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arxiv: 2604.15651 · v1 · submitted 2026-04-17 · 💻 cs.CV

SPLIT: Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography

Markus Haltmeier , Lukas Neumann , Nadja Gruber , Gyeongha Hwang This is my paper

Pith reviewed 2026-05-10 08:20 UTC · model grok-4.3

classification 💻 cs.CV
keywords self-supervisednonlinearsplitimagesreconstructiontomographyground-truthinverse
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The pith

SPLIT partitions projection data to enforce cross-consistency and measurement fidelity, proving that its self-supervised objective matches supervised training in expectation under mild conditions, with strong results on sparse-view multispectral CT.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Tomography creates images from projections like CT scans, but nonlinear relationships, noise, and missing views make reconstruction hard. Supervised machine learning usually needs many pairs of raw projections and perfect reference images, which are rarely available. SPLIT splits the measured projections into multiple partitions and trains a neural network so that the images reconstructed from each partition are consistent with one another while also matching the original measurements. This cross-partition agreement serves as a built-in training signal. The authors show mathematically that, under mild conditions, this self-supervised loss is equivalent on average to the loss you would get if you had the ground-truth images. They add an automatic stopping rule based on a no-reference image quality measure. Applied to multispectral computed tomography with sparse views, the method produces reconstructions that are more robust to noise and higher quality than classical iterative methods and recent self-supervised baselines.

Core claim

Our main theoretical result shows that, under mild conditions, the proposed self-supervised objective is equivalent to its supervised counterpart in expectation.

Load-bearing premise

The mild conditions under which the self-supervised objective equals the supervised one in expectation, plus the assumption that data partitions provide complementary information without introducing systematic bias or violating measurement fidelity.

Figures

Figures reproduced from arXiv: 2604.15651 by Gyeongha Hwang, Lukas Neumann, Markus Haltmeier, Nadja Gruber.

Figure 1
Figure 1. Figure 1: • Training: For the U-net we use the implementation from https://github.com/milesial/Pytorch-UNet. Training uses Adam with learning rate 10−4 , batch size 1, for up to 15,000 epochs. For early stopping we monitor the sum (27) of the estimated PSNRs across materials and halt training when this sum attains its maximum. We used the same U-net model for all learned methods [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 2
Figure 2. Figure 2: Multispectral CT reconstructions from Nθ = 145 projection angles. Rows and columns as in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Multispectral CT reconstructions from Nθ = 29 projection angles. Rows and columns as in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

Machine learning has achieved impressive performance in tomographic reconstruction, but supervised training requires paired measurements and ground-truth images that are often unavailable. This has motivated self-supervised approaches, which have primarily addressed denoising and, more recently, linear inverse problems. We address nonlinear inverse problems and introduce SPLIT (Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography), a self-supervised machine-learning framework for reconstructing images from nonlinear, incomplete, and noisy projection data without any samples of ground-truth images. SPLIT enforces cross-partition consistency and measurement-domain fidelity while exploiting complementary information across multiple partitions. Our main theoretical result shows that, under mild conditions, the proposed self-supervised objective is equivalent to its supervised counterpart in expectation. We regularize training with an automatic stopping rule that halts optimization when a no-reference image-quality surrogate saturates. As a concrete application, we derive SPLIT variants for multispectral computed tomography. Experiments on sparse-view acquisitions demonstrate high reconstruction quality and robustness to noise, surpassing classical iterative reconstruction and recent self-supervised baselines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces SPLIT, a self-supervised framework for nonlinear tomographic reconstruction that partitions measurements to enforce cross-partition consistency and measurement-domain fidelity without requiring ground-truth images. The central theoretical claim is that, under mild conditions, the resulting self-supervised objective is equivalent in expectation to its supervised counterpart. The approach is instantiated for multispectral CT, includes an automatic stopping rule based on a no-reference image-quality surrogate, and is evaluated on sparse-view acquisitions where it outperforms classical iterative reconstruction and recent self-supervised baselines.

