpasses previous similar approaches [32] in effectiveness. In fact, even though with no theoretical guarantees, exhaustive empirical evaluation in both synthetic and real experiments always resulted on a tight relaxation that allowed to recover a guaranteed globally optimal solution by exploiting duality theory.Thus, our approach allows for effectively solving the 3D registration with global optimality guarantees while running at a fraction of the time for the state-of-the-art alternative [34], based on a more computationally intensive Branch and Bound method.

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