Tessellating The Earth¶

Conference: ECCV 2026
arXiv: 2606.27514
Code: https://github.com/mvrl/TTE
Area: Remote Sensing
Keywords: Geographic Coordinate Encoding, Spherical Voronoi Tessellation, Learnable Spatial Decomposition, Global Semantic Tokens, Contrastive Learning

TL;DR¶

TTE replaces fixed base functions (spherical harmonics/Fourier features) with learnable spherical Voronoi tessellations to encode geographic coordinates. This allows anchor stations to automatically migrate toward highly discriminative regions during training to concentrate representation capacity. Additionally, it introduces global semantic tokens as cross-regional semantic sharing channels, achieving new state-of-the-art (SOTA) performance across multiple geospatial classification and regression tasks.

Background & Motivation¶

Longitude and latitude coordinates carry rich geographical information—climate, ecology, land cover, and infrastructure are all closely related to location. The goal of a location encoder is to learn a mapping from coordinates to high-dimensional embeddings, thereby characterizing the environmental features of a location using only coordinates. These embeddings serve as geographic priors in tasks such as species distribution modeling, crop yield estimation, and air quality prediction. Existing methods (e.g., SatCLIP, GeoCLIP) mostly adopt a contrastive learning framework to align the outputs of the location encoder and a satellite image encoder in an embedding space. The most critical design choice is location encoding—how to project longitude and latitude into a high-dimensional space. Current approaches use fixed base functions such as spherical harmonics, multi-scale Fourier features, or random Fourier features. However, their spatial structures are predetermined before training and cannot adaptively adjust based on data. Spherical harmonics distribute representation capacity uniformly across the entire sphere, allocating the same number of base functions to open oceans as to bustling cities. This indiscriminate allocation leaves coastlines and small landmasses chronically underrepresented.

The Key Challenge behind this uniform distribution is that the spatial complexity of the Earth's surface is far from uniform; fixed base functions cannot focus representation capacity where it is truly needed. While about 71% of the globe is covered by oceans, the landmasses of interest in most practical tasks account for only 29%. However, existing encoders cannot "selectively" allocate computational resources to land, coastlines, or ecological transition zones. The Key Insight proposed in this paper is: since adaptive spatial decomposition is required, why not turn Voronoi tessellation—a classic geometric partitioning tool—into a learnable process? By giving each Voronoi station both its own location and its own embedding vector, these stations can automatically migrate toward highly discriminative regions during contrastive training, achieving space-complexity allocation in a data-driven manner. Furthermore, the rich semantic knowledge (e.g., land cover types, vegetation cover) contained in the satellite image encoder is only available during training, while only coordinates are provided during inference. How to compress this visual semantics into a "concept vocabulary" that can be referred to at any time, allowing stations covering similar environments across different continents to share semantics, is the second key issue to be resolved.

Core Idea: Replace fixed location encoding base functions with learnable spherical Voronoi stations that automatically migrate to highly discriminative regions during contrastive training to concentrate representation capacity. Meanwhile, introduce a set of shared global semantic tokens to encode satellite image visual semantics into a compact concept dictionary via asymmetric attention distillation, enabling the encoder to query semantic information during inference using coordinates alone.

Method¶

Overall Architecture¶

The overall architecture of TTE is divided into two pathways. On the left is the location pathway (the only part used during inference): input latitude and longitude coordinates are mapped to a unit sphere, and pass through a learnable spherical Voronoi encoder to generate soft assignment weights. The embeddings of each station are weighted and summed to obtain the "Voronoi embedding". This embedding then passes through an attention mechanism with global semantic tokens to generate a "token representation", which is fused with the Voronoi embedding using equal weights (\(\alpha=0.5\)) before passing through LayerNorm to obtain the final location embedding. On the right is the image pathway (used only during training): co-located Sentinel-2 satellite images are processed by a frozen MoCo-ViT-L encoder to extract features, which then pass through an attention mechanism with the same set of tokens (but with a temperature fixed at 0.05, producing a peaky, near-one-hot distribution) to obtain the image-side token representation. Three loss functions—contrastive loss, reconstruction loss, and alignment loss—jointly optimize the entire system.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Coordinates (Lat/Lon)"] --> B["Spherical Voronoi Encoder<br/>K=4096 Learnable Stations"]
    B --> C["Soft Assignment Weights<br/>Cosine Distance × Learnable Temperature"]
    C --> D["Voronoi Embedding<br/>Weighted Sum of Station Embeddings → Residual MLP"]
    D --> E["Global Semantic Token Attention<br/>64 Learnable Concept Tokens"]
    E --> F["Equal-Weight Fusion α=0.5"]
    F --> G["LayerNorm → Final Location Embedding"]

