Spatial Feature Extractor (SFE)

• Spatial Feature Extractor (SFE) is a module designed to encode and preserve spatial relationships in structured data like images, geospatial coordinates, and spatiotemporal signals.
• It employs diverse methodologies, including multi-scale sinusoidal encodings, spectral similarity matrices, and attention-based deep encoders, to maintain invariances such as translation and rotation.
• SFEs transform high-dimensional spatial inputs into compact, robust representations, enabling effective downstream tasks like classification, clustering, and segmentation.

A Spatial Feature Extractor (SFE) is a computational architecture or algorithmic module designed to encode, emphasize, and preserve the spatial relationships and patterns in structured data such as images, geospatial coordinates, hyperspectral cubes, or spatiotemporal signals. In modern machine learning and data analysis, SFEs are implemented using diverse approaches—including learnable neural encoders, explicit mathematical transforms, or domain-specialized signal processing operators—with each tailored to its specific modality (e.g., 2D/3D vision, remote sensing, temporal-geospatial series, or neural signals). SFEs fundamentally bridge the gap between high-dimensional spatial input and the compact, information-rich representations required for efficient inference, classification, clustering, or downstream learning tasks. Their effectiveness is determined by their ability to preserve location- or neighborhood-sensitive information, maintain geometric invariances (such as translation, rotation, or geodesic distance), and/or enforce interpretability and robustness under noise or data sparsity.

1. Core Mathematical Principles Underpinning SFEs

SFEs operate on explicit or implicit representations of spatial organization. Core mathematical techniques vary across modalities and applications:

• Multi-scale Sinusoidal and Spherical Encodings: For geocoordinate representation, SFEs such as Sphere2Vec use a concatenation of multi-scale spherical Fourier basis features. The position encoder $\phi_S(p)$ for a point $p=(\lambda, \varphi)\in S^2$ (longitude, latitude) involves sinusoids of input coordinates, potentially augmented by cross-terms or Cartesian grid features. Linear-scale variants (e.g., sphereC, sphereM) preserve spherical distances and avoid projection distortions. The essential mathematical guarantee is that, for the minimal SFE (the "xyz" basis), the inner product preserves great-circle distance: $$\langle \phi_1(p_1), \phi_1(p_2) \rangle = \cos(d_S(p_1,p_2)/R)$$ where $d_S(\cdot, \cdot)$ is spherical arc length (Mai et al., 2023).
• Spectral Similarity and Local Windowing: In hyperspectral imaging, SFEs such as Local Spectral Similarity (LSS) form a similarity matrix by measuring vector distances (often Euclidean or fractional norms) between a center spectrum and its spatial neighbors within a sliding window. Edge intensity is derived via order statistics (median, max) or convolution with spatial kernels (Sobel, Laplacian) to capture elemental spatial gradients while integrating spectral patterns (Sahadevan et al., 2019).
• Isotropic and Rotation-Invariant Operators: Methods like Invariant Attribute Profiles (IAPs) within hyperspectral spatial-spectral pipelines employ banks of radially symmetric convolutional kernels—ensuring local invariance to rotation—followed by spatial aggregation over superpixels. This separates structural cues from location or orientation, addressing variances due to illumination or scene composition (Hong et al., 2019).
• Self-Supervised and Attention-Based Deep Encoders: In high-resolution remote sensing or vision-based reinforcement learning, SFEs are instantiated as learnable modules (e.g., masked Vision Transformers, convolutional attention heads) enforcing spatial localization via non-overlapping construction or explicit attention weighting tied to the input grid (Muzeau et al., 2024, Pham et al., 14 Apr 2025).
• Graph Autoencoders for Spatial Networks: For spatial relational data, SFEs operate on spatially structured graphs (e.g., distance and mobility between entities or POI types). Two-branch graph convolutional autoencoders encode neighborhood and semantic relations to yield compact per-entity embeddings capturing both spatial proximity and latent functional attributes (Wang et al., 2021).

