Geometry-Driven Recursive Plane Fitting

• The paper introduces a recursive plane fitting method using voxel hashing, octree subdivision, and RANSAC-based estimation to model planar features with high precision.
• It employs distribution-based validity checks and recursive outlier handling to ensure accurate segmentation and suppression of false plane mergers.
• Experimental results demonstrate over 20% ATE improvement and reduced computational load, enhancing SLAM performance and volumetric interface reconstruction.

Geometry-driven recursive plane fitting is a set of computational strategies for representing, segmenting, and analyzing spatial structures by recursively estimating planar features within discretized geometric domains. This methodology is critical in fields such as online LiDAR odometry and volume-of-fluid interface reconstruction, where precise mapping of surfaces or interfaces directly affects localization, data assimilation, and numerical stability.

1. Data Structures for Recursive Plane Fitting

In online mapping applications, the environment is partitioned using voxel-based spatial hashing. Each voxel (root cell) of size $S_{max}$ serves as the entry in a hash table, and contains an octree for adaptive subdivision based on local point density and geometric complexity. Octree nodes at depth $\ell$ have a cubical volume of side $\frac{S_{max}}{2^\ell}$, with subdivision proceeding to a maximum depth $D_{max}$. Each node, whether leaf or internal, holds a linked list of points and, where applicable, a probabilistic planar feature fitted from its contained points. Incoming point clouds, transformed into a consistent world frame, are placed into voxels and then recursively pushed into finer subcells, allowing re-partitioning without redundant memory operations (Xi et al., 18 Jan 2026).

2. Geometry-driven Plane Model Estimation and Refinement

Plane fitting within voxels utilizes robust geometric estimation via Random Sample Consensus (RANSAC). The goal is to identify planes

$\pi: n^T x + d = 0, \|n\| = 1$

using point sets $P_{in} \subset \mathbb{R}^3$. RANSAC randomly selects point triplets to fit candidate planes, evaluates the inlier set $I = \{p \in P_{in} : |n^T p + d| < d_{th}\}$ over $K$ iterations, and selects the maximal support model. If the inlier ratio $|I|/|P_{in}| > p_{th}$ holds, refinement follows: Principal Component Analysis computes the centroid $q$ and scatter matrix $A$, whose smallest eigenvalue $\lambda_{min}$ and corresponding eigenvector $u_3$ yield the normal $n = u_3$ and offset $d = -n^T q$. Acceptance occurs only if $\lambda_{min} < \lambda_{th}$, suppressing spurious fits in non-planar regions. This process is coupled with propagation of input point covariances, resulting in explicit plane parameter uncertainty via Jacobian-based linearization (Xi et al., 18 Jan 2026).

3. Recursive Outlier Handling and Octree Traversal

Recursive plane fitting relies on an outlier detect-and-reuse mechanism within the octree. For each node, RANSAC segregates inliers and outliers. When a valid plane is identified, a distribution-based region-growing validity check ensures coherence of the inlier cluster; non-coherent clusters are treated as outliers. Outliers and rejected points are recursively pushed into deeper octree levels, incrementally capturing finer surface detail and subdividing curved or complex structures. Stopping conditions include depletion of outliers and reaching $D_{max}$ (Xi et al., 18 Jan 2026). This hierarchical strategy guarantees that large, nearly-planar surfaces are modeled at coarse levels, while smaller or intricate features are represented at finer granularity.

4. Distribution-based Validity and False Merger Suppression

To prevent erroneous merging of physically distinct planes, a point-distribution validity check is applied post-RANSAC. Inlier points are projected onto their estimated plane using local principal axes. A grid—resolution determined by the voxel size and hyperparameter—bins projected coordinates, forming a 2D histogram. Clusters (connected components in 4-neighborhood) are identified; the largest cluster is accepted if its size exceeds a threshold fraction of total points. Otherwise, the node is deemed invalid, and all points revert to outliers. The final plane is refit on the accepted cluster, ensuring true physical segmentation and suppressing artifacts from over-segmentation or cross-plane fusion (Xi et al., 18 Jan 2026).

5. Integration with Measurement Models and Pose Estimation

In SLAM or odometry, recursive plane fitting outputs probabilistic plane models used for scan-to-map matching. Each incoming LiDAR point is transformed and associated with candidate planes in the map, using point-to-plane residuals: $d_i = n_i^T (\hat{p}_i - q_i)$. Noise: $$\sigma_i^2 = J_{w_i} \operatorname{diag}(\Sigma_{n_i,q_i}, \Sigma_{\hat{p}_i}) J_{w_i}^T$$ is derived from plane and point uncertainty. The likelihood

$$p_i = \frac{1}{\sqrt{2\pi}\sigma_i}\exp \left(-\frac{d_i^2}{2\sigma_i^2}\right)$$

is maximized to select the best matching plane. These residuals are integrated into iterative EKF pose estimation, tightly coupling mapping uncertainty with state optimization (Xi et al., 18 Jan 2026).

6. Experimental Performance and Computational Properties

Empirical results indicate that geometry-driven recursive plane fitting, as implemented in R-VoxelMap, yields average trajectory error (ATE) improvements of $>20\%$ over prior VoxelMap variants in major SLAM benchmarks (KITTI, M2DGR, NTU VIRAL), with reductions in end-to-end positional drift on challenging environments (degraded corridor, small-FoV) from $6$–$15$ m to $1$–$2.5$ m (Xi et al., 18 Jan 2026). Runtime for full map maintenance, including extra RANSAC operations and reduced covariance update load, is improved (from $44.4$ ms/frame and $4.09$ GB on VoxelMap to $40.2$ ms/frame and $3.16$ GB on R-VoxelMap for a $3.7$ km run), establishing computational efficiency and memory economy. Stability tests over multiple seeds confirm these gains.

7. Extensions to Polyhedral Interface Positioning: Recursive Face-based Plane Fitting

In volume-of-fluid (VoF) reconstruction for arbitrary polyhedra, recursive plane fitting employs the Gaussian divergence theorem to compute the truncated volume

$V(s) = \int_{\Omega(s)} 1\, dV$

where $$\Omega(s) = \{x \in \Omega^* : \langle x - x_0, n \rangle \leq s \}$$, with the truncating plane $\Gamma(s)$. Face-wise coefficients ($\alpha_k^0,\alpha_k^1$) and edge-wise coefficients ($\beta_{k,m}^0, \beta_{k,m}^1$) are precomputed, allowing $V(s)$ to be expressed as a sum over faces and edges. Within intervals (“brackets”) defined by sorted vertex projections, clipped areas are quadratic and $V(s)$ becomes a cubic polynomial. Combining bracketing and higher-order root-finding (using Taylor expansion and cubic solves) enables robust, efficient interface positioning without global topological reconstruction (Kromer et al., 2021). The method demonstrates $O(1-2)$ average truncations per instance across convex and non-convex polyhedra, outperforming previous algebraic or Newton-only schemes and delivering significant computational advantages.

Table: Comparative Summary of Geometry-driven Recursive Plane Fitting Applications

Application Domain	Recursive Strategy	Key Benefit
LiDAR Odometry (R-VoxelMap)	Octree + outlier reuse + distribution check	$>20$\% ATE reduction, no runtime penalty
VoF Polyhedral Interface	Gaussian-divergence + bracketed cubic root-finding	$O(1-2)$ truncations, topological independence

These approaches reflect the efficacy and generality of geometry-driven recursive plane fitting for high-precision spatial modeling across diverse scientific domains.