Outlier feature matches and loop-closures that survived front-end data association can lead to catastrophic failures in the back-end optimization of large-scale point cloud based 3D reconstruction. To alleviate this problem, we propose a probabilistic approach for robust back-end optimization in the presence of outliers. More specifically, we model the problem as a Bayesian network and solve it using the Expectation-Maximization algorithm. Our approach leverages on a long-tail Cauchy distribution to suppress outlier feature matches in the odometry constraints, and a Cauchy-Uniform mixture model with a set of binary latent variables to simultaneously suppress outlier loop-closure constraints and outlier feature matches in the inlier loop-closure constraints. Furthermore, we show that by using a Gaussian-Uniform mixture model, our approach degenerates to the formulation of a state-of-the-art approach for robust indoor reconstruction. Experimental results demonstrate that our approach has comparable performance with the state-of-the-art on a benchmark indoor dataset, and outperforms it on a large-scale outdoor dataset. Our source code can be found on the project website.
@cite_10 achieve the state-of-the-art performance for indoor reconstruction with robust loop-closure. However, they assume no outlier feature matches in the odometry and inlier loop-closure constraints. We relax this assumption to achieve robust feature matching. More specifically, @cite_10 estimates a switch variable @cite_7 for each loop-closure constraint using line processes @cite_26 . Outlier loop-closures are disabled by setting the respective switch variables to zero. Additional switch prior terms are imposed and chosen empirically @cite_7 to prevent a trivial solution of removing all loop-closure constraints. In comparison, our approach does not require the additional prior terms. We estimate the posterior of a loop-closure being an inlier constraint in the Expectation step shown in Sec. . The EM approach is also used by @cite_29 . However, they solve a robust pose-graph optimization problem without coping with the feature matches for reconstruction.
@cite_29 @cite_26 @cite_10 @cite_7
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In this paper, we model the robust loop-closure pose-graph SLAM problem as a Bayesian network and show that it can be solved with the Classification Expectation-Maximization (EM) algorithm. In particular, we express our robust pose-graph SLAM as a Bayesian network where the robot poses and constraints are latent and observed variables. An additional set of latent variables is introduced as weights for the loop-constraints. We show that the weights can be chosen as the Cauchy function, which are iteratively computed from the errors between the predicted robot poses and observed loop-closure constraints in the Expectation step, and used to weigh the cost functions from the pose-graph loop-closure constraints in the Maximization step. As a result, outlier loop-closure constraints are assigned low weights and exert less influences in the pose-graph optimization within the EM iterations. To prevent the EM algorithm from getting stuck at local minima, we perform the EM algorithm multiple times where the loop constraints with very low weights are removed after each EM process. This is repeated until there are no more changes to the weights. We show proofs of the conceptual similarity between our EM algorithm and the M-Estimator. Specifically, we show that the weight function in our EM algorithm is equivalent to the robust residual function in the M-Estimator. We verify our proposed algorithm with experimental results from multiple simulated and real-world datasets, and comparisons with other existing works. Consider a Markov chain on an infinite tree T = (V,E) rooted at . In such a chain, once the initial root state ( ) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let µj denote the resulting measure for ( ) = j. µj is defined on configurations = ( (x))x2V 2 A V , where A is some finite set. Let µ n denote the restriction of µ to the sigma-algebra generated by the variables (x) where x is at distance exactly n from . Letting n = maxi,j2A dTV (µ n ,µ n ), where dTV denotes total variation distance, we say that the reconstruction problem is solvable if liminfn!1 n > 0. Reconstruction solvability roughly means that the n’th level of the tree contains a non-vanishing amount of information on the root of the tree, as n ! 1. In this paper we study the problem of robust reconstruction. Let be a non degenerate distribution on A and > 0. Let be chosen according to µ n and 0 be obtained from by letting for each node independently (v) = 0 (v) with probability 1 and 0 (v) to be an independent sample from otherwise. We denote by µ n [, ] the resulting measure on 0 . The measure µ n [, ] is a perturbation of the measure µ n . Letting n(, ) = maxi,j2A dTV (µ n [, ],µ n [, ]), we say that the reconstruction problem is -robust-solvable if liminfn!1 n(, ) > 0 for all 0 1, where 2(M) is the second largest eigenvalue of M (in absolute value), then for all non-degenerate , the reconstruction problem is -robust-solvable. We prove a converse and show that the reconstruction problem is not -robust-solvable if B| 2(M)| 2 < 1. This proves a conjecture by the second author and Peres. We also consider other models of noise and general trees. It is essential for a robot to be able to detect revisits or loop closures for long-term visual navigation. A key insight is that the loop-closing event inherently occurs sparsely, i.e., the image currently being taken matches with only a small subset (if any) of previous observations. Based on this observation, we formulate the problem of loop-closure detection as a sparse, convex 1-minimization problem. By leveraging on fast convex optimization techniques, we are able to efficiently find loop closures, thus enabling real-time robot navigation. This novel formulation requires no offline dictionary learning, as required by most existing approaches, and thus allows online incremental operation. Our approach ensures a global, unique hypothesis by choosing only a single globally optimal match when making a loop-closure decision. Furthermore, the proposed formulation enjoys a flexible representation, with no restriction imposed on how images should be represented, while requiring only that the representations be close to each other when the corresponding images are visually similar. The proposed algorithm is validated extensively using public real-world datasets. The neighbor-embedding (NE) algorithm for single-image super-resolution (SR) reconstruction assumes that the feature spaces of low-resolution (LR) and high-resolution (HR) patches are locally isometric. However, this is not true for SR because of one-to-many mappings between LR and HR patches. To overcome or at least to reduce the problem for NE-based SR reconstruction, we apply a joint learning technique to train two projection matrices simultaneously and to map the original LR and HR feature spaces onto a unified feature subspace. Subsequently, the k -nearest neighbor selection of the input LR image patches is conducted in the unified feature subspace to estimate the reconstruction weights. To handle a large number of samples, joint learning locally exploits a coupled constraint by linking the LR-HR counterparts together with the K-nearest grouping patch pairs. In order to refine further the initial SR estimate, we impose a global reconstruction constraint on the SR outcome based on the maximum a posteriori framework. Preliminary experiments suggest that the proposed algorithm outperforms NE-related baselines.