Certifiably optimal rotation and pose estimation based on the Cayley map

We present novel, convex relaxations for rotation and pose estimation problems that can a posteriori guarantee global optimality for practical measurement noise levels. Some such relaxations exist in the literature for specific problem setups that assume the matrix von Mises-Fisher distribution (a.k.a., matrix Langevin distribution or chordal distance) for isotropic rotational uncertainty. However, another common way to represent uncertainty for rotations and poses is to define anisotropic noise in the associated Lie algebra. Starting from a noise model based on the Cayley map, we define our estimation problems, convert them to Quadratically Constrained Quadratic Programs (QCQPs), then relax them to Semidefinite Programs (SDPs), which can be solved using standard interior-point optimization methods; global optimality follows from Lagrangian strong duality. We first show how to carry out basic rotation and pose averaging. We then turn to the more complex problem of trajectory estimation, which involves many pose variables with both individual and inter-pose measurements (or motion priors). Our contribution is to formulate SDP relaxations for all these problems based on the Cayley map (including the identification of redundant constraints) and to show them working in practical settings. We hope our results can add to the catalogue of useful estimation problems whose solutions can be a posteriori guaranteed to be globally optimal.

Certifiably Optimal Outlier-Robust Geometric Perception: Semidefinite Relaxations and Scalable Global Optimization

We propose the first general and scalable framework to design certifiable algorithms for robust geometric perception in the presence of outliers. Our first contribution is to show that estimation using common robust costs, such as truncated least squares (TLS), maximum consensus, Geman-McClure, Tukey's biweight, among others, can be reformulated as polynomial optimization problems (POPs). By focusing on the TLS cost, our second contribution is to exploit sparsity in the POP and propose a sparse semidefinite programming (SDP) relaxation that is much smaller than the standard Lasserre's hierarchy while preserving empirical exactness, i.e., the SDP recovers the optimizer of the nonconvex POP with an optimality certificate. Our third contribution is to solve the SDP relaxations at an unprecedented scale and accuracy by presenting [Formula: see text], a solver that blends global descent on the convex SDP with fast local search on the nonconvex POP. Our fourth contribution is an evaluation of the proposed framework on six geometric perception problems including single and multiple rotation averaging, point cloud and mesh registration, absolute pose estimation, and category-level object pose and shape estimation. Our experiments demonstrate that (i) our sparse SDP relaxation is empirically exact with up to 60%- 90% outliers across applications; (ii) while still being far from real-time, [Formula: see text] is up to 100 times faster than existing SDP solvers on medium-scale problems, and is the only solver that can solve large-scale SDPs with hundreds of thousands of constraints to high accuracy; (iii) [Formula: see text] safeguards existing fast heuristics for robust estimation (e.g., [Formula: see text] or Graduated Non-Convexity), i.e., it certifies global optimality if the heuristic estimates are optimal, or detects and allows escaping local optima when the heuristic estimates are suboptimal.

Improved Visual Localization via Graph Filtering

Vision-based localization is the problem of inferring the pose of the camera given a single image. One commonly used approach relies on image retrieval where the query input is compared against a database of localized support examples and its pose is inferred with the help of the retrieved items. This assumes that images taken from the same places consist of the same landmarks and thus would have similar feature representations. These representations can learn to be robust to different variations in capture conditions like time of the day or weather. In this work, we introduce a framework which aims at enhancing the performance of such retrieval-based localization methods. It consists in taking into account additional information available, such as GPS coordinates or temporal proximity in the acquisition of the images. More precisely, our method consists in constructing a graph based on this additional information that is later used to improve reliability of the retrieval process by filtering the feature representations of support and/or query images. We show that the proposed method is able to significantly improve the localization accuracy on two large scale datasets, as well as the mean average precision in classical image retrieval scenarios.

Rotation Averaging with the Chordal Distance: Global Minimizers and Strong Duality

In this paper we explore the role of duality principles within the problem of rotation averaging, a fundamental task in a wide range of applications. In its conventional form, rotation averaging is stated as a minimization over multiple rotation constraints. As these constraints are non-convex, this problem is generally considered challenging to solve globally. We show how to circumvent this difficulty through the use of Lagrangian duality. While such an approach is well-known it is normally not guaranteed to provide a tight relaxation. Based on spectral graph theory, we analytically prove that in many cases there is no duality gap unless the noise levels are severe. This allows us to obtain certifiably global solutions to a class of important non-convex problems in polynomial time. We also propose an efficient, scalable algorithm that outperforms general purpose numerical solvers by a large margin and compares favourably to current state-of-the-art. Further, our approach is able to handle the large problem instances commonly occurring in structure from motion settings and it is trivially parallelizable. Experiments are presented for a number of different instances of both synthetic and real-world data.

Incremental 3-D pose graph optimization for SLAM algorithm without marginalization

Pose graph optimization algorithm is a classic nonconvex problem which is widely used in simultaneous localization and mapping algorithm. First, we investigate previous contributions and evaluate their performances using KITTI, Technische Universität München (TUM), and New College data sets. In practical scenario, pose graph optimization starts optimizing when loop closure happens. An estimated robot pose meets more than one loop closures; Schur complement is the common method to obtain sequential pose graph results. We put forward a new algorithm without managing complex Bayes factor graph and obtain more accurate pose graph result than state-of-art algorithms. In the proposed method, we transform the problem of estimating absolute poses to the problem of estimating relative poses. We name this incremental pose graph optimization algorithm as G-pose graph optimization algorithm. Another advantage of G-pose graph optimization algorithm is robust to outliers. We add loop closure metric to deal with outlier data. Previous experiments of pose graph optimization algorithm use simulated data, which do not conform to real world, to evaluate performances. We use KITTI, TUM, and New College data sets, which are obtained by real sensor in this study. Experimental results demonstrate that our proposed incremental pose graph algorithm model is stable and accurate in real-world scenario.