Spatial Foreground Predictor (SFP)

• SFP is a framework for spatially discriminating and modeling foreground signals, enabling efficient extraction by focusing on informative data subsets.
• It uses domain-specific techniques such as grouped convolutions in LiDAR, Gaussian process kernels in 21 cm mapping, and template-driven modeling in LISA.
• Performance improvements include increased detection success, reduced computational overhead, and enhanced signal fidelity in diverse scientific applications.

A Spatial Foreground Predictor (SFP) is a class of frameworks and modules designed for the spatial discrimination, modeling, and prediction of foreground content in scientific data or sensor measurements, where spatial structure is critical to downstream tasks. SFPs have emerged within disparate domains, including 3D point cloud tracking, 21 cm cosmology, and gravitational wave data analysis, but share the unifying goal of improving the extraction or prediction of informative foreground signals embedded in spatially redundant or background-dominated data.

1. Conceptual Motivation and Objectives

Spatial Foreground Predictors address challenges where the foreground (of interest) constitutes a small fraction of a high-dimensional spatial field, and naive processing of all spatial elements results in wasted computation or poor signal recovery. SFPs improve signal modeling, computational efficiency, and downstream performance by either masking, reweighting, or explicitly modeling spatial variation in foreground structure.

In LiDAR-based 3D SOT, the majority of BEV (bird's-eye view) pillars are background or empty; spatial redundancy impairs efficiency and accuracy. In 21 cm intensity mapping, line-of-sight (LoS) foreground contamination varies across the field; global modeling conflates spectral smoothness scales and leads to excess residuals. In LISA data, the anisotropic Galactic foreground must be accurately mapped and subtracted to reveal weak cosmological signals. In each case, SFPs provide a principled mechanism for foreground/background separation or for parameterizing spatial structure in a hierarchical or template-driven way (Zhou et al., 19 Nov 2025, Diao et al., 16 Jul 2024, Criswell et al., 30 Oct 2024).

2. Mathematical Formulations and Information-Theoretic Principles

SFP architectures are generally motivated by information theory or Bayesian inference:

• Redundancy Filtering (LiDAR SOT): The feature map $\mathbf{X} \in \mathbb{R}^{H \times W \times C}$ has a small foreground occupation probability $p \ll 1$. The total Shannon entropy is $H(\mathbf{X}) = H_b(p)HW + pH_{\mathrm{fg}}HW$, where $H_b(p)$ is the binary entropy. Since $p$ is small, masking out background elements is nearly information-lossless (Zhou et al., 19 Nov 2025).
• Bayesian Modeling (21 cm, LISA): Foreground structure is modeled either by locally adaptive Gaussian process kernels (21 cm), or as a sum over spatial templates (LISA). These approaches enable spatial variation in parameters and explicitly trace how information about the foreground is encoded across the sky or field (Diao et al., 16 Jul 2024, Criswell et al., 30 Oct 2024).

3. Architectures and Algorithms

Spatial Foreground Predictors vary by application:

LiDAR Point Cloud Tracking SFP

• Input: Concatenated BEV features from template and search frames.
• Architecture: Two-stage grouped convolution module:
  1. $3 \times 3$ grouped convolution $\to$ BatchNorm $\to$ ReLU.
  2. $1 \times 1$ grouped convolution $\to$ Sigmoid, yielding $\mathbf{Y}_\mathrm{pred} \in [0,1]^{H \times W}$.

• Modulation: The search features $\mathbf{F}_s$ are reweighted: $$\hat{\mathbf{F}}_s(x,y,c) = \mathbf{F}_s(x,y,c) \times \mathbf{Y}_\mathrm{pred}(x,y)$$.

• Objective: Mean squared error against Gaussian center-biased heatmaps derived from ground-truth object boxes.

