MTD-Map: Single-Stage Long-Term LiDAR Map Maintenance Framework via Mixture Transition Distribution

Abstract: While robust map maintenance has advanced significantly, existing studies have focused on specific tasks, especially dynamic object removal or change detection. In this paper, we take a holistic view of the map maintenance problem and propose MTD-Map, a single-stage framework that handles both dynamic object removal and change detection without separate task-specific modules. MTD-Map employs an explicit representation that compactly encodes the direction and duration of occupancy transitions through Mixture Transition Distribution (MTD) modeling. We develop a recursive MTD formulation that encodes historical occupancy patterns into an augmented state to capture high-order temporal dependencies. Furthermore, a stability-driven adaptive strategy balances noise suppression with the preservation of quasi-static structures. Extensive experiments verify that MTD-Map robustly removes dynamic objects and achieves competitive change detection performance, subsequently reducing computational costs. Our project page is available at: https://taeyoung96.github.io/mtd_map/.

• The paper introduces a single-stage Mixture Transition Distribution framework that unifies dynamic object removal and change detection through constant-memory, stability-aware voxel updates.
• MTD-Map achieves the highest harmonic accuracy on SemanticKITTI sequences 00 and 07, reaching 97.33% and 94.13%, while delivering competitive results across HeLiMOS and MOE.
• The method improves change-detection F1 scores to 0.643 for appearing structures and 0.755 for disappearing structures, and processes the LT-ParkingLot pipeline in 402.9 seconds—about 25% of LT-mapper’s runtime and 8% of ELite’s.

Overview

MTD-Map is a single-stage framework for long-term LiDAR map maintenance that unifies dynamic object removal (DOR) and change detection (CD) within one probabilistic voxel-wise update, developed by researchers at Hyundai Motor Company's Robotics Lab (2606.29469). The central observation motivating the work is that prior lifelong mapping systems treat DOR and CD as separate tasks with modular pipelines, imposing binary "appearing/vacating" decisions that discard the continuous spectrum of persistence. The paper's key representational choice is a bipolar encoding of occupancy transitions—capturing both the direction and duration of change—which contrasts with the directionless, unipolar ephemerality representation of ELite (2606.29469).

The framework draws on the Mixture Transition Distribution (MTD) model from time-series statistics, which approximates high-order Markov dependencies as a linear mixture of first-order transition kernels. Whereas a full $k$-order Markov chain requires storing $k$ discrete states per voxel and incurs exponential parameter growth, MTD-Map reformulates the mixture into an immediate response term and an aggregate historical term summarized by a single long-term occupancy probability $\pi$, yielding constant memory per voxel.

Method

Each voxel maintains an augmented state $$\mathcal{S}_t^{(v)} = [p_t, \pi_t, \beta_t, t_{last}]$$: instantaneous occupancy probability, long-term occupancy probability, temporal stability, and last update time. The prediction step replaces the static MTD mixing weight $\lambda_1$ with a stability-modulated power-law decay weight $w_t$ computed from elapsed time and an effective time constant $\tau_e(\beta_t)$ that grows with stability, so established structures decay slowly while unstable voxels adapt rapidly. For large temporal gaps between sessions, the prior switches between $p_{t-1}$ and $\pi_{t-1}$ depending on long-term occupancy to avoid over-committing to obsolete structures.

The measurement update is a unified Bayesian filter: instantaneous occupancy is updated in log-odds space against the adaptive prior; long-term occupancy follows a stability-modulated exponential moving average, which the authors note is equivalent to an infinite-lag MTD with geometric lag weights; and stability $\beta_t$ is reinforced when observations match predictions within tolerance $k$0, scaled down by a penalty factor $k$1 under strong conflict, or linearly decayed otherwise. This coupling—prediction fidelity driving stability, which in turn modulates the balance between short- and long-term terms—is what allows one mechanism to suppress transient dynamics while preserving quasi-static entities.