2. Algorithmic Design Patterns and Implementation Strategies

Concrete SFE implementations are shaped by the modality and downstream task:

• Geospatial SFE (Sphere2Vec): The architecture consists of a position encoder mapping $(\lambda, \varphi)$ to a high-dimensional spherical embedding via scale-spaced sin/cos basis functions. Multiple variants exist: $sphereC_S$ (dimension $3S$), $sphereM_S$ ($5S$), and their grid-augmented variants ($6S$, $8S$). The encoder output is refined by a small MLP, yielding a representation $Enc(p) = NN(\phi_S(p))$, which is combined with non-spatial features (e.g., CNN image embeddings) for classification (Mai et al., 2023).
• Hyperspectral SFE (LSS, IAPs, S3FSE):
  • LSS operates by forming, for each pixel, a local similarity matrix and distilling edge strengths via order statistics or spatial convolutions.
  • IAPs convolve each spectral band with multiscale isotropic filters, aggregate via superpixels, and fuse with frequency-domain features.
  • S3FSE learns a joint low-dimensional subspace from stacked spectral and spatial descriptors (e.g., Gabor texture, morphology), enforcing graph-based neighborhood structure and class separability via joint Laplacian regularized objectives and $\ell_{2,1}$ sparsity (Sahadevan et al., 2019, Hong et al., 2019, Zhang et al., 2019).
• Deep Learning SFE (SAFE, IFE, SFE-Net):
  • SAFE applies masked Siamese Vision Transformers with SAR-specific data augmentation and contrastive-clustering loss to yield robust spatial features under speckle, intensity, or resolution variations (Muzeau et al., 2024).
  • Interpretable Feature Extractors (IFE) in deep RL split front-end processing into a Human-Understandable Encoding (spatially aligned, non-overlapping conv + softmax attention mask) and an Agent-Friendly Encoding (standard CNN or residual branches), fusing the two by gating the main feature stream with the interpretable mask (Pham et al., 14 Apr 2025).
  • SFE-Net for EEG-based emotion recognition uses improved bicubic spatial interpolation, symmetric channel folding, and 3D-CNNs on each fold, enhancing spatial representation via ensemble fusion (Deng et al., 2021).

3. Geometric Preservation and Theoretical Guarantees

Robust geometrical encoding is central to high-performance SFE, especially for global geospatial learning:

• Spherical Distance Preservation: Sphere2Vec, by including specific sinusoidal terms dictated by discrete Fourier analysis on the sphere, provably encodes exactly the great-circle (geodesic) distance between two points in embedding space. For $S=1$, the mapping yields $$\langle \phi_1(p_1), \phi_1(p_2) \rangle = \cos(d_S(p_1,p_2)/R)$$ and $$\|\phi_1(p_1)-\phi_1(p_2)\| = 2 \sin(d_S(p_1,p_2)/(2R))$$, ensuring monotonic retrieval and non-aliasing over the sphere (Mai et al., 2023, Mai et al., 2022). By contrast, grid- or 3D-Euclidean-based encoders fail to maintain this property, introducing significant distortion in high-latitude or data-sparse regimes.
• Translation and Rotation Invariance: Isotropic filter banks in IAPs endow invariance to spatial rotations, and superpixel aggregation imparts robustness to position and local scene changes. In LSS, choosing fractional norm distances (e.g., $p < 1$) enhances small-discrepancy contrast in high-dimensional spaces (Hong et al., 2019, Sahadevan et al., 2019).
• Spatial Alignment and Interpretability: Non-overlapping convolutions and explicitly normalized attention mechanisms ensure that extracted masks or spatial maps remain strictly aligned to the original input. The single-head attention in HUE of IFE guarantees that each attention score spatially corresponds to an interpretable region in the input (Pham et al., 14 Apr 2025).

4. Application Domains and Empirical Results

SFEs are validated across diverse datasets and modalities:

