Enabling self-supervised learned primal dual with Noise2Inverse

Abstract: X-ray computed tomography reconstruction is an ill-posed inverse problem, particularly in low-dose and sparse-angle settings where measurements are noisy and incomplete. While learned reconstruction methods such as the Learned Primal-Dual algorithm achieve strong performance, they typically rely on supervised training with access to ground-truth data, which is often unavailable in practice. In this work, we propose a self-supervised reconstruction method by extending the Noise2Inverse framework to the Learned Primal-Dual algorithm. The resulting approach, called Noise2Inverse Learned Primal-Dual (N2I-LPD), enables training of a learned iterative reconstruction operator without ground-truth images by exploiting the statistical independence of noise in distinct measurements with respect to angular rotation of the CT-scan. We compare the proposed method with classical reconstruction methods, as well as neural network-based approaches such as a U-Net trained within the same N2I framework. The results demonstrate that N2I-LPD achieves improved reconstruction quality, highlighting the potential of combining learned reconstruction operators with self-supervised training strategies for practical CT imaging scenarios where ground-truth data is unavailable.

• The paper introduces N2I-LPD, an end-to-end self-supervised Learned Primal-Dual reconstruction method that trains on independently split noisy sinograms without ground-truth images.
• N2I-LPD max achieved 29.62 dB PSNR on simulated walnut CT data with 5% Gaussian noise, outperforming N2I U-Net models, FBP, and oracle-tuned TV reconstruction in PSNR.
• The method preserved finer structural detail than TV regularization and generalized qualitatively to cone-beam data, although its performance on real clinical noise and optimal split strategies remain open questions.

The reconstruction of X-ray computed tomography (CT) images from sparse or low-dose measurements is an ill-posed inverse problem, and modern data-driven methods such as the Learned Primal-Dual (LPD) algorithm achieve strong reconstruction quality when trained with paired ground-truth data. In clinical practice, however, such ground-truth images are rarely available, since obtaining them would require additional high-dose scans. The paper under review addresses this limitation by combining the self-supervised Noise2Inverse (N2I) framework with the LPD architecture, yielding a learned, unrolled reconstruction operator that can be trained directly from noisy measurements without reference images (2606.26991).

Background and problem setting

The measurement model is the standard linear inverse problem $g = Af + \varepsilon$, where $A$ is the discretized X-ray transform in a fan-beam geometry and $\varepsilon$ is assumed to be zero-mean Gaussian noise. The paper reviews the classical solution landscape: filtered backprojection (FBP), which is fast but amplifies noise and produces artifacts under sparse sampling, and variational regularization, specifically Total Variation (TV) minimization solved via the Primal-Dual Hybrid Gradient (PDHG) algorithm. Learned reconstruction operators are categorized into post-processing schemes of the form $\Lambda_\theta \circ A^\dagger$ and unrolled iterative schemes, with LPD as the representative example of the latter. LPD replaces the proximal operators of PDHG with learned dual updates $\Gamma_{\theta_i^d}$ and primal updates $\Lambda_{\theta_i^p}$ across $I$ unrolled iterations, implemented here as residual networks (2606.26991).

The Noise2Inverse framework and its extension

Noise2Inverse exploits the fact that projection noise at distinct angles is statistically independent. The sinogram is partitioned into $K$ sub-sinograms by angular splitting, each is reconstructed independently with FBP, and input–target pairs are formed from complementary subsets of sub-reconstructions. Because input and target contain independent noise realizations, the trained network converges to the regression onto the noise-free image, up to an irreducible noise-variance term. The paper restates this as a proposition: under mean-zero, element-wise independent noise and a linear reconstruction operator, the expected prediction error decomposes into a supervised error term plus a noise-variance term, generalizing the original N2I proof to learned reconstruction operators $\mathcal{R}_\theta$.

The central methodological contribution is the adaptation of LPD to split data. Three modifications are required: the forward and adjoint operators must be replaced by split operators $\mathcal{A}_k$ and $A$0; the split sub-sinograms must be recombined into a single sinogram for the dual network via an explicit combining function that restores angular ordering; and the backprojected dual iterate is split, reconstructed per subset with FBP, and averaged before the primal update. The resulting algorithm, N2I-LPD, defines a learned reconstruction operator trained end-to-end with the $A$1 loss on N2I-style pairs.

