Hierarchical Voxel Map Overview

Updated 4 December 2025

Hierarchical voxel maps are multiscale spatial data structures that recursively partition 3D space using diverse encoding methods for efficient query and memory management.
They integrate various base formats—such as raw grids, distance fields, SVO, and SVDAG—to optimize performance in applications like ray tracing, SLAM, and semantic mapping.
Advanced construction techniques, including bottom-up aggregation and pointer induction, enable significant compression, real-time processing, and improved cache utilization.

A hierarchical voxel map is a multiscale spatial data structure that encodes 3D space or volumetric attributes through recursive, level-wise subdivision or aggregation of voxels, with each level implementing a (potentially distinct) encoding for efficient memory usage, query, and computation. Hierarchical voxel maps serve as the foundation for a range of domains, including high-performance volumetric rendering, real-time robotic mapping, multi-resolution semantic fusion, SLAM, and zero-shot navigation. State-of-the-art designs exploit base-format heterogeneity, spatial sparsity, hierarchical semantic annotation, and data-independent hashing to meet application-specific trade-offs in speed, compression, or accuracy.

1. Formal Structure and Mathematical Characterization

Hierarchical voxel maps are generally modeled as a stack of levels, indexed by $\ell=0,\dots,L$ . At each level $\ell$ , 3D space is partitioned into axis-aligned super-voxels of prescribed dimensions $D_\ell^x\times D_\ell^y\times D_\ell^z$ . Each super-voxel is either (a) a pointer to the next, finer, level ( $\ell+1$ ) or (b) a leaf containing volumetric data such as color, distance, semantics, or higher-level summaries.

Base structure at level $\ell$ is defined by its format $F_\ell$ , which can differ between levels:

Raw grid ( $R(W,H,D)$ ): Uniform, dense $2^W \times 2^H \times 2^D$ array storing per-voxel data.
Distance field ( $D(W,H,D,M)$ ): Dense grid as above, but augmented with L1 distance fields.
Sparse Voxel Octree (SVO or $S(L)$ ): Binary-tree decomposition up to $L$ subdivisions with explicit masks and pointers.
Sparse Voxel DAG (SVDAG or $G(L)$ ): Octree with subtree de-duplication, enabling aggressive compression.

Storage at level $\ell$ is given by: $C_{\text{mem}}(F_\ell) = S_\ell \cdot \left[\text{overhead} + \Delta_\ell \cdot \text{entry size}\right]$ where $S_\ell$ is the number of sub-volumes and $\Delta_\ell = D_\ell^x D_\ell^y D_\ell^z$ . The total memory usage is

$C_{\text{total}} = \sum_{\ell=0}^{L} C_{\text{mem}}(F_\ell)$

with parent-child relationships encoded by pointers or offsets.

The hybrid (hierarchical) voxel map is the sequence $(F_0, F_1, \ldots, F_L)$ , each $F_\ell$ potentially with a distinct spatial format, yielding a large design space for tailoring to application demands (Arbore et al., 18 Oct 2024).

2. Construction and Update Algorithms

Construction of a hierarchical voxel map proceeds bottom-up:

Voxelization: Source data (e.g., triangle mesh, point cloud) is quantized into the finest-level grid, typically in Morton order (Z-order curve) for spatial locality.
Aggregation: At each coarser level, voxels/supervoxels are aggregated, with rules dictated by the base format—mean, mask, surfel fit, etc.
Pointer induction: Non-empty regions create pointers/offsets to child buffers.
Application-specific processing:
- For SVO/SVDAG, deduplication and queue-driven bottom-up clustering are performed.
- For surfel maps (Choi et al., 3 Dec 2025), Level-1 voxels aggregate centroids and fit planes via eigendecomposition over occupied Level-0 voxel centroids.

A typical recursive construction (in pseudocode):

function construct_levelℓ(lowerXYZ):
  is_empty ← true
  for child_offset in 0...Δℓ-1:
    child_lower = lowerXYZ + child_offset * size_of_levelℓ
    if ℓ < L:
      (subvol, sub_empty) = construct_levelℓ+1(child_lower)
      ...
    else:
      ...
  ...
  return (A, is_empty)

Specialized update paths (e.g., real-time erosion/re-integration (Gregorio et al., 2017)) employ weight-tracking and pose-adjusted fusion.

3. Base Format Diversity, Hybridization, and Spatial Indexing

Each level's format $F_\ell$ is selected based on application-relevant metrics:

Sparsity and homogeneity: SVO/SVDAG provide aggressive culling for empty or identical sub-regions.
Ray-marching efficiency: Distance field formats enable rapid skipping of homogeneous regions.
Memory overhead: Raw formats are compact but lack spatial efficiency.
Indirection cost: SVDAG achieves best-in-class compression at the cost of pointer-chasing.

Hybrid compositions, or "format sequences" $[F_0, ..., F_L]$ , are chosen to balance traversal efficiency, memory, and query speed. For instance, $R(4,4,4)~G(8)$ efficiently handles large, dense scenes by combining a coarse grid with an aggressively de-duplicating SVDAG (Arbore et al., 18 Oct 2024).

Spatial locality in memory and $O(1)$ access is achieved by Z-order (Morton) hashes (Choi et al., 3 Dec 2025), significantly improving data cache utilization during traversal and registration.

