Topo-Omni: 3D Navigation & Meta-Optics

• Topo-Omni is a framework that integrates omnidirectional 3D perception, hierarchical topological mapping, and token-efficient LLM reasoning for visual-language navigation in robotic agents.
• It employs persistent homology for robust room segmentation and multi-resolution spatial attention to reduce computational tokens while boosting navigation accuracy and performance.
• The concept extends to meta-optics, where topology-optimized designs enable arbitrary angle-dependent phase control for aberration correction and convergent focusing.

Topo-Omni refers to the integration of omnidirectional 3D perception, hierarchical topological mapping, and token-efficient LLM reasoning for visual-language navigation in robotic agents operating across complex environments. The system leverages raw sensory fusion, persistent-homology-based segmentation, and Topo-Omni’s structured reasoning pipeline to substantially improve both agent performance and computational memory efficiency, particularly in multi-room and cluttered indoor settings (Liu et al., 18 Mar 2026). Topo-Omni also denotes a class of meta-optical devices discovered via topology optimization (TO), capable of arbitrary angle-dependent phase control for aberration correction and convergent focusing (Lin et al., 2017). This entry emphasizes Topo-Omni in the context of embodied AI and visual-language navigation.

1. Omnidirectional Sensor Fusion and Dynamic Scene Graph Construction

Topo-Omni’s 3D mapping stack fuses data from a rotating LiDAR and panoramic RGB camera to produce a globally registered point cloud $\mathcal{P}_{map}$. Motion compensation is handled via a LiDAR-Inertial backend applying:

$$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$

Here, $T_{w b_k}\in SE(3)$ is the global pose, $\omega_k$ and $v_k$ are body-frame velocities, and $T_{b c}$ is the extrinsic calibration. Raw points $P_i^{raw}$ are rendered into an equirectangular panorama and labeled using a mask generator ($\mathcal{M}_t$) to assign semantic labels stabilized by temporal majority voting.

The semantically fused cloud is incrementally organized into a five-layer Dynamic Scene Graph (DSG) $\mathcal{G}=(\mathcal{V},\mathcal{E})$:

• L1: TSDF-based mesh geometry or triangle mesh
• L2: Object nodes with $\mathbf{p}_j$ (centroid) and $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$0 (semantic tag)
• L3: Place nodes from a Generalized Voronoi Diagram, annotated by clearance $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$1
• L4: Room nodes as macro-clusters of places
• L5: Building node aggregating all rooms

Meshes reside in scalable TSDF volumes, object nodes are indexed in a spatial-hash, and place nodes form a sparse connectivity graph modeling collision-free corridors.

2. Topological Room Segmentation via Persistent Homology

To partition the environment into meaningful topological compartments (rooms), Topo-Omni applies persistent homology to the place-graph $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$2. Each node $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$3 is weighted by clearance $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$4. The weighted filtration is:

$$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$5

As $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$6 decreases, the system monitors the zeroth Betti number $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$7, corresponding to the number of connected components. By selecting the plateau where $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$8 persists, the threshold $$P_i = T_{w b_k}\,\exp\bigl(\widehat{\omega}_k\,(t_i-t_k)\bigr)\,T_{b c}\,P_i^{raw} + v_k\,(t_i-t_k)$$9 is determined:

$T_{w b_k}\in SE(3)$0

The $T_{w b_k}\in SE(3)$1 connected components at $T_{w b_k}\in SE(3)$2 define $T_{w b_k}\in SE(3)$3 rooms, which instantiate L4 nodes.

Subsequent graph edge refinement combines:

• (a) Geometric pruning: Remove $T_{w b_k}\in SE(3)$4 if a point $T_{w b_k}\in SE(3)$5 along $T_{w b_k}\in SE(3)$6 is nearer to $T_{w b_k}\in SE(3)$7 than threshold $T_{w b_k}\in SE(3)$8
• (b) Visual-LLM (VLM) verification for short-range edges: Accept $T_{w b_k}\in SE(3)$9 if $\omega_k$0

This yields a physically and semantically faithful topological segmentation.

