We study the problem of robust subspace recovery (RSR) in the presence of adversarial outliers. That is, we seek a subspace that contains a large portion of a dataset when some fraction of the data points are arbitrarily corrupted. We first examine a theoretical estimator that is intractable to calculate and use it to derive information-theoretic bounds of exact recovery. We then propose two tractable estimators: a variant of RANSAC and a simple relaxation of the theoretical estimator. The two estimators are fast to compute and achieve state-of-the-art theoretical performance in a noiseless RSR setting with adversarial outliers. The former estimator achieves better theoretical guarantees in the noiseless case, while the latter estimator is robust to small noise, and its guarantees significantly improve with non-adversarial models of outliers. We give a complete comparison of guarantees for the adversarial RSR problem, as well as a short discussion on the estimation of affine subspaces.
An example of a method with guarantees for general position outliers is RandomizedFind. It searches for linearly dependent @math -subsets of points, and thus linearly dependent outliers present a problem @cite_2 . Another similar method is a RANSAC-type algorithm that sub-samples @math points until a linearly dependent subset is found @cite_9 . This procedure does not work with adversarial outliers for a similar reason. Indeed, suppose that outliers all lie on a 1-dimensional subspace. Then, if one samples a subset with just two of these outliers, one would return a corrupted estimate of the subspace. We will later discuss modifications of the RANSAC algorithm that work for these degenerate cases up to certain limits.
@cite_9 @cite_2
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In this paper we consider the problem of outlier removal for large scale multiview reconstruction problems. An efficient and very popular method for this task is RANSAC. However, as RANSAC only works on a subset of the images, mismatches in longer point tracks may go undetected. To deal with this problem we would like to have, as a post processing step to RANSAC, a method that works on the entire (or a larger) part of the sequence. In this paper we consider two algorithms for doing this. The first one is related to a method by Sim & Hartley where a quasiconvex problem is solved repeatedly and the error residuals with the largest error is removed. Instead of solving a quasiconvex problem in each step we show that it is enough to solve a single LP or SOCP which yields a significant speedup. Using duality we show that the same theoretical result holds for our method. The second algorithm is a faster version of the first, and it is related to the popular method of L 1 -optimization. While it is faster and works very well in practice, there is no theoretical guarantee of success. We show that these two methods are related through duality, and evaluate the methods on a number of data sets with promising results.1 Outliers pose a problem in all real structure from motion systems. Due to the use of automatic matching methods one has to expect that a (sometimes very large) portion of the detected correspondences can be incorrect. In this paper we propose a method that estimates the relative translation between two cameras and simultaneously maximizes the number of inlier correspondences. Traditionally, outlier removal tasks have been addressed using RANSAC approaches. However, these are random in nature and offer no guarantees of finding a good solution. If the amount of mismatches is large, the approach becomes costly because of the need to evaluate a large number of random samples. In contrast, our approach is based on the branch and bound methodology which guarantees that an optimal solution will be found. While most optimal methods trade speed for optimality, the proposed algorithm has competitive running times on problem sizes well beyond what is common in practice. Experiments on both real and synthetic data show that the method outperforms state-of-the-art alternatives, including RANSAC, in terms of solution quality. In addition, the approach is shown to be faster than RANSAC in settings with a large amount of outliers. In this paper, we propose an efficient algorithm, called vector field consensus, for establishing robust point correspondences between two sets of points. Our algorithm starts by creating a set of putative correspondences which can contain a very large number of false correspondences, or outliers, in addition to a limited number of true correspondences (inliers). Next, we solve for correspondence by interpolating a vector field between the two point sets, which involves estimating a consensus of inlier points whose matching follows a nonparametric geometrical constraint. We formulate this a maximum a posteriori (MAP) estimation of a Bayesian model with hidden latent variables indicating whether matches in the putative set are outliers or inliers. We impose nonparametric geometrical constraints on the correspondence, as a prior distribution, using Tikhonov regularizers in a reproducing kernel Hilbert space. MAP estimation is performed by the EM algorithm which by also estimating the variance of the prior model (initialized to a large value) is able to obtain good estimates very quickly (e.g., avoiding many of the local minima inherent in this formulation). We illustrate this method on data sets in 2D and 3D and demonstrate that it is robust to a very large number of outliers (even up to 90 ). We also show that in the special case where there is an underlying parametric geometrical model (e.g., the epipolar line constraint) that we obtain better results than standard alternatives like RANSAC if a large number of outliers are present. This suggests a two-stage strategy, where we use our nonparametric model to reduce the size of the putative set and then apply a parametric variant of our approach to estimate the geometric parameters. Our algorithm is computationally efficient and we provide code for others to use it. In addition, our approach is general and can be applied to other problems, such as learning with a badly corrupted training data set. This paper presents a novel improved RANSAC algorithm based on probability and DS evidence theory to deal with the robust pose estimation in robot 3D map building. In this proposed RANSAC algorithm, a parameter model is estimated by using a random sampling test set. Based on this estimated model, all points are tested to evaluate the fitness of current parameter model and their probabilities are updated by using a total probability formula during the iterations. The maximum size of inlier set containing the test point is taken into account to get a more reliable evaluation for test points by using DS evidence theory. Furthermore, the theories of forgetting are utilized to filter out the unstable inliers and improve the stability of the proposed algorithm. In order to boost a high performance, an inverse mapping sampling strategy is adopted based on the updated probabilities of points. Both the simulations and real experimental results demonstrate the feasibility and effectiveness of the proposed algorithm.