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Unsupervised Deep Unrolling Networks for Phase Unwrapping
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Proposes U3Net, an unsupervised deep unrolling network for phase unwrapping trained end-to-end on noisy wrapped phases only, using a noise-resistant recorruption-based self-reconstruction loss in the gradient domain plus self-distillation.
Problem
Phase unwrapping recovers true phase from noisy wrapped measurements, but supervised deep learning needs ground-truth phases that are often impractical to acquire; internal unsupervised methods are slow due to per-sample test-time training.
Approach
Treats the gradients and wrapped gradients of the wrapped phase as noisy labels for true phase gradients, with sparse outliers handled by an error variable; unrolls a variational PU model (proximal gradient with Nesterov acceleration and a sparse error term) into a DUN trained with a recorruption-based self-reconstruction loss (noise pair ±∇U with outer wrap) and a self-distillation loss.
Key results
- On MoGR and RME, U3Net achieves best NRMSE in 8 of 10 settings without any ground truth, outperforming unsupervised PUDIP and conventional LS/QGPU.
- On InSAR data, U3Net ranks first across noise levels 5, 10, and 20 dB, e.g. 0.82% at 10 dB and 0.46% at 20 dB.
- U3Net has the smallest parameter count among DL methods and runs about 10 ms per 256x256 sample versus roughly 99.7 s total for PUDIP's test-time training.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace Lsr with L (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 0.69 -> 25.45 | Hurt |
| remove the ±∇U noise recorruption terms (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 0.69 -> 6.16 | Hurt |
| replace Lsr with L and remove the ±∇U terms (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 0.69 -> 25.85 | Hurt |
| remove the self-distillation loss (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 0.69 -> 1.14 | Hurt |
| add the sparse error term E during unrolling (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 7.16 -> 0.69 | Helped |
| use accelerated gradient descent instead of standard gradient descent (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 2.14 -> 0.69 | Helped |
| add the condition-aware module (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 0.98 -> 0.69 | Helped |
| use W(∇Y) instead of ∇Y (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 2.88 -> 0.69 | Helped |
| use the physics-encoded DUN instead of UFormer (ablation) | U3Net · MoGR, SNR=0 | NRMSE(%): 1.55 -> 0.69 | Helped |
| use U3Net instead of Restormer | phase-unwrapping models · InSAR, SNR=20 | NRMSE(%): 0.91 -> 0.46 | Helped |
| use U3Net instead of Restormer | phase-unwrapping models · MoGR, SNR=0 | NRMSE(%): 0.45; 0.69 | Mixed |
Surprising findings
- At the lowest SNR (0 dB), the unsupervised U3Net is worse than supervised UFormer and Restormer, attributed to more outlier points.
- Using a plain gradient loss without the outer wrap or noise re-corruption degrades performance sharply (NRMSE above 20% at 0 dB).
Limitations
- Assumes known noise statistics, as with other unsupervised and synthetic-data supervised techniques.
- Performance drops at very low SNR (0 dB) relative to supervised transformers because of increased outlier points.
Experimental setup
Datasets: MoGR (simulated, synthetic Gaussian mixtures with ramps) RME (random matrix enlargement, synthetic) InSAR phase patches from Texas digital elevation model
Benchmarks: MoGR RME InSAR phase data
Metrics: NRMSE (%) parameters (M) FLOPs (G) inference time (ms)
Baselines: LS QGPU Ryu et al. PhaseNet2.0 PUDIP SQD-LSTM EESANet TriNet UFormer Restormer PU-GAN PUNet
Builds on: Deep image prior Recorruption-based denoising (Noise2Noise-style recorruption) Proximal gradient descent with Nesterov acceleration UFormer Restormer
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