    H["Satellite Image<br/>(Training Only)"] --> I["Frozen ViT-L/16<br/>MoCo Pre-trained"]
    I --> J["Image Token Attention<br/>T=0.05 Peaky Distribution"]
    J -->|"Alignment Loss KL"| E
    J --> K["Reconstruction Loss L2"]

Key Designs¶

1. Learnable Spherical Voronoi Tessellation: Adapting Capacity to Data

The biggest issue with fixed base functions is that they allocate the same number of primitives regardless of where the Earth's surface is complex or simple. The core design of this paper is to replace these fixed bases with a set of learnable Voronoi stations. \(K=4096\) stations are placed on the unit sphere, with each station having three learnable parameters: a location vector \(s_k\) (movable on the sphere, normalized back to \(\mathbb{S}^2\) after each gradient step), a temperature \(\tau_k\) (controlling the sharpness of Voronoi boundaries, where stations with larger values have smaller receptive fields), and an embedding vector \(e_k\) (\(\mathbb{R}^{384}\)). For a query coordinate \(x\), its soft assignment weight to station \(k\) is calculated via softmax weighted by cosine similarity:

\[w_k(x) = \frac{\exp(\tau_k \cdot (s_k \cdot x))}{\sum_j \exp(\tau_j \cdot (s_j \cdot x))}\]

The final Voronoi embedding is the soft-weighted sum of all station embeddings \(f(x) = \sum_k w_k(x) \cdot e_k\), which is then mapped to 512 dimensions through a two-layer residual block MLP. Stations are initialized on Fibonacci grid points filtered for land (excluding Antarctica), with the temperature initialized to 45. Driven by contrastive training, stations automatically migrate to highly discriminative regions such as coastlines and ecological transition zones during training. Interestingly, instead of staying strictly on land, many stations move offshore, allowing their receptive fields to specifically cover narrow coastal strips rather than sharing representation capacity across both land and sea. Station distribution in ocean areas remains sparse. This fully data-driven, non-uniform distribution is impossible to achieve with prior fixed-base methods.

2. Global Semantic Tokens: Sharing Representation for the Same Ecoregion Across Continents

Voronoi encoders are inherently local—each station only senses its immediate vicinity. There is no direct semantic sharing channel between a station covering the Amazon rainforest and another covering the African rainforest. However, intuitively, the semantic concept of "rainforest" should be shared across continents. This paper introduces \(R=64\) learnable global semantic tokens \(\mathbf{r}_r \in \mathbb{R}^{512}\) to act as a shared "concept dictionary". The location and image pathways each use a linear projection to map embeddings to 64-dimensional attention logits, which, after temperature-scaled softmax, yield token weights. The token vectors are then weighted and summed to obtain the context representation.

The key lies in the asymmetry temperature design: the image pathway uses a fixed low temperature \(T_{\text{img}}=0.05\), producing a peaky, near-one-hot attention distribution (each image strongly activates only 1–2 tokens); the location pathway uses a learnable \(T_{\text{loc}}\) (linearly annealed from 0.5 to 0.2) to learn to predict which semantic tokens should be activated from coordinates. This asymmetric design forces token semantics to specialize. Ultimately, each token successfully learns to represent specific environmental concepts such as "snow/ice cover," "arid desert," "rocky terrain," and "coastal areas." Importantly, this specialization emerges in a completely unsupervised manner without any labels. The final location embedding is generated by fusing a linear projection of the Voronoi embedding with the token representation using equal weights (\(\alpha=0.5\)), followed by LayerNorm, which preserves local spatial structure while integrating global semantics.

