OAM Radar Imaging via Space-Frequency Diversity

Abstract

This repository presents an end-to-end deep learning framework for sparse target reconstruction in Orbital Angular Momentum (OAM) radar imaging. The central challenge in OAM-based imaging is the inherent ill-posedness of the observation matrix $\Phi$, which manifests as a severely high condition number and a critically low effective rank — rendering direct inversion or naive learning-based approaches unstable.

We address this at the physical level through two complementary strategies: (1) Virtual Array Shifting, which introduces per-frequency lateral displacements of the transmit/receive array to break the structural symmetry of the measurement model, and (2) Phase-Randomized Sampling, which injects fixed random phase offsets per receiver to reduce mutual coherence across measurement rows. Together, these interventions reduce the condition number of $\Phi$ from $\sim 10^{13}$ to $\sim 10^{8}$ and increase the effective rank from 7 to over 196, eliminating the central imaging blind spot that plagues conventional OAM configurations.

Built on this physically stabilized observation model, a compact End-to-End Reconstruction Network achieves a median PSNR of 70.00 dB on held-out test data, converging within 30 training epochs.

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Key Innovations

1. Physical-Level Observation Matrix Optimization

Rather than compensating for a poorly conditioned forward model through increased network capacity, this work optimizes $\Phi$ directly:

Strategy	Mechanism	Effect
Virtual Array Shifting	Per-frequency lateral displacement of the UCA	Breaks rotational symmetry; diversifies row-space of $\Phi$
Phase-Randomized Sampling	Fixed random phase offset $\delta_n \sim \mathcal{U}(0, 2\pi)$ per receiver	Reduces pairwise mutual coherence between measurement rows
Wideband Multi-Frequency	8 frequencies spanning ±25% bandwidth	Multiplies effective rank by factor $F$; fills spectral gaps

Before optimization: condition number $\kappa(\Phi) \approx 10^{13}$, effective rank $\approx 7$. After optimization: condition number $\kappa(\Phi) \approx 10^{8}$, effective rank $> 196$.

2. End-to-End Reconstruction Network

The network (src/model.py) maps complex measurement vectors $\mathbf{y} \in \mathbb{C}^M$ directly to target images $\hat{\mathbf{x}} \in [0,1]^{N \times N}$:

• Input normalization: per-sample L2 normalization with a learned scale parameter, making the network invariant to scene energy variation.
• Feature extractor: three-layer MLP with LayerNorm and GELU activations (hidden_dim=768).
• Spatial decoder: transposed convolution upsampling $6 \times 6 \to 12 \times 12 \to 24 \times 24$ with residual refinement blocks.
• Loss function: combined L1 + L2 loss ($\alpha = 0.5$), promoting both sparsity and smoothness.

3. Physics Diagnostic Tooling

scripts/diagnose_physics.py is the core methodology validation tool. It performs a systematic parameter sweep over array configurations and reports:

• Singular value spectrum and effective rank
• Condition number
• Mutual coherence $\mu$ of column-normalized $\Phi$
• Per-mode Bessel function coverage over the target region
• $\ell = 0$ mode redundancy (pairwise cosine similarity)

This script should be run before any training to verify that the chosen physical parameters yield a computationally tractable observation matrix.

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Performance

Evaluated on 1,000 held-out test samples (fixed seed, no augmentation):

Metric	Value
PSNR (Median)	70.00 dB
PSNR (Mean ± Std)	63.06 ± 14.54 dB
MAE (Mean)	0.0004
RMSE (Mean)	0.0057
Samples evaluated	1,000

PSNR is capped at 70 dB to prevent trivially easy near-empty samples from dominating aggregate statistics.

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Visual Results

Singular Value Spectrum

Normalized singular value spectrum comparing the static baseline vs. the proposed virtual-shift configuration. The proposed method yields consistently higher singular values across all indices, indicating improved conditioning (κ = 2.87×10⁸ vs. 3.66×10⁸).

Reconstruction Comparison

Ground truth (left) vs. network reconstruction (right) on a representative held-out test sample. Sharp edges, circular boundaries, and central-region targets are faithfully recovered.

Training Convergence

Validation PSNR and training loss over 30 epochs. PSNR peaks at 77.9 dB at epoch 25; training loss drops from 1.2×10⁻¹ to 9.0×10⁻⁵.

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Project Structure

.
├── src/
│   ├── physics.py        # OAM radar geometry; generates observation matrix Φ
│   ├── dataset.py        # Synthetic sparse target dataset with augmentation
│   ├── model.py          # End-to-end reconstruction network and combined loss
│   └── train.py          # Training loop with mixed-precision and LR warmup
├── scripts/
│   ├── diagnose_physics.py      # Physical stability diagnostic (run before training)
│   ├── evaluate.py              # Quantitative evaluation: PSNR, MAE, RMSE
│   ├── baseline_tikhonov.py     # Tikhonov regularization baseline (Table II)
│   └── plotting/
│       ├── visualize.py         # Qualitative reconstruction visualization
│       ├── plot_paper_figures.py
│       └── plot_convergence.py
├── outputs/
│   ├── best_model.pth           # Best checkpoint (saved by PSNR)
│   └── training_history.json    # Per-epoch loss and PSNR curves
├── results/
│   └── evaluation_metrics.json
├── figures/
│   ├── fig1_singular_values.pdf     # SVD spectrum figure (paper Fig. 1)
│   ├── reconstruction_sample.png    # Reconstruction comparison (paper Fig. 2)
│   └── convergence_profile.png      # Training convergence (paper Fig. 3)
└── requirements.txt

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Getting Started

Installation

pip install -r requirements.txt

Requirements (requirements.txt):

torch>=2.0.0
numpy>=1.24.0
scipy>=1.10.0
matplotlib>=3.7.0
tqdm>=4.65.0
Pillow>=9.5.0

For GPU support, install PyTorch with the appropriate CUDA version from pytorch.org.

Step 1 — Validate Physical Configuration

Before training, verify that the observation matrix is computationally tractable:

python scripts/diagnose_physics.py

Key indicators to check:

• Condition number $\kappa < 10^{9}$
• Effective rank (SV > 1% of max) $\geq 100$
• Mutual coherence $\mu < 0.5$
• $\ell = 0$ pairwise cosine similarity $< 0.9$

Step 2 — Train

python src/train.py

The best checkpoint is saved to outputs/best_model.pth. Training history is written to outputs/training_history.json.

Default configuration: 30 epochs, batch size 64, AdamW with cosine annealing and 10-epoch linear warmup, 30,000 training samples.

Step 3 — Evaluate

python scripts/evaluate.py --checkpoint_path outputs/best_model.pth

Step 4 — Visualize

python scripts/visualize.py

Output is saved to figures/reconstruction_sample.png.

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Citation

If you use this work, please cite:

@misc{chen2026oam,
  author    = {Jiayan Chen},
  title     = {OAM Radar Imaging via Space-Frequency Diversity},
  year      = {2026},
  note      = {GitHub repository},
  url       = {https://github.com/Dryoung95/OAM-Spatial-Diversity}
}

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License

This project is licensed under the MIT License.

MIT License

Copyright (c) 2026 Jiayan Chen

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