Multi-scale Octree Surfel Maps

• Multi-scale Octree Surfel Maps are a hierarchical spatial representation that organizes planar surfels within an octree to capture varying geometric resolutions for 3D mapping and rendering.
• They employ statistical accumulation and incremental update schemes to achieve robust LiDAR-Inertial odometry and efficient texture-preserving compression.
• The system optimizes spatial queries and rendering through multi-scale association, loop closure, and MAP estimation, leading to improved accuracy and lower bitrates compared to traditional methods.

Multi-scale octree surfel maps provide a hierarchical, surface-oriented spatial representation that blends efficient storage, multi-resolution processing, and robust geometric abstraction for 3D mapping, rendering, odometry, and compression. In these approaches, surfels—disk-like local approximations to surfaces—are organized within an octree, with varying spatial granularities corresponding to different levels in the tree. Two primary and distinct paradigms—real-time mapping for Lidar-Inertial Odometry (SLICT) (Nguyen et al., 2022) and 3D content compression/streaming (TeSO) (Hu et al., 9 Aug 2025)—define the current frontier.

1. Hierarchical Octree Organization

At the foundation, the multi-scale surfel map uses an octree, a tree data structure in which each node corresponds to a cubic voxel at a particular scale. The root node represents the coarsest spatial granularity, with child nodes recursively subdividing space into finer voxels.

In SLICT (Nguyen et al., 2022), each node at depth $d_i \in \{0,1,\dotsc,D_{\max}\}$ maps to a voxel of size $s_i=2^{d_i} \ell$, with leaves corresponding to the finest resolution $\ell$. Each node contains a single surfel aggregating all 3D points historically mapped to its cubic volume.

In TeSO (Hu et al., 9 Aug 2025), the octree is pruned so that leaf nodes represent those cubes for which a planar surfel yields an approximation error below an explicit threshold $\tau$ (measured via D1-PSNR). Nodes at different levels, $l=0,\dots,L$, correspond to cube sizes $b(l) = B/2^l$, where $B$ is the root edge length, enabling a wide range of granularity. Pruning is data-driven: only subdivide when a cube cannot be accurately modeled by a single planar surfel.

2. Surfel Representation and Statistical Accumulation

A surfel is canonically parameterized as a local planar patch:

• Center position $p \in \mathbb{R}^3$
• Oriented normal $n \in \mathbb{R}^3$, $\|n\|=1$
• Radius of influence $s_i=2^{d_i} \ell$0
• Additional attributes (e.g., planarity, color/texture, statistics)

SLICT computes surfel attributes for each node $s_i=2^{d_i} \ell$1 using cumulative statistics of contained points $s_i=2^{d_i} \ell$2:

• $s_i=2^{d_i} \ell$3
• First-order moment $s_i=2^{d_i} \ell$4
• Second-order (corrected) moment $s_i=2^{d_i} \ell$5
• Mean: $s_i=2^{d_i} \ell$6
• Covariance: $s_i=2^{d_i} \ell$7
• Surface normal: $s_i=2^{d_i} \ell$8 (eigenvector of smallest variance)
• Planarity: $s_i=2^{d_i} \ell$9

TeSO constructs each surfel $\ell$0 at a given octree leaf as:

• $\ell$1: mean position of member points within the cubic voxel
• $\ell$2: mean of normals, normalized
• $\ell$3: maximum Euclidean distance from $\ell$4 to any member point

For texture, each surfel receives an $\ell$5 grid of RGB values built as a local planar projection with interpolation from nearest original points.

3. Incremental Construction and Efficient Operations

Incremental Update (SLICT): Surfels and their statistics are updated as new points arrive using Welford’s algorithm. Merging two child nodes $\ell$6 into a parent $\ell$7 involves:

• $\ell$8
• $\ell$9
• $\tau$0
• $\tau$1

Propagation proceeds up the tree, enabling efficient, on-the-fly updates for point insertion or removal in $\tau$2 time.

Construction (TeSO): The surfel octree is assembled via a multi-pass, coarse-to-fine grouping. At each candidate level $\tau$3:

• Points are partitioned into non-empty cubes.
• For each cube, fit a candidate surfel and evaluate error (D1-PSNR).
• If error is below threshold $\tau$4 (or at maximum level), record the surfel and remove points from subsequent passes.
• Otherwise, points are considered at a finer level in the next pass.

