Absolute pose estimation is a fundamental problem in computer vision, and it is a typical parameter estimation problem, meaning that efforts to solve it will always suffer from outlier-contaminated data. Conventionally, for a fixed dimensionality d and the number of measurements N, a robust estimation problem cannot be solved faster than O(N^d). Furthermore, it is almost impossible to remove d from the exponent of the runtime of a globally optimal algorithm. However, absolute pose estimation is a geometric parameter estimation problem, and thus has special constraints. In this paper, we consider pairwise constraints and propose a globally optimal algorithm for solving the absolute pose estimation problem. The proposed algorithm has a linear complexity in the number of correspondences at a given outlier ratio. Concretely, we first decouple the rotation and the translation subproblems by utilizing the pairwise constraints, and then we solve the rotation subproblem using the branch-and-bound algorithm. Lastly, we estimate the translation based on the known rotation by using another branch-and-bound algorithm. The advantages of our method are demonstrated via thorough testing on both synthetic and real-world data
@cite_19: We describe a new framework for globally solving the 3D-3D registration problem with unknown point correspondences. This problem is significant as it is frequently encountered in many applications. Existing methods are not fully satisfactory, mainly due to the risk of local minima. Our framework is grounded on the Lipschitz global optimization theory. It achieves a guaranteed global optimality without any initialization. By exploiting the special structure of the problem itself and of the 3D rotation space SO(3), we propose a box-and-ball algorithm, which solves the problem efficiently. The main idea of the work can be applied to many other problems as well.
@cite_5: Point cloud alignment is a common problem in computer vision and robotics, with applications ranging from 3D object recognition to reconstruction. We propose a novel approach to the alignment problem that utilizes Bayesian nonparametrics to describe the point cloud and surface normal densities, and branch and bound (BB) optimization to recover the relative transformation. BB uses a novel, refinable, near-uniform tessellation of rotation space using 4D tetrahedra, leading to more efficient optimization compared to the common axis-angle tessellation. We provide objective function bounds for pruning given the proposed tessellation, and prove that BB converges to the optimum of the cost function along with providing its computational complexity. Finally, we empirically demonstrate the efficiency of the proposed approach as well as its robustness to real-world conditions such as missing data and partial overlap.
@cite_15: 
@cite_34: Recently there has been a surge of interest to use branch-and-bound (bnb) optimisation for 3D point cloud registration. While bnb guarantees globally optimal solutions, it is usually too slow to be practical. A fundamental source of difficulty is the search for the rotation parameters in the 3D rigid transform. In this work, assuming that the translation parameters are known, we focus on constructing a fast rotation search algorithm. With respect to an inherently robust geometric matching criterion, we propose a novel bounding function for bnb that allows rapid evaluation. Underpinning our bounding function is the usage of stereographic projections to precompute and spatially index all possible point matches. This yields a robust and global algorithm that is significantly faster than previous methods. To conduct full 3D registration, the translation can be supplied by 3D feature matching, or by another optimisation framework that provides the translation. On various challenging point clouds, including those taken out of lab settings, our approach demonstrates superior efficiency.
@cite_12: The Iterative Closest Point (ICP) algorithm is one of the most widely used methods for point-set registration. However, being based on local iterative optimization, ICP is known to be susceptible to local minima. Its performance critically relies on the quality of the initialization and only local optimality is guaranteed. This paper presents the first globally optimal algorithm, named Go-ICP, for Euclidean (rigid) registration of two 3D point-sets under the @math error metric defined in ICP. The Go-ICP method is based on a branch-and-bound scheme that searches the entire 3D motion space @math . By exploiting the special structure of @math geometry, we derive novel upper and lower bounds for the registration error function. Local ICP is integrated into the BnB scheme, which speeds up the new method while guaranteeing global optimality. We also discuss extensions, addressing the issue of outlier robustness. The evaluation demonstrates that the proposed method is able to produce reliable registration results regardless of the initialization. Go-ICP can be applied in scenarios where an optimal solution is desirable or where a good initialization is not always available.
Another topic that is closely related to the absolute pose estimation problem is point set registration @cite_15 @cite_12 . The only difference is that in the point set registration, there is no existing point correspondence. Similarly, the search for globally optimal solutions is a hot topic in the field of point set registration, and the branch-and-bound algorithm has also been broadly applied in recent related studies. One of the most successful algorithms for this purpose may be the algorithm proposed in @cite_19 and its subsequent versions @cite_15 @cite_12 @cite_34 . These works are all based on rotation search theory, and for @math optimization in particular, a more systematic scheme called the nested branch-and-bound is applied. Moreover, the decoupling methods presented in @cite_5 improve efficiency, which inspires us to decouple the rotation and translation subproblems by means of pairwise constraints.