4. Hierarchical Voxel Maps in Robotic and Semantic Mapping

Robotic mapping and navigation systems exploit hierarchical voxel representations to support multi-level reasoning, semantic fusion, or belief propagation.

Topometric Hierarchy (Storey-Region-Volume): 3D occupancy grids are segmented into vertical “columns,” then volumetrically clustered, then further merged into semantic regions and floors ("storeys"). Passages/edges are explicit, and furnishings/structural inhomogeneity are robustly handled (He et al., 2021).
SkipList-based Voxel Trees: Alternative to octrees, the Tree of SkipLists allows $O(\log N)$ average insertion and direct multi-resolution querying (2D, 2.5D, 3D) (Gregorio et al., 2017). The layered skiplist organization is naturally hierarchical but avoids pointer bloat.
Hierarchical Semantic Voxel Belief Maps: For zero-shot navigation, fine-grained 3D grids store multi-level semantic features (CLIP embeddings) per voxel, enabling context-conditioned fusion of LLM priors, real-time observations, and path-planning. Hierarchy appears in the semantic scoring pipeline, even if the grid itself is single-resolution (Zhou et al., 27 May 2025).

5. Applications in Perception, Rendering, and Planning

Hierarchical voxel maps are central to a wide spectrum of computational tasks:

Ray Tracing and Volumetric Rendering: Hybrid formats support efficient traversal and culling, providing Pareto-optimal trade-offs between compression and throughput; e.g., SVDAG-based methods realize frame-time improvements of $>2.8\times$ with $>10\times$ memory reduction over dense grids for $2048^3$ scenes (Arbore et al., 18 Oct 2024).
3D Semantic Occupancy Prediction: Hierarchical multi-resolution grids permit selective refinement (HVFR), where only “important” voxels are subdivided (e.g., 2×, 4×). Pixel-to-voxel fusion is executed via deformable attention across levels, maximizing sensor fusion accuracy while containing FLOPs (Seong et al., 29 Dec 2024).
SLAM and Odometry: Precomputed surfel representations at coarse levels support $O(1)$ correspondence lookup and eliminate online plane fitting, yielding real-time registration and mapping (Choi et al., 3 Dec 2025).
Zero-Shot Navigation: Probabilistic belief maps on voxel grids allow Bayesian updating of target object probability, integrating LLM-inferred semantic priors and online observation likelihoods for global, path-dependent planning (Zhou et al., 27 May 2025).

6. Performance, Compression, and Implementation Trade-offs

Hierarchical voxel maps achieve substantial compression, speed, and representational efficiency:

Compression–Throughput Frontier: Hybrid formats—e.g., $R(3^3)~G(8)$ —define empirical Pareto frontiers; San Miguel $2048^3$ : raw grid $55.3$ GiB/8.7 ms vs. hybrid $4.2$ GiB/3.1 ms (Arbore et al., 18 Oct 2024).
Real-time Considerations: O(1) hash-based access and lazy update of surfels drastically reduce runtime for high-throughput pipelines (Choi et al., 3 Dec 2025).
Semantic Fidelity and Navigation Efficacy: Matthews Correlation Coefficient (MCC) of $0.97$–$0.99$ for semantic segmentation (He et al., 2021); SPL improvement of over $46\%$ in zero-shot navigation tasks using hierarchical belief voxel maps (Zhou et al., 27 May 2025).
Parameterization: Fine control through voxel size $r$ , base format selection at each level, maximum skip-list depth, block sizes for sparse convolutional processing, and differential fusion heuristics.

7. Current Research Directions and Open Challenges

Contemporary research investigates further optimizations:

Metaprogramming and Adaptive Transformations: Automated generators for hybrid format construction and ray-intersection code, with transformations for spatial reordering, deduplication, and stack-elision (Arbore et al., 18 Oct 2024).
Sparse and Efficient Fusion: Multi-sensor fusion implementations (camera/LiDAR) now exploit adaptive hierarchical refinement coupled with attention-based mechanisms for differential focus on critical regions (Seong et al., 29 Dec 2024).
Probabilistic and Semantic Hierarchies: Integration of hierarchical semantic attributes with real-time Bayesian updating, providing context- and observation-dependent inferences for navigation in previously unseen environments (Zhou et al., 27 May 2025).

A plausible implication is that further performance gains and application-generalization will arise from tighter coupling between base format automated selection, memory layout optimizations, and higher-level task semantics, especially as neural rendering and learning-based planning increasingly rely on hierarchical spatial representations.

Key References:

"Hybrid Voxel Formats for Efficient Ray Tracing" (Arbore et al., 18 Oct 2024)
"MR-Occ: Efficient Camera-LiDAR 3D Semantic Occupancy Prediction Using Hierarchical Multi-Resolution Voxel Representation" (Seong et al., 29 Dec 2024)
"Surfel-LIO: Fast LiDAR-Inertial Odometry with Pre-computed Surfels and Hierarchical Z-order Voxel Hashing" (Choi et al., 3 Dec 2025)
"Hierarchical Topometric Representation of 3D Robotic Maps" (He et al., 2021)
"SkiMap: An Efficient Mapping Framework for Robot Navigation" (Gregorio et al., 2017)
"BeliefMapNav: 3D Voxel-Based Belief Map for Zero-Shot Object Navigation" (Zhou et al., 27 May 2025)