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Efficient Noise Calculation in Deep Learning-based MRI Reconstructions
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Derives an unbiased, Jacobian-sketching estimator (complex random-phase probes with JVPs) for voxel-wise noise variance in deep MRI reconstructions, matching Monte-Carlo references at a fraction of the cost.
Problem
Noise propagation from k-space acquisition noise to reconstructed images is rarely quantified for deep learning MRI reconstructions; explicit Monte-Carlo estimation of voxel-wise variance is computationally and memory intensive, and the full network Jacobian is intractable to form.
Approach
Linearize the reconstruction around the noise-free input so the image covariance is J_f A^H Sigma_k A J_f^H; the diagonal is an expectation of (Sigma v) conj(v) style products, estimated unbiasedly with S complex random-phase probe vectors pushed through Jacobian-vector products instead of forming the Jacobian.
Key results
- Across six DL reconstruction methods at R=8, alpha=1, the proposed estimator achieves PCC of 99.4-99.9% and NRMSE of 0.4-1.8% versus the 3,000-trial Monte-Carlo reference.
- Per knee slice for an unrolled network, the proposed method takes 1.3 s and 1.1 MB versus 54.0 s and 3513.3 MB for the empirical Monte-Carlo baseline.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Jacobian sketching to estimate variance instead of empirical Monte-Carlo simulation | Deep MRI reconstruction methods · Knee dataset, standard unrolled architecture, per slice | Time: 54.0 s -> 1.3 s | Helped |
| Use Jacobian sketching to estimate variance instead of empirical Monte-Carlo simulation | Deep MRI reconstruction methods · Knee dataset, standard unrolled architecture, per slice | Storage: 3513.3 MB -> 1.1 MB | Helped |
| Use Jacobian sketching instead of the naive row-by-row Jacobian calculation | Deep MRI reconstruction methods · Knee dataset, standard unrolled architecture, per slice | Time: 880.2 s -> 1.3 s | Helped |
| Use complex random-phase vectors instead of complex Gaussian vectors | Jacobian diagonal estimator · Variance estimation | Estimator variance | Helped |
| Increase sketch size from 100 to 1,900 probing vectors (ablation) | Jacobian sketching · fastMRI brain, E2E-VarNet, R=8, α=1 | NRMSE | Helped |
| Apply the estimator at extreme noise levels (α=50–200) | Jacobian sketching · MRI with extreme k-space noise | Agreement with empirical variance maps | Hurt |
Surprising findings
- Complex random-phase probes give strictly lower estimator variance than complex Gaussian probes, per the derived variance expressions.
Limitations
- Linear (first-order Taylor) approximation may under- or over-estimate variance when noise is comparable to or exceeds the signal (very low SNR).
- Captures only diagonal (voxel-wise) aleatoric variance from k-space noise, not off-diagonal covariance or epistemic uncertainty.
Experimental setup
Datasets: Stanford knee dataset fastMRI brain dataset
Benchmarks: Stanford knee dataset fastMRI brain dataset
Metrics: Pearson correlation coefficient (PCC) normalized root-mean-square error (NRMSE) runtime storage
Baselines: Monte-Carlo empirical variance naive row-by-row Jacobian computation
Builds on: E2E-VarNet MoDL Noise2Recon VORTEX SSDU
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