3. Hierarchical LLM Reasoning and Multi-Resolution Attention

Topo-Omni transforms the DSG into an agent-centric, egocentric 3D-octant representation. At time $\omega_k$1, for robot pose $\omega_k$2, each node at $\omega_k$3 is mapped to local coordinates $\omega_k$4, and assigned one of eight octants by the sign pattern:

$\omega_k$5

The scene is then structured into three tiers:

• Tier 1 (Foveal): Objects within 3 m, full semantic and coordinate resolution
• Tier 2 (Peripheral): Same-room but $\omega_k$63 m, summarized at functional-group granularity
• Tier 3 (Global Memory): Other rooms, collapsed to room-level descriptors only

This strategy substantially reduces token usage by compressing off-target or distant information.

The LLM prompt is structured as a four-stage hierarchical chain-of-thought (H-CoT):

1. Room Filtering: Select reachable rooms matching the linguistic goal.
2. Orientation: Determine likely octant for target localization within selected room.
3. Functional Group Inference: Infer probable functional group in the identified region.
4. Object Localization: Enumerate final candidate objects for navigation.

Action primitives (e.g., go_to_room, turn_to(octant), go_near, stop) are proposed by the Actor module and checked by the Critic for consistency with DSG constraints.

4. Performance Metrics and Computational Efficiency

Empirical results in 3-room environments (referring-expression generation over 44 objects) demonstrate that Topo-Omni’s hierarchical interface increases referring accuracy from 77.27% (flat list) to 93.18%, with visual identification (VI) and visual disambiguation (VD) sub-task gains of +9.09% and +22.73%, respectively.

In synthetic 4-room settings (object counts 5–50), multi-resolution prompting reduces token usage on average by ~61.7%. Savings by prompt tier are:

Tier	Low	Mid	High
Foveal	–0.25%	–41.03%	–47.59%
Peripheral	–26.83%	–44.64%	–50.50%
Global Memory	–55.94%	–64.81%	–69.98%

Navigation success in cluttered, cross-room trials increased by up to 11.68% (e.g., D9: Flat = 16%, Topo-Omni = 80%). Inference latency was reduced from 12.4 s to 3.8 s per decision, a vital factor for real-time deployment (Liu et al., 18 Mar 2026).

Ablation indicates that removing multi-resolution summarization causes token count to scale linearly with object count, exceeding context limits beyond approximately 30 objects. In the hierarchical pipeline, token usage per reasoning stage is kept at 80–200 tokens, compared to 300–800 tokens per stage for flat enumeration.

5. Comparative Advantages, Limitations, and Applications

Topo-Omni achieves significant advances over flat-list and non-topological approaches by leveraging persistent-homology-based segmentation and spatial abstraction. Key benefits include:

• Hardware-agnostic omnidirectional 3D scene mapping for both ground and aerial robots.
• Robust topological room segmentation using persistent homology, enabling more stable navigation and spatial reference.
• Multi-resolution spatial attention and hierarchical prompting, dramatically reducing LLM prompt length.
• Seamless integration with actor–critic navigation and auxiliary tool calls for spatial relation disambiguation.

Applications include language-driven navigation and search tasks in IoT labs, homes, offices, and large-inventory warehouses, as well as deployment in multi-level buildings and environments with dense clutter. This methodology is particularly valuable for domains requiring both high accuracy in spatial referencing and token efficiency, such as mobile robotics and human-robot interaction under compute constraints (Liu et al., 18 Mar 2026).

6. Connection to Topology Optimization in Meta-optics

A parallel usage of the term “Topo-Omni” appears in computational optics, where topology-optimized multi-layered metalenses are synthesized to impart arbitrary angle-dependent phase profiles, enabling advanced functionalities like aberration correction and angularly convergent focusing. In this context, the density-based TO framework encodes the device as a design field $\omega_k$7, enforced by filter-projection and regularization to yield near-binary microstructures. The forward design loop relies on solving Maxwell’s equations with adjoint-based gradients and converges when phase transfer (figure-of-merit $\omega_k$8) approaches unity.

These “Topo-Omni” meta-optics overcome the generalized Snell’s law limit on phase control in single-layer metasurfaces, enabling compact, single-piece, volumetric lenses for applications in imaging, LiDAR, and spectroscopic sensing (Lin et al., 2017). A plausible implication is the shared utility of topological reasoning and optimization in both robotics and optics when engineering robust, generalizable functional structures.

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References

• "OmniVLN: Omnidirectional 3D Perception and Token-Efficient LLM Reasoning for Visual-Language Navigation across Air and Ground Platforms" (Liu et al., 18 Mar 2026)
• "Topology Optimized Multi-layered Meta-optics" (Lin et al., 2017)