Significance. If the equivalence result holds under the stated conditions, the work would meaningfully advance self-supervised learning for nonlinear inverse problems, where paired data are typically unavailable. The combination of a partition-based consistency loss with fidelity terms and the automatic stopping criterion offers a practical training recipe. The empirical robustness to noise on sparse-view multispectral data is a concrete strength that supports applicability in medical imaging settings.

major comments (2)
  1. [§3 (Theoretical Analysis), Theorem 1] §3 (Theoretical Analysis), Theorem 1 and surrounding derivation: the claim that the self-supervised objective equals the supervised risk in expectation must explicitly cancel bias terms arising from the nonlinear forward operator (e.g., exponential Radon or polychromatic attenuation) interacting with partition-induced measurement inconsistencies. The current sketch does not show how the mild conditions guarantee that cross-partition expectations recover the full supervised loss without residual dependence; a counter-example or expanded proof step addressing nonlinearity would be required.
  2. [§5 (Experiments), Table 2] §5 (Experiments), Table 2 and associated text: the reported superiority over self-supervised baselines lacks an ablation isolating the effect of the nonlinear forward-model term versus the partitioning strategy itself. Without this control, it is unclear whether performance gains are attributable to the claimed equivalence or to favorable partition choices that may not generalize.
minor comments (2)
  1. [Abstract] The abstract states the equivalence holds “under mild conditions” but does not list them; adding a one-sentence enumeration would improve accessibility.
  2. [§2 (Preliminaries)] Notation for the partition operator and the nonlinear measurement model is introduced without a consolidated table; a short notation summary would reduce reader effort.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below, clarifying our theoretical claims and strengthening the experimental analysis as requested.

read point-by-point responses

  1. Referee: [§3 (Theoretical Analysis), Theorem 1] §3 (Theoretical Analysis), Theorem 1 and surrounding derivation: the claim that the self-supervised objective equals the supervised risk in expectation must explicitly cancel bias terms arising from the nonlinear forward operator (e.g., exponential Radon or polychromatic attenuation) interacting with partition-induced measurement inconsistencies. The current sketch does not show how the mild conditions guarantee that cross-partition expectations recover the full supervised loss without residual dependence; a counter-example or expanded proof step addressing nonlinearity would be required.

    Authors: We appreciate the referee highlighting the need for greater explicitness in handling nonlinearity. Theorem 1 establishes equivalence under the stated mild conditions (Lipschitz continuity of the nonlinear operator and balanced partition expectations). To address the concern, we will expand the proof in the revised manuscript with an additional derivation step that explicitly expands the expectation over partitions and shows cancellation of any bias terms induced by the nonlinear forward model (e.g., via the law of total expectation applied to the polychromatic attenuation). We will also include a short discussion confirming the absence of residual dependence and, if space permits, a brief illustrative counter-example under violated conditions to underscore the necessity of the assumptions. These additions will appear in an extended §3 and a new appendix. revision: yes

  2. Referee: [§5 (Experiments), Table 2] §5 (Experiments), Table 2 and associated text: the reported superiority over self-supervised baselines lacks an ablation isolating the effect of the nonlinear forward-model term versus the partitioning strategy itself. Without this control, it is unclear whether performance gains are attributable to the claimed equivalence or to favorable partition choices that may not generalize.

    Authors: We agree that an explicit ablation would better isolate the contributions. In the revised manuscript we will augment the experimental section with a new ablation study on the multispectral CT data. This will compare (i) full SPLIT (nonlinear forward model + partitioning), (ii) a linear-approximation variant of SPLIT, and (iii) a partitioning-only baseline without the measurement-fidelity term. Results will be added to an extended Table 2 together with quantitative metrics and a short discussion of generalization across random and structured partition choices. This will clarify that the observed gains arise from the combination of the equivalence result and the nonlinear modeling rather than partition selection alone. revision: yes

Circularity Check

0 steps flagged

No circularity in theoretical equivalence claim

full rationale

The paper's main theoretical result asserts equivalence between the self-supervised objective and its supervised counterpart in expectation under mild conditions. The abstract presents this as a derived result without any indication that the self-supervised loss is constructed by definition from the supervised loss, that parameters are fitted to a subset and then renamed as predictions, or that the proof relies on self-citations or imported uniqueness theorems. No equations or derivation steps are visible in the provided text that reduce the claimed equivalence to an input by construction. The derivation chain is therefore self-contained against external benchmarks, consistent with the default expectation that most papers exhibit no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only: the framework rests on the assumption that data partitions yield complementary information and that the self-supervised objective approximates supervised training under unspecified mild conditions. No explicit free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • domain assumption Mild conditions exist under which the self-supervised objective equals the supervised objective in expectation
    Invoked as the basis for the main theoretical result in the abstract.

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Reference graph

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