Surfel-LIO: Efficient LiDAR-Inertial Mapping

• Surfel-LIO is a LiDAR–inertial odometry method that aggregates raw LiDAR points into surfels to encode local geometry for efficient scan-to-map registration.
• The approach employs hierarchical data structures like voxel grids and octrees along with IMU pre-integration to optimize residual construction and data association.
• Experimental results indicate that Surfel-LIO achieves superior runtime efficiency with frame rates up to 690 FPS while maintaining competitive localization accuracy.

Surfel-LIO denotes a family of LiDAR–inertial odometry (LIO) techniques that utilize surfel (surface element) representations as the primary geometric primitive for map construction, data association, and optimization. These methods exploit the aggregation of raw LiDAR points into minimal surface patches—surfels—that encode local geometry (centroid, normal, planarity, covariance) and facilitate efficient and robust scan-to-map registration. Contemporary Surfel-LIO systems are distinguished by their pre-computation of surfel parameters, hierarchical or multi-resolution surfel organization (e.g., voxel grids, octrees, or tetrahedral lattices), and hierarchical data association pipelines. These innovations achieve superior runtime efficiency while maintaining or exceeding the localization accuracy of direct point-based systems.

1. Surfel Representation and Data Structures

A surfel is typically defined as a tuple $(\mathbf{c}, \mathbf{n}, \mathbf{C}, \rho)$, where $\mathbf{c}$ is the centroid, $\mathbf{n}$ the surface normal (often the eigenvector associated with the smallest eigenvalue of the local covariance $\mathbf{C}$), and $\rho$ a planarity or confidence score. Surfel statistics are accumulated over point clouds falling within a spatial cell, which can be implemented via voxels, octrees, or other spatial partitioning structures.

• In Surfel-LIO’s hierarchical voxel hashing (hVox) design, the mapping space is divided into two levels: L₀ “micro-voxels” collect centroids by running Welford’s algorithm, while L₁ “macro-voxels” aggregate these into surfels. Each L₁ cell spans a $3\times3\times3$ block of L₀’s and maintains a surfel updated lazily to amortize computation (Choi et al., 3 Dec 2025).
• Octree-based multi-scale surfel maps support operations at variable resolution, enabling simultaneous global and local registration while preventing computational explosion; nodes store sufficient statistics for incremental merging and splitting (Nguyen et al., 2022).
• In multi-resolution lattice systems, sparse tetrahedral lattices with adaptively chosen cell sizes support four concurrently maintained surfel resolutions (Quenzel et al., 17 Nov 2025).

Pre-computation and caching of surfel parameters at mapping time eliminates the need for runtime local PCA or plane fitting during scan matching. Efficient indexing via Z-order (Morton) codes or octree traversal ensures $O(1)$ or $O(\log N)$ correspondence retrieval, in contrast to neighbor enumeration in direct k-d-tree-based methods.

2. Pipeline: LiDAR–Inertial Odometry with Surfels

Surfel-LIO pipelines typically implement an iterated extended Kalman filter (IEKF) or sliding-window optimization framework:

1. IMU Pre-integration. State propagation exploits raw IMU data to provide a motion prior and to undistort LiDAR scans (Choi et al., 3 Dec 2025, Nguyen et al., 2022).
2. Scan Undistortion & Map Association. Each deskewed LiDAR point is transformed to the world frame and mapped to a corresponding surfel (via Z-order hashing, octree traversal, or other spatial indexing).
3. Residual Construction. For each association, a point-to-surfel residual is computed using

$$r_i = \mathbf{n}_i^\top (\mathbf{R}\mathbf{p}_i^{L} + \mathbf{p} - \mathbf{c}_i).$$

Optional extensions include anisotropic covariance weighting (e.g., $\|f-\mu\|_{\Sigma_\perp}$) (Nguyen et al., 2022).

1. Hierarchical Data Association. Hierarchical pipelines prioritize associations to large, planar, high-confidence surfels, falling back to smaller scales and (if needed) traditional point-to-plane constraints (Huang et al., 2023).

For continuous-time LIO, trajectory states are either interpolated linearly (Nguyen et al., 2022) or via non-uniform cumulative $\mathrm{SE}(3)$ B-splines with adaptive knot placement (Quenzel et al., 17 Nov 2025), supporting accurate modeling of rolling-shutter effects and scan deskewing at fine temporal resolution.

3. Computational Efficiency and Algorithmic Improvements

Surfel-LIO systems address two historical computational bottlenecks in LIO:

• Neighbor Search. By using Z-order hashing, only one hash-table lookup is required per correspondence, with high L1 cache hit rates. There is no need to inspect multiple neighboring voxels or perform radius/k-nearest neighbor searches (Choi et al., 3 Dec 2025).
• Plane Parameter Estimation. Surfels are incrementally updated offline or in a batched fashion, and are never re-fitted during each optimization iteration (unlike PCA approaches in LOAM-like pipelines). All residuals and Jacobians are assembled in $O(1)$ per correspondence, facilitating the use of multi-threaded backends.

