OptMap: Geometric Map Distillation via Submodular Maximization

TL;DR

OptMap tackles real-time, application-specific geometric map distillation from large LiDAR datasets by formulating map informativeness as a submodular maximization problem. It introduces Continuous Exemplar-Based Clustering (CEBC) as a scalable reward that rewards representativeness and diversity while enabling input-set reduction, and pairs it with dynamically reordered streaming to mitigate input-order bias. The approach is augmented with parallel map loading, tight initial bounds, and optional position/time constraints, enabling near-optimal distillation with minimal computation. Experiments across diverse datasets and open-source ROS packages demonstrate CEBC's effectiveness and the practicality of real-time map summarization for planning and localization tasks.

Abstract

Autonomous robots rely on geometric maps to inform a diverse set of perception and decision-making algorithms. As autonomy requires reasoning and planning on multiple scales of the environment, each algorithm may require a different map for optimal performance. Light Detection And Ranging (LiDAR) sensors generate an abundance of geometric data to satisfy these diverse requirements, but selecting informative, size-constrained maps is computationally challenging as it requires solving an NP-hard combinatorial optimization. In this work we present OptMap: a geometric map distillation algorithm which achieves real-time, application-specific map generation via multiple theoretical and algorithmic innovations. A central feature is the maximization of set functions that exhibit diminishing returns, i.e., submodularity, using polynomial-time algorithms with provably near-optimal solutions. We formulate a novel submodular reward function which quantifies informativeness, reduces input set sizes, and minimizes bias in sequentially collected datasets. Further, we propose a dynamically reordered streaming submodular algorithm which improves empirical solution quality and addresses input order bias via an online approximation of the value of all scans. Testing was conducted on open-source and custom datasets with an emphasis on long-duration mapping sessions, highlighting OptMap's minimal computation requirements. Open-source ROS1 and ROS2 packages are available and can be used alongside any LiDAR SLAM algorithm.

Figures (15)

• Figure 1: OptMap is a geometric map distillation algorithm which provides size-constrained, informative geometric maps for real-time autonomy algorithms. The two examples above demonstrate OptMap's flexibility in providing application-specific maps with different requirements.
• Figure 2: The OptMap pipeline consists of three stages: (i) descriptor set generation, (ii) dynamically ordered streaming submodular maximization, and (iii) map loading. The first stage occurs either offline or alongside SLAM. Dynamically reordered streaming submodular maximization and map loading take place on parallel threads after receiving an OptMap function call specifying the desired map size as well as additional optional position and time constraints.
• Figure 3: The descriptors of LiDAR point clouds collected along a time-parameterized trajectory are assumed to be samples along a corresponding trajectory in $\mathcal{S}^{255}$. Point clouds and therefore descriptors are assumed to be sampled with unequal distances.
• Figure 4: Section of a full and reduced set generated by \ref{['alg:reduced_set_selection']}. All elements in the full set (blue outlines) are no further than $\mathcal{E}$ apart, and all elements of the reduced set (orange interior) are no further than $2\mathcal{E}$ apart. The weight assigned to $r_{2}$ is the sum of weights from its source set $\hat{w}_{2} = w_{2}+w_{3}+w_{4}$. The closest element of $S$ to $e_{2}$ and $r_{2}$ does not have to be the same element of the solution $S$.
• Figure 5: Sensor coverage selection problem with a maximium limit of four selected sensors. Sensors in the solution set are blue, and sensors not included in the solution are orange. From left to right: optimal solution, solution generated by the greedy algorithm which does not consider input order, streaming solution with the input order indicated by the numbers in the middle of the sensor footprints, and the streaming solution with dynamic reordering (DR) which modifies the input order according to approximate element values. Assumed that $OPT$ is known a priori and only one solution is maintained by the streaming variants.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 1
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