In contrast to local optimisation, global optimisation searches over the entire paramet- ric domain. Optimising over the entire domain does not guarantee that the optimal solution is found, however; for that, a globally-optimal algorithm is required. In this section, the global optimisation problem is formulated and methods for optimising it in across the entire parametric domain are briefly outlined. The fundamental property of global optimisation algorithms is that a good parameter initialisation is not required.
The global optimisation problem for geometric sensor data alignment uses the same formulation as (3.85), except parameter initialisations are not required. As for the local optimisation formulation, functions to compute the first and second order partial derivatives with respect to the parameters can also be supplied to the solver if required. Non-convexity and non-differentiability are the primary motivators for global op- timisation. Due to the non-convexity of the objective function, local optimisation can at best find a local optimum, typically the optimum closest in some sense to the initial parameter vector. While objective functions can be designed to smooth out the func- tion landscape and widen the basin of convergence of the correct solution, this cannot mitigate the problem entirely. The more the objective function is smoothed, the less it represents the underlying geometric alignment problem. Thus, while it might be easier to find the solution, the solution may no longer be relevant to the alignment problem. Non-differentiability presents a different challenge. Most good local optimisation algo- rithms need to compute exact or approximate gradients in order to converge to a local optimum. As a result, continuous and differentiable functions can be handled well by local optimisers, but discrete and non-differentiable functions cannot.
As observed in Section 3.5, the alignment problem is a constrained nonlinear opti- misation problem. There are many approaches to solving this problem globally. It can be helpful to think of geometric alignment algorithms as jointly solving for the trans- formation and correspondences. Hence, there are two options for a global algorithm: to lead with search over the transformation space or the correspondence space.
The first approach has transformation search lead the correspondence search. As foreshadowed in Section 3.5, a naïve approach is to run multiple instances of a local optimisation algorithm using different initial parameters. These initialisations can cover the parametric domain systematically or randomly, however it can be useful to use an optimally self-avoiding quasi-random distribution [David et al., 2002]. More sophisticated approaches in this class apply methods such as particle filtering [Sandhu et al., 2010], genetic algorithms [Silva et al., 2005] or Kalman filtering [Moreno-Noguer et al., 2008] to intelligently select the parameter initialisations.
While any local optimisation algorithm can be made global in this way, specifically global algorithms tend to approach the problem in a different way. In particular, many methods search over the parametric domain implicitly by instead searching explicitly over the correspondences. Hence in these approaches correspondence search leads the transformation search. RANdom SAmple Consensus (RANSAC), introduced by Fis- chler and Bolles [1981], is a robust but non-deterministic global method for solving the consensus set maximisation problem (3.62). As such, it requires a set of putative cor- respondences and is therefore not solving the geometric alignment problem. Nonethe- less, it is a fundamental algorithm and the underlying principle has been extended to correspondence-free alignment. RANSAC maximises the consensus set by stochasti- cally generating a minimal set, computing the transformation from this set, computing the cardinality of the consensus set given this transformation, updating the parameters if a better cardinality was found, and repeating. The minimal sets differ depending on the data and the problem. For example, three 3D–3D point correspondences are a minimal set for point-set registration and three 2D–3D point correspondences are a minimal set for camera pose estimation. The RANSAC algorithm can be modified to work without correspondences by introducing another stochastic step, where a minimal set is selected at random from dataset X2 after one has been selected at random from datasetX1 [Grimson, 1990]. This approach scales very poorly with the number of data elements and is not practical for most problems.
The most common way to pre-compute a correspondence set for RANSAC is using feature correspondences [Rusu et al., 2009]. Within a single modality, this can be an effective strategy, particularly if the features are robust and reproducible. However, for feature extraction across multiple modalities, such as 2D–3D alignment, this is a non-trivial unsolved problem. Even in a single modality, factors such as variable
sampling densities, repetitive features and occlusions make the correspondence prob- lem challenging. However, one class of RANSAC-based approaches does not require feature extraction: congruent set methods. These methods extract all approximately- congruent near-minimal sets directly from the raw sensor data and use RANSAC to find the alignment using these sets. The 4-Points Congruent Sets method (4PCS) [Aiger et al., 2008], which provides a way to rapidly extract coplanar 4-point sets, its extension Super4PCS [Mellado et al., 2014], which exploits a clever data structure to achieve linear-time performance, and the 2-Points+Normal Sets (2PNS) method [Ra- poso and Barreto, 2017], which uses normals to reduce the number of points needed, are examples of these methods.
The primary disadvantage of all of these global optimisation methods is that they do not necessarily converge to the global optimum. While they are not restricted to finding the local optimum in the neighbourhood of a parameter initialisation, they may instead find a local optimum somewhere else in the parametric domain. To guarantee that the global optimum has been found, globally-optimal methods, such as branch- and-bound, must be used.