Timing profiles demonstrate substantial runtime gains:

Operation	Fast-LIO2	Faster-LIO	Surfel-LIO
NN search (µs/pt)	1.42	2.76	0.05
Plane estimation	0.17	0.61	0.01
Map update (ms/fr)	0.34	0.03	0.03

For M3DGR sequences:

• Surfel-LIO achieves 531 FPS (AVIA) and 690 FPS (Mid-360), compared with 125/282 FPS (Fast-LIO2) and 184/353 FPS (Faster-LIO), with comparable translation RMSE (0.36–0.40 m) (Choi et al., 3 Dec 2025).

4. Multi-scale and Continuous-time Extensions

Multi-scale surfel representations—either via hierarchical voxels, octrees, or tetrahedral lattices—enable simultaneous exploitation of coarse (for early large-surface matches) and fine (for precise registration) geometric structure:

• Multi-scale octree surfel maps in SLICT (Nguyen et al., 2022) and multi-resolution lattices in LIO-MARS (Quenzel et al., 17 Nov 2025) provide implicitly scalable data structures.
• Continuous-time LIO is addressed by employing B-spline trajectory models with non-uniform, scan-synchronized knots, yielding continuous pose trajectories and natural handling of asynchronous sensor data (Quenzel et al., 17 Nov 2025).

Advanced Surfel-LIO approaches (e.g., LIO-MARS) also incorporate Gaussian mixture model (GMM) alignment over surfels, motion compensation for rolling-shutter LiDAR via unscented transform (UT)-based surfel de-skewing, and intra-scan segmentation for accurate motion modeling at the sub-scan level.

5. Optimization, Factor Graphs, and Loop Closure

Most Surfel-LIO implementations employ either an IEKF or batch sliding-window optimization:

• Residual stacking includes IMU pre-integration, surfel-based point-to-plane terms, soft zero-velocity or relative-pose constraints, and marginalization of older states (Nguyen et al., 2022, Choi et al., 3 Dec 2025, Quenzel et al., 17 Nov 2025).
• SLICT (Nguyen et al., 2022) builds a factor graph over a sliding window, optimizing pose, velocity, and bias variables for each node with Ceres.
• Loop closure is handled by marginalizing out keyframes and augmenting the global pose graph with loop factors derived by ICP-based relative pose estimation between temporally distant, spatially proximate frames; the final pose-graph is optimized by Gauss–Newton and the consistent trajectory is re-inserted into the surfel map.

The optimization cost functions typically integrate all the above terms; for LIO-MARS (Quenzel et al., 17 Nov 2025): $$J = \sum_i J_{\mathrm{surfels}}(T(t_i)) + \sum_j J_{\mathrm{IMU}}(T(t_j), \Delta r_j, \Delta v_j) + \lambda\sum_k\|\cdot\|^2.$$

6. Experimental Results and Comparative Performance

Benchmark evaluations across datasets such as M3DGR (Choi et al., 3 Dec 2025), NTU VIRAL (Huang et al., 2023), and Oxford Newer College (Nguyen et al., 2022, Quenzel et al., 17 Nov 2025) indicate that Surfel-LIO methods offer both superior runtime and state-of-the-art accuracy:

• Surfel-LIO (Choi et al., 3 Dec 2025) matches or slightly outperforms Fast-LIO2 and Faster-LIO in translation RMSE, but achieves 3–4x higher frame rates.
• LOG-LIO (Huang et al., 2023) attains lowest mean translation RMSE across most sequences in both M2DGR and NTU VIRAL, with only minimal runtime overhead compared to non-surface-element methods.
• SLICT (Nguyen et al., 2022) achieves sub-decimeter accuracy and the lowest ATE in the majority of NTU VIRAL and Newer College sequences, with O(log N) scaling and efficient support for loop closure.
• LIO-MARS (Quenzel et al., 17 Nov 2025) demonstrates leading accuracy (mean RMS error ≈0.10–0.30 m across benchmarks), robust performance across platforms, and efficient multi-threaded execution.

7. Limitations and Future Directions

Limitations of current Surfel-LIO frameworks include:

• Reliance on local planarity—performance can degrade in highly curved or non-planar environments (Choi et al., 3 Dec 2025).
• Fixed or coarse voxel sizing—single-resolution cells may not adapt optimally to varying scene geometries.
• Evaluations are predominantly on ground robot and select LiDAR sensor modalities.

Potential future research avenues include:

• Incorporation of richer geometric primitives (e.g., edge-surfels, corner features).
• Deployment of adaptive or octree-based multi-resolution voxelization.
• Integration of global loop closure and surfel-based global bundle adjustment.
• Generalization to aerial, handheld, and multi-LiDAR platforms, and extension to non-spinning and solid-state sensors (Choi et al., 3 Dec 2025).

Surfel-LIO’s combination of hierarchical, pre-computed surface element representation and highly optimized spatial indexing offers compelling avenues for real-time, robust odometry and mapping in diverse environments. For detailed implementations and code, public repositories are available from the original authors (Choi et al., 3 Dec 2025, Nguyen et al., 2022, Huang et al., 2023).