# Fast Loop Closure Selection Method with Spatiotemporal Consistency for Multi-Robot Map Fusion
## 4 Proposed Method
### Topologically Related Loop Closures  
Following the definition of consistency proposed in [25], we know those accurate measurements \(\mathbf{Z}^{ab}\) are consistent with each other and further satisfy the consistency metric (3). In terms of topology, accurate inter-robot loop closures are also all topologically correct with each other, which refers to the sequence of measurements related to the overlapped regions of robot trajectories. However, perceptual aliasing is nearly inevitable in natural environments and can lead to false loop closures that will be mixed into the measurement set as part of the measurement. This issue drives us to distinguish between true and false measurements at the initial stage. The potential fact is that the false measurements may be inconsistent or topologically incorrect. The definition of consistency allows us to determine which subset is trusted from the perspective of entire measurement set in the inter-robot \(\mathbf{Z}^{ab}\), whereas it cannot be used to evaluate the degree of coincidence between two trajectories. It is certain that the presence of measurements \(\mathbf{Z}^{ab}\) means that the robots repeatedly observed the same location at different moments or by different robots. The distribution of inter-robot loop closures, to a certain extent, agrees with the distribution of regions where the two robot trajectories overlap. Accordingly, we seek to determine which portions of the two trajectories are overlapped, where the overlapped portions only consist of topologically related links (sequences of loop closures that relate coincident parts of the robot trajectories). For the sake of brevity, we embody each of these portions as a cluster. The measurements \(\mathbf{Z}^{ab}\) consist of the several clusters such that  
\[\mathbf{Z}^{ab}=\left\{\bigcup_{c=1}\mathbf{Z}^{ab}_{c}\mid\mathbf{Z}^{ab}_{m}\cap\mathbf{Z}^ {ab}_{n}=\varnothing,\forall(m\
eq n)\right\} \tag{5}\]  
Assuming that if there are no false constraints in the measurement set, the constraints within each cluster agree with each other and are topologically correct, the inter-clusters are mutually consistent, and the true subset of the inter-robot measurements should consist of all clusters. If the front-end introduces false measurements, the subsets can only be composed of partial constraints within clusters. Once the measurement set is divided into multiple clusters by the proposed method, it allows to search the entire measurement set in a block-wise way, instead of the traditional one-constraint-by-constraint, and then segmentally identify the regions of loop closures. In this paper, we demonstrate that the proposed clustering method is feasible and practical since the idea of segmentation conforms to the characteristics of the continuity of similar scenes. Now, we explain how the proposed method divides the measurement set into multiple clusters consisting of topologically related loop closures. As previously described, the cluster is initialized and maintained according to the timestamp of measurements, which belongs to the time category. We initialize a cluster \(\mathbf{Z}^{ab}_{c}=\left\{\mathbf{z}^{ab}_{ik}\right\}\) with the measurementgenerated by the front-end. Next, the formulation (6) is introduced to evaluate whether the new measurement \(\mathbf{z}_{jl}^{ab}\) belongs to an existing cluster or reinitializes a new cluster. \[\left\{\exists\mathbf{z}_{jl}^{ab}\in\mathbf{Z}_{c}^{ab}:\left\|t_{i}-t_{j}\right\|\leq t _{g}\ \oplus\ \left\|t_{k}-t_{l}\right\|\leq t_{g}\right\} \tag{6}\]  
where \(t_{g}\) is the threshold to distinguish cluster area, which is set by the running frequency of the front-end place recognition. The operator \(\oplus\) is AND operation in the Boolean logic function. Specifically, once the first measurement \(\mathbf{z}_{ik}^{ab}\) generates in the front-end, it assembles a new cluster.