21 cm Gaussian Process SFP

• No-Pooling (NP) Model: Each LoS is assigned independent GP kernel parameters for smooth and polarization-leakage foregrounds, leading to local adaptation.
• Hierarchical GP (HGP): The spatial field is divided into superpixels sharing kernel parameters with hyperpriors regularizing the population.
• Inference: Bayesian posterior sampling using NUTS; posterior predictive mean and covariance for component separation.

LISA Galactic Foreground SFP

• Spectral–Spatial Decomposition: Foreground power $$S_\mathrm{GW}(f, \hat{\Omega}) = S(f)P(\hat{\Omega})$$ is modeled with $$P(\hat{\Omega}) \approx \sum_i a_iT_i(\hat{\Omega})$$, where $T_i$ are physically motivated templates.
• Data Modeling: The time- and frequency-dependent covariance matrix includes convolution with the response of the instrument to each template.
• Inference: Hierarchical Bayesian posterior sampling over template amplitudes and spectral parameters.

4. Training Objectives and Statistical Inference

Domain	Principal SFP Objective	Loss/Bayesian Metric
LiDAR SOT	Attention heatmap over BEV grid	$\mathcal{L}_\mathrm{pred}$ (MSE vs. heatmap)
21 cm	LoS-/superpixel-specific GP kernels	Marginal likelihood; residual std; PS recovery
LISA	Template amplitudes and spectral shape	Likelihood of TDI data; posterior spectrum/map

LiDAR SFPs are optimized with supervised MSE regression; 21 cm and LISA SFPs use full Bayesian inference, often with HMC/NUTS implemented in JAX/NumPyro and stringent convergence diagnostics ($\widehat R \leq 1.01$).

5. Performance, Computational Trade-Offs, and Domain-Specific Impact

LiDAR SFP: On nuScenes, introducing SFP alone (with all else fixed) increases mean Success from 59.38% to 60.01% and Precision from 71.63% to 72.20%, boosting throughput from 48 FPS to 55 FPS. When paired with an information bottleneck module, the overall framework reaches 90 FPS at 61.04%/73.68% (Zhou et al., 19 Nov 2025).

21 cm SFP: The NP3 model achieves up to a 30% reduction in residual standard deviation compared to global-kernel baselines; HGP3 offers nearly identical gains while reducing parameter count by $\sim256\times$ and improving convergence. Power spectrum and wavelet scattering analyses show substantially improved recovery of cosmological signals (Diao et al., 16 Jul 2024).

LISA SFP: Spectrum recovery errors are typically below a few percent, with the angular power spectrum $C_\ell$ of the reconstructed foreground map accurate to within tens of percent. The recovered modulation curve matches the true modulation to within $\lesssim 1\%$ in shape and phase (Criswell et al., 30 Oct 2024).

SFP methods consistently reduce spatial/modeling redundancy, improve resource utilization, and provide higher-fidelity foreground/background separation, enabling more sensitive downstream analyses.

6. Implementation Considerations and Limitations

• LiDAR SFPs are computationally lightweight, relying on grouped convolutions and simple elementwise masking.
• 21 cm SFP (NP) incurs a high-dimensional parameter space ($\sim4000\times$ more kernel params), necessitating data chunking and posing overfitting/convergence risks; HGP achieves a tractable compromise with spatial pooling, though superpixel choices introduce artifacts.
• LISA SFP is sensitive to the choice of template basis and prior; efficient GPU-based sampling is recommended due to the high dimensionality of the template amplitude vector.

Computational efficiency gains are domain-specific, and there exists a trade-off between modeling flexibility (fine spatial adaptation) and statistical/computational tractability.

7. Outlook and Ongoing Research

Current work targets scaling hierarchical models to higher resolution and larger fields using sparse/approximate Gaussian processes, optimizing template banks and prior structures for LISA, and further reducing information loss in attention-based foreground selection. In all domains, SFPs represent a modular, principled strategy to leverage spatial structure for efficient, accurate foreground recovery and signal analysis (Zhou et al., 19 Nov 2025, Diao et al., 16 Jul 2024, Criswell et al., 30 Oct 2024).