3. Joint Optimization with Triple Losses: Contrastive + Reconstruction + Alignment

The training objective consists of three loss functions. The primary loss is a symmetric contrastive loss (identical to SatCLIP) that aligns co-located image and location embeddings while pushing negative pairs in a batch apart, with a learnable temperature parameter. The reconstruction loss requires tokens to reconstruct image features—minimizing the \(L_2\) distance between the image-pathway token representation \(z_{\text{img}}\) and the original ViT features. This is critical to prevent token collapse, ensuring the 64 tokens indeed span the main directions of the image feature space. The alignment loss uses KL divergence to align the token attention distribution of the location pathway as closely as possible to that of the image pathway (with a stop-gradient on the image target), enabling the location pathway to infer semantic concepts from coordinates alone.

Loss & Training¶

The total loss is \(\mathcal{L} = \mathcal{L}_{\text{con}} + 100 \cdot \mathcal{L}_{\text{recon}} + 0.1 \cdot \mathcal{L}_{\text{align}}\). The reconstruction loss has a higher weight to ensure semantic relevance, while the alignment loss has a lower weight to allow the location pathway to retain geographical adaptability while following semantic guidance. Training uses the AdamW optimizer with different learning rates for different parameter groups (the station locations are fastest at 1e-3, and projection heads are slowest at 5e-5). After a linear warmup of 5 epochs, cosine annealing is applied down to 1e-6. The effective batch size is 6,384 (over 2× H100 GPUs using FP32), training for a total of 300 epochs in approximately 10 hours. Ours (TTE) adds only 4.0M trainable parameters (1.3% of the total parameter count), making it almost identical in size to SatCLIP.

Key Experimental Results¶

Main Results¶

Task (Classification)	ACC	Prev. SOTA	Task (Regression)	R²	Prev. SOTA
Biome	77.8	TaxaBind 72.3 (+5.5)	Temperature	0.946	SINR 0.942
EcoRegion	67.4	TaxaBind 72.9	Elevation	0.839	SatCLIP 0.666 (+0.173)
Country	94.8	SINR 88.3 (+6.5)	Population	0.790	TaxaBind 0.712
Classification Avg.	80.0	Second Best 77.2 (+2.8%)	Regression Avg.	0.777	Second Best 0.732 (+0.045)

iNaturalist-2018 Geographic Prior	Top-1	Top-3	Top-5	Top-10
Image Only	66.1	83.3	88.0	92.2
+ SatCLIP	75.1	88.7	91.9	94.5
+ TaxaBind	75.1	89.7	93.0	95.6
+ TTE	76.2	90.0	93.2	95.7

Ablation Study¶

Configuration	Classification Avg. ↑	Regression Avg. ↑	Key Observations
Full TTE	80.0	0.777	Baseline
Fixed Stations (No Location Update)	67.1 (-12.9)	0.652 (-0.125)	Worst Degradation, below SatCLIP
Remove Tokens (No Semantic Branch)	76.3 (-3.7)	0.739 (-0.038)	Significant contribution from tokens
No Alignment Loss	78.3 (-1.7)	0.763 (-0.014)	Classification is more affected
No Reconstruction Loss	78.8 (-1.2)	0.766 (-0.011)	Primarily on EcoReg and Housing
Symmetric Soft Attention	77.8 (-2.2)	0.752 (-0.025)	Peaky distribution outperforms symmetric

Key Findings¶

The contribution of learnable station locations far exceeds other components: fixing the stations causes classification to drop by 12.9% and regression to drop by 0.125 on average, falling entirely below SatCLIP. This strongly supports the core claim that "uniformly distributing capacity is the actual bottleneck."
The optimal number of tokens is 64 (too few underrepresents the semantic space, too many dilutes the alignment signal), and the optimal number of stations is 4,096 (8,192 actually leads to slight degradation). Both parameters exhibit clear saturation curves in the experiments.
Station migration exhibits a counter-intuitive geometric strategy: when migrating toward coastlines, stations do not stay on land but instead move offshore. This allows their receptive fields to specifically cover narrow coastal zones, avoiding sharing representation capacity between coastal and inland areas.
As a purely parametric encoder, TTE even outperforms the retrieval-augmented RANGE method (76.2 vs 75.2) on iNaturalist-2018, which requires an external database search. This demonstrates that the tokens have accumulated sufficient cross-regional semantic priors.

