Convex Global 3D Registration with Lagrangian Duality

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.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Global Optimality via Tight Convex Relaxations for Pose Estimation in Geometric 3D Computer Vision

In this thesis, we address a set of fundamental problems whose core difficulty boils down to optimizing over 3D poses. This includes many geometric 3D registration problems, covering well-known problems with a long research history such as the Perspective-n-Point (PnP) problem and generalizations, extrinsic sensor calibration, or even the gold standard for Structure from Motion (SfM) pipelines: The relative pose problem from corresponding features. Likewise, this is also the case for a close relative of SLAM, Pose Graph Optimization (also commonly known as Motion Averaging in SfM). The crux of this thesis contribution revolves around the successful characterization and development of empirically tight (convex) semidefinite relaxations for many of the aforementioned core problems of 3D Computer Vision. Building upon these empirically tight relaxations, we are able to find and certify the globally optimal solution to these problems with algorithms whose performance ranges as of today from efficient, scalable approaches comparable to fast second-order local search techniques to polynomial time (worst case). So, to conclude, our research reveals that an important subset of core problems that has been historically regarded as hard and thus dealt with mostly in empirical ways, are indeed tractable with optimality guarantees.Artificial Intelligence (AI) drives a lot of services and products we use everyday. But for AI to bring its full potential into daily tasks, with technologies such as autonomous driving, augmented reality or mobile robots, AI needs to be not only intelligent but also perceptive. In particular, the ability to see and to construct an accurate model of the environment is an essential capability to build intelligent perceptive systems. The ideas developed in Computer Vision for the last decades in areas such as Multiple View Geometry or Optimization, put together to work into 3D reconstruction algorithms seem to be mature enough to nurture a range of emerging applications that already employ as of today 3D Computer Vision in the background. However, while there is a positive trend in the use of 3D reconstruction tools in real applications, there are also some fundamental limitations regarding reliability and performance guarantees that may hinder a wider adoption, e.g. in more critical applications involving people's safety such as autonomous navigation. State-of-the-art 3D reconstruction algorithms typically formulate the reconstruction problem as a Maximum Likelihood Estimation (MLE) instance, which entails solving a high-dimensional non-convex non-linear optimization problem. In practice, this is done via fast local optimization methods, that have enabled fast and scalable reconstruction pipelines, yet lack of guarantees on most of the building blocks leaving us with fundamentally brittle pipelines where no guarantees exist

Initialization of 3D Pose Graph Optimization using Lagrangian duality

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-of-the-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.University of Malaga travel grant, the Spanish grant program FPU14/06098 and the project PROMOVE (DPI2014-55826-R), funded by the Spanish Government and the "European Regional Development Fund". Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Fast Global Optimality Verification in 3D SLAM

J. Briales, J. Gonzalez-Jimenez, "Fast Global Optimality Verification in 3D SLAM", in Int. Conf. on Intelligent Robots and Systems (IROS), Daejeon, Korea, IEEE/RSJ, pp. 4630-4636, 2016Graph-based SLAM has proved to be one of the most effective solutions to the Simultaneous localization and Mapping problem. This approach relies on nonlinear iterative optimization methods that in practice perform both accurately and efficiently. However, due to the non-convexity of the problem, the obtained solutions come with no guarantee of global optimality and may get stuck in local minima. The application of SLAM to many real-world applications cannot be conceived without additional control tools that detect possible suboptimalities as soon as possible in order to take corrective action and avoid catastrophic failure of the entire system. This paper builds upon the state-of-the-art framework [1] in verification for this problem and introduces a novel superior formulation that leads to a much higher efficiency. While retaining the same high effectiveness, the verification times of our proposal reduce up to >50x, paving the way for faster verification in critical real applications or in embedded low-power systems.We support our claims with extensive experiments with real and simulated data.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Spanish grant program FPU14/06098 and the project PROMOVE (DPI2014-55826-R), funded by the Spanish Government and the "European Regional Development Fund"

A Tighter Relaxation for the Relative Pose Problem Between Cameras

This paper tackles the resolution of the Relative Pose problem with optimality guarantees by stating it as an optimization problem over the set of essential matrices that minimizes the squared epipolar error. We relax this non-convex problem with its Shor’s relaxation, a convex program that can be solved by off-the-shelf tools. We follow the empirical observation that redundant but independent constraints tighten the relaxation. For that, we leverage equivalent definitions of the set of essential matrices based on the translation vectors between the cameras. Overconstrained characterizations of the set of essential matrices are derived by the combination of these definitions. Through extensive experiments on synthetic and real data, our proposal is empirically proved to remain tight and to require only 7 milliseconds to be solved even for the overconstrained formulations, finding the optimal solution under a wide variety of configurations, including highly noisy data and outliers. The solver cannot certify the solution only in very extreme cases, e.g.noise 100 pix and number of pair-wise correspondences under 15. The proposal is thus faster than other overconstrained formulations while being faster than the minimal ones, making it suitable for real-world applications that require optimality certification.Open Access funded by Universidad de Malaga / CBUA