The sections that follow discuss work on robustness to try overcome many of the errors associated with non-Gaussian measurements being introduced as Gaussian measurement likelihood models. Several robustness efforts have focused identi- fying and removing ”bad” measurement factors from the factor graph. The as- sumption of bad factors implies incorrect measurements were made, either by bad data association or otherwise, and should be removed from the inference prob- lem. Many methods in SLAM either avoid ”bad” measurements with highly- engineerined front-end processes, or preprocessing of an existing factor graph be- fore actual variable inference is done. For example, Latif et al. [132] show the value of finding consensus at the front-end stage, delaying loop closure constraints until several new constraints agree and adding them to the factor graph as a batch of new constraints. This section takes a brief look at some of these methods.
2.7.1
Null-hypothesis Approaches
Switch variables were proposed in 2012 by Sunderhauf et al. [220] as binary slack variables into the optimization that can enable or disable each measurement. Mea- surements which are inconsistent with the rest of the graph are discarded through multiplication by zero. An additional variable, being one or zero, is added to that factor in the least square sum objective function, and introduced as part of the inference procedure. A user specified penalty is used to ensure some attempt is made to keep the factor active during the optimization process. Switch variables are comparable to a null-hypothesis approach [181], and has the disadvantage of ignoring information and relying heavily selecting the correct penalty values.
Further Olson et al. [181] points out that switch variables increase the num- ber of variables and increases fill-in during inference which may result in notable computational performance loss. Furthermore, the null hypothesis approach may easily discard lonely, but valid, measurements and thereby ignore true data.
More recently, Graham [76] suggests using an expectation maximization (EM) approach to smoothly transition poorly matched measurements to assumed ”out- liers” by adjusting their measurement covariance. The EM algorithm is used to iterate between covariance weight selection and optimal variable assignments and thereby suppresses outlier-like measurements and emphasizing the majority of constraints which form consensus.
Moving closer to benefits offered by the symbolic structure of the Bayes tree, the hybrid continuous-discrete inference by Segal et al. [205,206] uses discrete states to enable or disable measurements, much like switch variables. The difference, how- ever, is that the likelihood of enabling or disabling a factor is encapsulated by a discrete belief. By explicitly splitting posterior belief and optimal variable assign- ment computations, a best fit solution is found by searching for the posterior belief over all the discrete variable, in multiple passes over the Bayes tree, relative to the initialized state. Segal’s work suggests an underlying synergy between ambiguity through belief and consensus amongst multiple hypotheses.
2.7.2
Max-Component Approach (Max-product)
Olson et al. [180,181] proposed the max-mixtures approach which selects the local maximal weighted Gaussian from a mixture of Gaussians before continuing with a parametric optimization routine. Their approach is akin to max-product inference, which greatly simplifies the inference problem by discarding all but the most likely hypothesis for each factor before inference, using a local if statement. Once each
factor has locally selected the most likely Gaussian hypothesis, a usual solution (as discussed above) is used.
2.7.3
Multi-hypothesis Approaches
Rather than retrofitting a single parametric solution with a null-hypothesis, an- other approach, as recently suggested by Huang et al. [97], suggest solving mul- tiple parametric problems in parallel and then picking the most likely solution. The FastSLAM approach of Thrun et al. [226] (mentioned earlier) also maintains parallel trajectories, but keep all permutations in separate particles with very high dimension.
While multi-hypothesis approaches seem appealing, one should not underesti- mate the complexity associated with tracking all possible hypotheses in a system. Consider a case where several data association uncertainties, such as loop closures to objects, is to be deferred to the back-end solution. Following only the forward trajectory in time, each new binary decision introduces a doubling in the possible permutation of choices.
Therefore, if ten binary associations are to be tracked, there are 1024 possible hypotheses. FastSLAM had skirted this problem by requiring the front-end pro- cess to not request all hypotheses, assuming a local solution where only the most dominant modes in the current trajectory are being tracked, but the user must make the decision as to which modes are dominant. This behavior is a contrast to our proposed method where the back-end inference solution determines through consensus which modes are dominant, purely based on all uncertainty modeled in the factor graph.
Consider the factor graph view of parametric multi-hypothesis methods. Each possible solution permutation (hypothesis) is explicitly solved with a slightly dif- ferent factor graph. If all possible permutations are tracked, the correct solution will be contained in one of the many available solutions. It is important to note that using a method like RANSAC on such a collection of solutions is not valid, since each solution represents a different problem and not a Monte Carlo style variation on the same problem which may express consensus.
Intuitively, the factor graph encodes a random field where the belief interpreta- tion can capture all the uncertainty in the system. The Bayes tree is a data structure which precisely encodes the type of structure needed for multi-hypothesis track- ing, by aggregating all information during the upward pass from leaves to root. Consensus should occur as information is combined from sibling cliques. Fur- thermore, the inference algorithm selected modes are passed back down the tree
with the full posterior result already available during the downward pass.
The matrix permanent method introduced by Atanasov et al. [8] for semantic based mapping is considered a multi-hypothesis approach where object recogni- tion with semantic labels are used to help discard false loop closure proposals and help detect other possible loop closures. The matrix permanent computation is used to search across all possible hypotheses and extract the consensus set, in- cluding discrete labels on object semantics. The matrix permanent computation is equally expensive as considering all possible combinations and does not support incremental inference offered by the Bayes tree approach.