Highlights & Insights¶

Using spherical Voronoi to replace fixed bases for geographic encoding is a highly elegant design: Cosine distance naturally corresponds to angular distance on the sphere, and softmax allocation ensures that all stations receive non-zero gradients. The migration of ideas from DeRF to geographic representation learning is clearly defined, yielding substantial benefits with a minimal parameter increment (1.3%).
Asymmetric attention distillation is the most ingenious detail: The image pathway uses a very low temperature to force token specialization, while the location pathway uses slow annealing to learn to predict this specialized distribution. Ablation experiments demonstrate that this "teacher-student" framework outperforms symmetric soft attention.
The token visualization is remarkable—the model learns semantic concepts like "snow/ice", "desert", "rock/terrain", and "coastal areas" without any explicit supervision, and the attention maps exhibit global consistency across different continents.
The code and model weights are open-sourced, providing a strong drop-in replacement baseline for SatCLIP in the remote sensing community.

Limitations & Future Work¶

Currently, training relies solely on single-temporal Sentinel-2 imagery without integrating a temporal dimension, leading to poor performance on time-sensitive tasks (e.g., housing price prediction, where there is a temporal mismatch between 1990 census data and contemporary satellite imagery).
Although the global semantic tokens (64 tokens) are effective, they cannot fully replace direct image retrieval—RANGE's retrieval-augmentation improves biome accuracy from 77.8% to 84.2%, indicating that token compression discards some fine-grained visual information.
It lags behind TaxaBind on EcoRegion, as the latter utilizes iNaturalist species observation alignments. This suggests that highly fine-grained ecological classification may require specialized training signals rather than general visual contrastive learning.
The differences between station initialization strategies are negligible (Fibonacci grid initialization vs. SatCLIP/RANGE clustering initialization yields similar results), indicating that the training process is sufficiently robust, but also suggesting that more compact initializations could be explored to reduce convergence time.

vs SatCLIP: SatCLIP uses spherical harmonics + SIREN as a fixed-base location encoding. TTE replaces this with learnable Voronoi, boosting Biome from 68.9 to 77.8, and Elevation from 0.666 to 0.839, intuitively demonstrating the value of adaptive spatial decomposition.
vs GeoCLIP: GeoCLIP uses hierarchical random Fourier features, which is also a fixed-base method. TTE achieves 94.8 vs 81.3 on Country classification, showing a highly significant gap.
vs RANGE: RANGE's retrieval-augmentation (searching an external database during inference) provides stronger representations but introduces deployment constraints. TTE's purely parametric encoder outperforms RANGE on iNat-2018, demonstrating that well-designed self-supervised strategies can bridge some of the advantages brought by retrieval.
Conceptual Origin: The idea of Voronoi station migration originates from DeRF (learnable Voronoi decomposition in Neural Radiance Fields). The design of global semantic tokens is mechanistically isomorphic to ViT's register tokens and the inducing points of the Set Transformer, but serves a different objective—cross-modal knowledge distillation.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ Replacing fixed bases with learnable spherical Voronoi for geographic encoding is a highly original design, and the asymmetric attention distillation is also very clever.
Experimental Thoroughness: ⭐⭐⭐⭐⭐ Covers 8 geographic tasks + species classification. The ablation study is comprehensive (component-by-component ablation, station and token count sweep, multiple initializations, architectural variants, and comparison with retrieval-augmentation).
Writing Quality: ⭐⭐⭐⭐⭐ Problem motivation is clear, method description is self-consistent, and figures (Voronoi distribution, token attention, station migration dynamics) coordinate well with the exposition.
Value: ⭐⭐⭐⭐⭐ Substantially improves geographic encoding quality with only a 1.3% parameter increase. The code is open-sourced, realistically driving the development of Geospatial AI.