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Bagged Deep Image Prior for Recovering Images in the Presence of Speckle Noise
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Proposes Bagged-DIP-based projected gradient descent with Newton-Schulz inverse updates for multilook, undersampled complex-valued speckle-noise image recovery, with an MSE upper bound for the DIP-based MLE.
Problem
Recovering a complex-valued image from undersampled (m<n) multilook measurements corrupted by multiplicative speckle noise, where the likelihood-based estimator must be solved under a deep image prior hypothesis.
Approach
Averages several DIP projections built from patch-wise independent DIPs with different patch sizes (Bagged-DIP) inside PGD, so complex DIPs can be used without overfitting the noisy gradient step, and approximates the matrix inverse with a single Newton-Schulz step warm-started from the previous iteration.
Key results
- Bagged-DIP PGD averages 19.24/0.406 (12.5%, L=25) to 29.78/0.856 (50%, L=100) PSNR/SSIM across 8 images, outperforming DIP-M^3 by about 1.1 to 1.5 dB on average across looks.
- Single Newton-Schulz step reproduces the exact-inverse PGD result, while gradient time per step drops from about 52.8 s to about 1e-4 s at 128x128.
- Empirical scaling checks: doubling m gave an average 3.99 dB gain (theory predicts about 4.5 dB) and doubling L gave 1.42 dB (theory predicts about 1.5 dB).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Bagged-DIPs instead of DIP-simple | PGD for undersampled speckle-noise image recovery · m/n=12.5%, 25%, and 50%; L=25, 50, and 100 | average PSNR (dB): 18.03 -> 19.24; 19.25 -> 20.83; 20.15 -> 21.78; 22.00 -> 22.86; 23.42 -> 24.95; 24.79 -> 26.24; 25.62 -> 27.21; 26.81 -> | Helped |
| Use Bagged-DIPs instead of DIP-M³ | PGD for undersampled speckle-noise image recovery · m/n=12.5%, 25%, and 50%; L=25, 50, and 100 | average PSNR (dB): 17.81 -> 19.24; 18.96 -> 20.83; 19.94 -> 21.78; 21.69 -> 22.86; 23.39 -> 24.95; 25.08 -> 26.24; 26.01 -> 27.21; 27.81 -> | Helped |
| Average three DIP estimates with different patch sizes | Bagged-DIPs · Simulated image recovery; three-estimate comparison | PSNR (dB): between 0.5dB and 1dB | Helped |
| Use one Newton-Schulz step instead of exact matrix inversion | Bagged-DIP-based PGD · L=50, m/n=0.5 | reconstruction performance | No clear effect |
| Use Newton-Schulz approximation instead of exact matrix inversion | PGD matrix-inversion step · 32 × 32, 64 × 64, and 128 × 128 images | time (seconds): ∼ 0.3 -> ∼ 7e-5; ∼ 1.2 -> ∼ 8e-5; ∼ 52.8 -> ∼ 1e-4 | Helped |
| Use DnCNN-UB with training data and a well-conditioned A=I measurement setup instead of Bagged-DIP-based PGD | Speckle despeckling comparison · m/n=50%; L=25, 50, and 100; most test images | PSNR (dB): between 1-3dB better; with the exception of the cameraman | Mixed |
| Increase the number of looks | Bagged-DIP-based PGD · Simulated image recovery | average PSNR (dB): 1.42 dB | Helped |
Surprising findings
- A DnCNN despeckler trained on BSD400 with fully-observed well-conditioned measurements often beats the Bagged-DIP method by 1-3 dB, except on Cameraman.
- A single Newton-Schulz step is sufficient, while stopping inverse updates after 5, 10 or 20 iterations makes PGD diverge.
Limitations
- Algorithm faces GPU memory limits at 256x256 images, so experiments are limited to 128x128.
- Bagging uses only three patch-size estimates, and some images (e.g., Foreman) show individual estimates outperforming the average due to bias.
- Newton-Schulz step convergence holds only for small changes in x (threshold 0.12), requiring exact inverses in the first iterations.
Experimental setup
Datasets: Set11 (8 images cropped to 128x128) BSD400
Benchmarks: Set11 (Barbara, Peppers, House, Foreman, Boats, Parrots, Cameraman, Monarch)
Metrics: PSNR SSIM
Baselines: DIP-simple DIP-M^3 DnCNN-UB
Builds on: deep image prior projected gradient descent deep decoder Newton-Schulz iteration
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