• Geospatial Prediction and Geo-aware Classification: Sphere2Vec achieves up to 30.8% error rate reduction on synthetic spherical datasets and 0.6–1.3% absolute Top1 accuracy improvement on real-world tasks (BirdSnap, NABirds, Flickr, fMoW). Its gains are especially pronounced in polar and data-sparse domains, as confirmed by latitude band and MRR analyses (Mai et al., 2023).
• Hyperspectral Imaging: LSS yields Pratt Figure-of-Merit (FOM) of 0.92 versus <0.75 for multichannel gradient methods, with lower false-alarm counts and order-of-magnitude faster computation than HySPADE. IAPs improve Random Forest classification OA by 6-8% over baseline profiles on Houston datasets, particularly excelling in edge regions due to shift- and rotation-invariant design (Sahadevan et al., 2019, Hong et al., 2019).
• SAR, Vision, and Spatiotemporal Signals: SAFE demonstrates 69.6–98.6% few-shot SAR classification (MSTAR), outperforming deep CNN and manifold methods by 10–30 percentage points in low-data regimes; segmentation OA=77.2% and Kappa=66.3% are competitive with or better than other domain-specific feature extractors (Muzeau et al., 2024). In vision-based deep RL, IFE improves mean Human-Normalized Score from 139.8% to 157.2% over baselines in Atari benchmarks, providing spatially precise, human-interpretable attention maps (Pham et al., 14 Apr 2025). SFE-Net in EEG-based emotion recognition achieves 91.94–99.19% accuracy on DEAP and SEED, with major robustness gain from spatial folding and interpolation (Deng et al., 2021).
• Distributed Remote Sensing: DIFET applies scalable feature extraction (Harris, Shi-Tomasi, SIFT, SURF, FAST, BRIEF, ORB) on LandSat-8, extracting millions of robust, scale- and rotation-invariant points via Hadoop/HIPI, matching single-node results with near-linear speedup across clusters (Eken et al., 2018).

5. Practical Integration, Hyperparameter Considerations, and Limitations

Implementation efficiency and downstream integration are critical for high-dimensional, large-scale spatial data:

• Parameter Selection: For Sphere2Vec, $S=32$ scales with geometric spacing, MLP hidden widths of 512–1024, and dropout of 0.5 provide performance gains. For IAPs, filter radii $r=[2,4,6]$, superpixel counts $Q=2000$–$5000$, and Fourier frequency $m=0$–$3$ are standard. LSS operates effectively with windows $n=5$ and $p=2$ (Euclidean) or $p<1$ (fractional) (Mai et al., 2023, Sahadevan et al., 2019, Hong et al., 2019).
• Computational Complexity: The use of MapReduce frameworks (DIFET), local superpixel aggregation (IAPs), or batch-wise feature extraction (Sphere2Vec) enables practical scalability even for terabyte-scale jobs. Deep SFEs (SAFE, IFE) require GPU resources and careful orchestration of transformer or attention-based subcomponents (Eken et al., 2018, Muzeau et al., 2024).
• Integration with Downstream Models: SFEs can be employed as stand-alone modules (providing explicit spatial representations), joint feature branches (combined with visual or contextual signals), or implicitly trained encoders whose weights become part of a larger end-to-end system (for classification, segmentation, anomaly detection, or semantic alignment) (Mai et al., 2023, Hong et al., 2019, Pham et al., 14 Apr 2025).
• Limitations and Potential Extensions: Non-learned SFEs (e.g., hand-crafted kernels in IAPs or the base DFS encoding in Sphere2Vec) are susceptible to data modality shift unless tuned. Learnable deep SFEs may lack interpretability or be driven by task-specific objectives unless explicit spatial alignment or disentangling constraints are included. Hybrid approaches (e.g., modular fusion of interpretable and agent-friendly encoders, as in IFE) offer one remedy, while expanded domain adaptation and multi-modal fusions remain areas for future SFE development (Pham et al., 14 Apr 2025).

6. Comparative Table of Representative SFE Approaches

SFE Method	Core Mechanism	Domains
Sphere2Vec	Multi-scale spherical Fourier	Geospatial prediction
LSS	Local spectral similarity matrix	Hyperspectral edge det.
IAP (SIF/FIF)	Isotropic conv. + region agg.	Hyperspectral, HSI class.
DIFET	Keypoint/descriptor extraction	Remote sensing (large)
SAFE	Siamese masked ViT, contrastive	SAR, segmentation
IFE	Non-overlap conv. + soft-attn	Deep RL/vision
SFE-Net	Folding/3D CNN/voting	EEG/emotion

Each listed method selects and encodes spatial cues relevant for its domain by explicit geometric, algebraic, or data-driven strategies. All are empirically benchmarked on large-scale, high-dimensional tasks and demonstrate improved robustness or interpretability over naive alternatives.