Two post-processing stages complete the pipeline. Hierarchical spatial regularization clusters voxels via TRAVEL-style geometric grouping and rejects outliers using Mahalanobis distance to the cluster median feature vector, propagating representative states for local belief consistency. Map refinement then thresholds the augmented state into static/dynamic sets for DOR, applies a trilateral filter, and produces a temporal transition map $k$2 whose sign encodes change direction (appeared vs. disappeared quasi-static structures) and magnitude encodes persistence—a bipolar representation that disambiguates arrival from departure events without extra computation.

Experimental results

DOR evaluation uses SemanticKITTI sequences 00 and 07, HeLiMOS 6593, and MOE 02 at a fixed 0.1 m voxel size, reporting Static Accuracy (SA), Dynamic Accuracy (DA), and Harmonic Accuracy (HA). MTD-Map achieves the highest HA on KITTI 00 (97.33%) and KITTI 07 (94.13%), and second-best HA on HeLiMOS (90.06%) and MOE (95.39%). The results expose a trade-off among baselines: HMM-MOS and DUFOMap are conservative (e.g., HMM-MOS reaches only 5.84% DA on MOE), leaving ghost trails, while ERASOR and ELite are aggressive, attaining peak DA (99.38% and 99.76% on MOE) but degrading SA (94.55% and 82.93%). MTD-Map does not claim state-of-the-art DA; its stated contribution is consistency across environments rather than peak dynamic removal. Learning-based baselines such as SwiftMOS were excluded because they require per-sensor supervised training, so comparisons are limited to training-free methods.

For CD, evaluation on LT-ParkingLot (sequences 03→04) with manually annotated changing points shows that MTD-Map with regularization achieves an F1 of 0.643 for Positive Differences versus 0.164 for LT-mapper and 0.267 for ELite, and an F1 of 0.755 for Negative Differences versus 0.680 and 0.555 respectively. The regularization ablation is stark: without it, PD precision collapses to 0.107, indicating that cluster-level Mahalanobis filtering is responsible for most of the PD precision gain (0.618). The authors attribute baseline failures to viewpoint-dependent visibility artifacts (LT-mapper) and nearest-neighbor/iterative point-wise update limitations that miss low-density vehicle interiors (ELite).

On computational efficiency, the single-stage design processes the full LT-ParkingLot pipeline in 402.9 s—approximately 25% of LT-mapper's runtime and 8% of ELite's—because DOR and CD share one voxel-wise update rather than sequential point-wise passes. Additional qualitative experiments on a custom indoor dataset collected with the MobED platform across three consecutive sessions demonstrate that the augmented state encodes transient obstacles, quasi-static transitions, and stable structures distinctly within a single representation.

Limitations and open questions

The paper concedes several constraints. The framework assumes sequentially ordered sensor poses; handling out-of-order pose inputs for real-time deployment remains open. The CD evaluation relies on manually annotated ground truth for a single sequence pair of one outdoor dataset, since LT-ParkingLot lacks binary change labels, leaving broader quantitative CD validation unaddressed. Several hyperparameters—including the Mahalanobis threshold $k$3, which the authors describe as bounding structural variance rather than serving as a statistical confidence bound—are empirically chosen, and sensitivity analyses are not reported. Finally, the recursive MTD formulation treats $k$4 as a sufficient statistic for all history; the paper grounds this in infinite-lag MTD theory but does not quantify approximation error relative to explicit high-order modeling.

Conclusion

MTD-Map reframes DOR and CD as unified inference of occupancy transitions over varying timescales, realized through a recursive MTD formulation with constant memory, a stability-driven adaptive prior, and hierarchical spatial regularization. It delivers the best harmonic accuracy on two of four DOR benchmarks, substantially higher CD F1-scores than LT-mapper and ELite, and roughly an order-of-magnitude runtime reduction relative to ELite, at the cost of not matching peak dynamic accuracy of aggressive removal methods and depending on sequential pose inputs.