This most-to-least coarse sweep yields a sparse set of locally planar surfels at the minimal necessary resolution, greatly reducing primitive count relative to grid-aligned representations.

4. Multi-Scale Association, Map Query, and Rendering

Point-to-Surfel Association (SLICT):

• Given a query point $\tau$5, search across octree nodes at relevant depths $\tau$6 for which the node’s cube intersects $\tau$7.
• Surfel is eligible if $\tau$8 and $\tau$9.
• Accept if signed point-to-plane distance $l=0,\dots,L$0, where $l=0,\dots,L$1.

Rendering (TeSO):

• For each surfel, a disk of radius $l=0,\dots,L$2 in the tangent plane is rendered, with a “soft” $l=0,\dots,L$3-halo of width $l=0,\dots,L$4 to eliminate cracks.
• Texture mapping is performed by projecting pixel grid samples through the local surfel frame into RGB via nearest neighbor interpolation from original point cloud data.
• Ray–surfel intersection follows: solve for $l=0,\dots,L$5, where $l=0,\dots,L$6 is the ray origin and $l=0,\dots,L$7 the direction.

5. Optimization, Loop Closure, and MAP Estimation

SLICT’s Complete Framework:

• Maintains continuous-time odometry through joint optimization over sequence of discrete “key states” and IMU preintegration in a sliding window.
• The residual for each LiDAR-IMU correspondence uses temporally interpolated pose and surfel association.
• MAP objective couples IMU and surfel association terms:

$l=0,\dots,L$8

• Loop closure iteratively builds pose-graph constraints from rigid body ICP between temporally distant revisits; pose-graph optimization over the stack of odometry and loop edges (Nguyen et al., 2022).

6. Compression and Entropy Coding

TeSO Compression Pipeline:

• Geometry quantization: cube-local offset, normal (octahedral encoding), and radius quantized per level.
• Base octree occupancy: entropy coded with G-PCC for coarse levels.
• Learned entropy models (3D sparse-conv nets) contextually predict occupancy, split flags, and surfel attributes for finer octree levels, minimizing the cross-entropy loss.

$l=0,\dots,L$9

• Texture codes either as a single Morton-ordered 2D atlas (compressed with AV1) or using G-PCC color for each surfel’s texels.
• TeSO achieves encoding rates for 1M points $b(l) = B/2^l$00.3 sec (GEFORCE RTX4080), decoding times: geometry—0.7s (learned model), texture AV1—0.3s.

7. Computational Analysis and Application Contexts

System	Insertion/Removal Complexity	Query Complexity	Distinctive Features
SLICT	$b(l) = B/2^l$1	$b(l) = B/2^l$2 (multi-scale)	Real-time LiDAR-Inertial odometry, robustness via multi-scale PTS, loop closure, bundle adjustment (Nguyen et al., 2022)
TeSO	Pass over all points per level; parallelizable	$b(l) = B/2^l$3 (rendering)	High-fidelity streaming, compact 3D encoding, texture-preserving planar surfels, real-time decoding (Hu et al., 9 Aug 2025)

SLICT demonstrates that multi-scale association boosts odometry in degenerate environments, improving Absolute Trajectory Error by up to $b(l) = B/2^l$4 over single-scale direct methods at a moderate computational overhead (165 ms per cycle vs. 100 ms LiDAR arrival). TeSO delivers 30–50% better bitrates for the same perceptual quality (LPIPS) compared to G-PCC and 3D Gaussian baselines, with real-time decoding feasible on modern GPUs and precise control of rate-distortion through the surfel error threshold.

This suggests that the fusion of multi-scale spatial hierarchy and local surface approximation accommodates both production-grade mapping and high-performance 3D media streaming, illustrating the flexibility of octree surfel abstractions in contemporary 3D data systems.

• "SLICT: Multi-input Multi-scale Surfel-Based Lidar-Inertial Continuous-Time Odometry and Mapping" (Nguyen et al., 2022)
• "TeSO: Representing and Compressing 3D Point Cloud Scenes with Textured Surfel Octree" (Hu et al., 9 Aug 2025)