Effect of ReLU activation

The paper provides a short derivation concerning the ReLU output activation used to enforce non-negativity. Using the bound $A$2, the authors show that the training loss with a ReLU output upper-bounds the unrectified loss plus an additional term involving $A$3, while the effective noise in the comparison against the target is reduced by a factor of one half. This is consistent with their empirical observation that training loss is higher when ReLU is present. The analysis applies to the evaluation setting; the training-time behavior under ReLU is not covered by the stated proposition, and the derivation relies on the assumption that the reconstruction is already non-negative.

Experimental setup

Experiments use the walnut cone-beam CT dataset of 42 objects with effectively noise-free $A$4 reconstructions as ground truth [der2019cone]. Sinograms are simulated in ODL with 512 angles over $A$5, 496 detector elements, and 5% additive zero-mean Gaussian noise. Four network configurations are compared: U-Net min (25,522 parameters) and max (7,696,706 parameters), and LPD min (28,730 parameters, 5 unrolled iterations) and max (578,260 parameters, 10 iterations). All models are trained with ADAM, cosine annealing, gradient clipping, and MSE loss for 50,000 iterations, using the standard 3:1 N2I split. Notably, the authors report that a 1:1 split was unsuitable for N2I-LPD, though the reason remains unexplained. Baselines comprise FBP, and classical primal-dual TV reconstruction under both per-image optimal regularization (PD TV-I) and a uniform regularization parameter (PD TV-U).

Results

On 50 held-out test slices, N2I-LPD max achieves the best PSNR at 29.62 dB, with N2I-LPD min at 29.39 dB, compared to 29.00 dB for the per-image-tuned classical PD TV baseline and 28.67 dB for both N2I U-Net configurations. FBP performs far worse at 19.01 dB PSNR and 0.20 SSIM. The SSIM ordering is reversed: the TV-regularized methods attain the highest values (0.88–0.89), while the learned methods range from 0.68 to 0.76.

Method	PSNR (dB)	SSIM
FBP	19.01	0.20
PD TV-I	29.00	0.88
PD TV-U	28.31	0.89
N2I U-Net min	28.67	0.68
N2I U-Net max	28.67	0.70
N2I-LPD min	29.39	0.75
N2I-LPD max	29.62	0.76

The SSIM advantage of the TV baselines reflects their stronger smoothing; qualitatively, the N2I-LPD reconstructions preserve finer structural detail and avoid the piecewise-constant appearance characteristic of TV regularization. The implication is that a self-supervised unrolled operator can outperform a classical variational method that is, in the PD TV-I configuration, tuned with oracle knowledge of the ground truth — a favorable comparison for the proposed method, since no such tuning information is available in practice.

For generalizability, networks trained on simulated fan-beam data are evaluated on 2D sinograms constructed from the central-slice cone-beam projections of the walnut dataset. All networks produce quantitatively similar outputs, with pairwise PSNRs between network reconstructions of 36.73–41.54 dB, and N2I-LPD max gives the visually most accurate result in the highlighted discrepancy region. Difference maps between the N2I-LPD max reconstruction and FBP show no loss of significant structural detail, supporting the claim that the networks generalize without hallucinating structures. The authors concede that a perfectly matched ground truth is unavailable in this setting, since the dataset reference reconstructions use FDK rather than the FBP-based setup, so the generalizability evaluation is necessarily qualitative.

Limitations and open questions

Several limitations are acknowledged or evident. The evaluation is confined to simulated fan-beam data with Gaussian noise; the generalizability experiment, while encouraging, lacks a rigorous quantitative reference. The failure of the 1:1 split for N2I-LPD is reported but not explained, and the split-ratio question is left open — the authors note that more general $A$6 splitting strategies may affect both stability and quality. Architecturally, the dual network currently operates on the recombined full sinogram; assigning separate dual networks per split, or separate primal networks per sub-reconstruction instead of pre-averaging, could better exploit the independence assumptions and is left unexplored. Finally, the maximum feasible angular sparsification during training — where more splits yield more ill-posed sub-problems — remains an open question.

Conclusion

The paper demonstrates that the Noise2Inverse framework can be extended to the Learned Primal-Dual algorithm, producing a self-supervised, unrolled learned reconstruction operator requiring no ground-truth images. On simulated walnut data with 5% Gaussian noise, N2I-LPD outperforms both N2I U-Net baselines and oracle-tuned TV-regularized primal-dual reconstruction in PSNR while remaining competitive in SSIM, and the trained networks generalize qualitatively to previously unseen cone-beam data without introducing hallucinated structure. The work establishes a practical training recipe for learned iterative CT reconstruction in settings where paired ground truth is unobtainable.