Equivalence and Divergence of Bayesian Log-Odds and Dempster's Combination Rule for 2D Occupancy Grids

Continuous occupancy mapping methods that learn spatial representations from sensor observations support predictions at arbitrary locations, but training a single global model can become computationally expensive as the amount of training data grows during large-scale mapping. Constructing local continuous occupancy submaps reduces the size of individual learning problems, but relative pose errors can cause inconsistent predictions in overlapping regions. We address these challenges by formulating large-scale continuous occupancy mapping as a submap joining problem. Each local submap is represented by a relevance vector machine regressing a continuous average log-odds field. Repeated observations within each cell are summarized by their average log-odds and observation count, yielding a Gaussian likelihood with count-dependent precision and enabling Gaussian-conjugate inference. The resulting local fields provide predictive means, variances, and analytical spatial derivatives. We then formulate submap joining as a joint pose--field optimization problem, with consistency constraints weighted by the predictive uncertainty of overlapping local submaps. With uncertainty weights held fixed within each Gauss--Newton iteration, our joint pose--field system can be reduced to an equivalent pose-only system. After the optimized submap poses are obtained, the global field value at any covered location can be recovered in closed form through variance-weighted fusion of predictions from the corresponding local submaps, without training an additional global model. Experiments on simulated and real-world 2D LiDAR datasets demonstrate improved pose and occupancy-map accuracy over the evaluated baselines, reduced map storage and joining time, and improved map-construction scalability.

Systems often need to combine two numerical assessments of the same yes/no question. The appropriate formula depends on what the numbers represent and on how the sources are related. Averaging is correct when one of several alternative interpretations applies; multiplying odds is correct when probability reports are based on conditionally independent evidence and a common prior; and probabilities of alternative successful derivations require their dependence or shared evidence to be taken into account. We state the assumptions behind several common combination rules and derive the corresponding combined probabilities. Two groups of Monte Carlo experiments address different questions. First, controlled generating mechanisms verify that the derived rule recovers the correct probability in the situations for which its assumptions hold. Second, the same mechanisms measure the consequences of using a mismatched rule, using logarithmic score and threshold decisions with different costs. Distinct pooling rules can produce the same binary decision at threshold 1/2 while assigning substantially different probabilities, so binary accuracy alone can conceal important differences. We also give probabilistic interpretations of conflicting-evidence rules and show that, for overlapping derivations, retaining the identities of shared uncertain premises permits direct calculation of the probability that at least one derivation is available. Pairwise combination of proof probabilities loses information when there are three or more derivations.

Key part of robotics, augmented reality, and digital inspection is dense 3D reconstruction from depth observations. Traditional volumetric fusion techniques, including truncated signed distance functions (TSDF), enable efficient and deterministic geometry reconstruction; however, they depend on heuristic weighting and fail to transparently convey uncertainty in a systematic way. Recent neural implicit methods, on the other hand, get very high fidelity but usually need a lot of GPU power for optimization and aren't very easy to understand for making decisions later on. This work presents BayesFusion-SDF, a CPU-centric probabilistic signed distance fusion framework that conceptualizes geometry as a sparse Gaussian random field with a defined posterior distribution over voxel distances. First, a rough TSDF reconstruction is used to create an adaptive narrow-band domain. Then, depth observations are combined using a heteroscedastic Bayesian formulation that is solved using sparse linear algebra and preconditioned conjugate gradients. Randomized diagonal estimators are a quick way to get an idea of posterior uncertainty. This makes it possible to extract surfaces and plan the next best view while taking into account uncertainty. Tests on a controlled ablation scene and a CO3D object sequence show that the new method is more accurate geometrically than TSDF baselines and gives useful estimates of uncertainty for active sensing. The proposed formulation provides a clear and easy-to-use alternative to GPU-heavy neural reconstruction methods while still being able to be understood in a probabilistic way and acting in a predictable way. GitHub: https://mazumdarsoumya.github.io/BayesFusionSDF