  Nonlinear estimation in robotics and vision is typically plagued with
outliers due to wrong data association, or to incorrect detections from signal
processing and machine learning methods. This paper introduces two unifying
formulations for outlier-robust estimation, Generalized Maximum Consensus
(G-MC) and Generalized Truncated Least Squares (G-TLS), and investigates
fundamental limits, practical algorithms, and applications. Our first
contribution is a proof that outlier-robust estimation is inapproximable: in
the worst case, it is impossible to (even approximately) find the set of
outliers, even with slower-than-polynomial-time algorithms (particularly,
algorithms running in quasi-polynomial time). As a second contribution, we
review and extend two general-purpose algorithms. The first, Adaptive Trimming
(ADAPT), is combinatorial, and is suitable for G-MC; the second, Graduated
Non-Convexity (GNC), is based on homotopy methods, and is suitable for G-TLS.
We extend ADAPT and GNC to the case where the user does not have prior
knowledge of the inlier-noise statistics (or the statistics may vary over time)
and is unable to guess a reasonable threshold to separate inliers from outliers
(as the one commonly used in RANSAC). We propose the first minimally tuned
algorithms for outlier rejection, that dynamically decide how to separate
inliers from outliers. Our third contribution is an evaluation of the proposed
algorithms on robot perception problems: mesh registration, image-based object
detection (shape alignment), and pose graph optimization. ADAPT and GNC execute
in real-time, are deterministic, outperform RANSAC, and are robust up to 80-90%
outliers. Their minimally tuned versions also compare favorably with the state
of the art, even though they do not rely on a noise bound for the inliers.
