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

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

Canonical Dual Algorithms for Global Optimization with Applications

Canonical duality theory provides a unified framework which can transform a nonconvex primal minimization problem to a canonical dual maximization problem over a convex domain without duality gap. But the global optimality is guaranteed by a certain positive definite condition and such condition is not always satisfied. The goal of this thesis aims to explore possible techniques that can be used to solve global optimization problems based on the canonical duality theory. Firstly, an algorithmic framework for canonical duality theory is established, which shows that the canonical dual algorithms can be developed in four aspects under the positive definite condition explicitly or implicitly, namely, (i) minimizing the primal problem, (ii) maximizing the canonical dual problem, (iii) solving a nonlinear equation caused by total complementary function, and (iv) solving a nonlinear equation caused by canonical dual function. Secondly, we show that if there exists a critical point of the canonical dual problem in the positive definite domain, by solving an equivalent semidefinite programming (SDP) problem, the corresponding global solution to the primal problem can be obtained easily via off-the-shelf software packages. A specific canonical dual algorithm is given for each problem, including sum of fourth-order polynomials minimization, nonconvex quadratically constrained quadratic program (QCQP), and boolean quadratic program (BQP). Thirdly, we propose a canonical primal-dual algorithm framework based on the total complementary function. Convergence analysis is discussed from the perspective of variational inequalities (VIs) and contraction methods. Specific canonical primal-dual algorithms for sum of fourth-order polynomials minimization is given as well. And a real-world application to the sensor network localization problem is illustrated. Next, a canonical sequential reduction approach is proposed to recover the approximate or global solution for the BQP problem. By fixing some previously known components, the original problem can be reduced sequentially to a lower dimension one. This approach is successfully applied to the well-known maxcut problem. Finally, we discuss the canonical dual approach applied to continuous time constrained optimal control. And it shows that the optimal control law for the n-dimensional constrained linear quadratic regulator can be achieved precisely via one-dimensional canonical dual variable, and for the optimal control problem with concave cost functional, an approximate solution can be obtained by introducing a linear perturbation term.Ph

(Global) Optimization: Historical notes and recent developments

Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning

Causal geometries and third-order ordinary differential equations

We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation. Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal Lorentzian metric on the space of 2-jets of functions of one variable. When the Wuenschmann invariant vanishes, the degenerate metric descends to a proper conformal Lorentzian metric on the space of solutions. In the general case, when the Wuenschmann invariant is not zero, we define the notion of a causal geometry, and show that the space of solutions supports one. The Wuenschmann invariant is then related to the projective curvature of the indicatrix curve cut out by the causal geometry in the projective tangent space. When the Wuenschmann vanishes, the causal structure is then precisely the sheaf of null geodesics of the Chern conformal structure. We then introduce a Lagrangian and associated Hamiltonian from which the degenerate conformal Lorentzian metric are constructed. Finally, necessary and sufficient conditions are given for a rank three degenerate conformal Lorentzian metric in four dimensions to correspond to a third-order differential equation

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