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arxiv: 2304.11200 · v3 · submitted 2023-04-21 · 📊 stat.ME

A Plug-and-Play Method with Inpainting Network for Bayesian Uncertainty Quantification in Imaging

Xiaoyu Wang , Michael Tang , Audrey Repetti This is my paper

Pith reviewed 2026-05-24 09:26 UTC · model grok-4.3

classification 📊 stat.ME
keywords Bayesian uncertainty quantificationinpainting networkplug-and-play algorithmhypothesis testingMAP estimateMRI reconstructionCT reconstructionimage artefacts
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The pith

A convolutional neural network can replace hand-crafted inpainting in Bayesian uncertainty quantification for imaging

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

This paper develops a data-driven version of Bayesian uncertainty quantification by optimization for imaging problems. Instead of using hand-crafted mathematical definitions to inpaint potential local artefacts in the maximum a posteriori estimate, it employs a convolutional neural network for that step. The result is a plug-and-play algorithm built on primal-dual Condat-Vu iterations that tests whether those artefacts are present. Readers interested in scalable uncertainty assessment for medical image reconstruction from undersampled data would find this relevant because it avoids the need to design custom inpainting rules for each application.

Core claim

The paper establishes that the inpainting procedure in the BUQO algorithm can be performed by a convolutional inpainting neural network, yielding a plug-and-play method based on the primal-dual Condat-Vu iterations for solving the hypothesis test on the existence of local artefacts in the MAP estimate, as demonstrated on Fourier undersampling and Radon transform problems from magnetic resonance and computed tomography imaging.

What carries the argument

The convolutional inpainting neural network serving as the data-driven replacement for the mathematical inpainting operator in the minimization problem that formulates the hypothesis test.

If this is right

  • The method provides an efficient way to quantify uncertainty without sampling techniques.
  • It can be applied to different Fourier undersampling scenarios in MRI and to CT with the Radon operator.
  • The plug-and-play nature allows the inpainting network to be swapped or updated independently.
  • Simulations show its use in probing local structures in reconstructed medical images.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the network is trained on diverse enough data, the approach could extend to other inverse problems in imaging beyond the tested cases.
  • Discrepancies between the network output and traditional inpainting might affect the reliability of the uncertainty quantification in edge cases.
  • Future work could explore training the network jointly with the reconstruction process for better consistency.

Load-bearing premise

The trained inpainting network acts as a mathematically equivalent inpainting operator to the hand-crafted versions for the purpose of the hypothesis test.

What would settle it

A direct comparison where the new method and the original BUQO disagree on the presence of an artefact in a controlled simulation with known ground truth would indicate that the neural network does not preserve the required equivalence.

Figures

Figures reproduced from arXiv: 2304.11200 by Audrey Repetti, Michael Tang, Xiaoyu Wang.

Figure 1
Figure 1. Figure 1: Modified DnCNN architecture, where Conv is a [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (top) Examples of masks M with Ma = 150 and Ma = 250 angles (i.e., M/N = 0.52 and M/N = 0.76), respectively. (bottom) Ground truth (N = 128 × 128), and MAP estimates for iSNR = 30 dB. contrast to common PnP methods, which use neural networks as a regularizer, we propose to use a neural network to define the objective of the minimization problem. While both PnP algorithms and the use of NNs for image inpain… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of BUQO with PnP-BUQO. (left) MAP [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (left) MAP (for Ma = 150, iSNR=25 dB) with a checkerboard structure of interest. (center) PnP-BUQO output x ‡ and (right) inpainted output, with ρα = 0.018. inconclusive to reject the hypothesis, corresponding to Ma = 150 and iSNR=30 dB [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Structure confidence ρα (for a particular structure) with respect to the number of measurement angles, Ma (left) and the iSNR values (right).    [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Supporting data for experiments summarized in Figure 6 with [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Supporting data for experiments summarized in Figure 6 with [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

We contribute to an uncertainty quantification problem in imaging that evaluates a hypothesis test questioning the existence of local "artefacts" appearing in the maximum a posteriori (MAP) estimate (obtained from standard numerical tools). Such a method, called Bayesian uncertainty quantification by optimization (BUQO), was introduced a few years ago as an efficient and scalable alternative to sampling methods when per-pixel error-bars are not needed. BUQO formulates a hypothesis test for probing the existence of local structures in the MAP estimate as a minimization problem, that can be solved efficiently with standard optimization algorithms. In this context, BUQO requires a "mathematical" definition of the "local artefact". This definition can be interpreted as an inpainting of the structure. However, only simple hand-crafted techniques have been proposed so far due to the complexity of the problem. In this work, we propose a data-driven alternative to BUQO where the inpainting procedure in the algorithm is performed using a convolutional inpainting neural network (NN). This results in a plug-and-play algorithm, based on the primal-dual Condat-Vu iterations,where the inpainting procedure is performed with a NN. The proposed approach is assessed on two image reconstruction problems inspired by medicine. We specifically perform simulations on two Fourier undersampling problems (discrete and non-uniform) encountered in magnetic resonance imaging, as well as a computed tomography problem using the Radon measurement operator.

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 paper extends the BUQO framework for hypothesis testing of local artefacts in MAP estimates by replacing the hand-crafted inpainting operator with a convolutional neural network inpainter. The resulting plug-and-play algorithm embeds the NN inside primal-dual Condat-Vu iterations and is assessed on discrete and non-uniform Fourier undersampling (MRI) and Radon-transform (CT) reconstruction problems.

Significance. If the NN inpainting operator preserves the variational and monotonicity properties required by the original BUQO derivation, the method would allow data-driven handling of complex artefacts without manual design. The plug-and-play structure and medical-imaging simulations are practical strengths, but the significance hinges on whether the test statistic retains its interpretation.

major comments (2)
  1. [Method / algorithm description] The central substitution of the NN for the mathematical inpainting operator (described in the method section following the BUQO recall) is presented without any derivation or stated conditions showing that the trained network satisfies firm nonexpansiveness, monotonicity, or the same fixed-point characterization as the original hand-crafted operator. This equivalence is load-bearing for both convergence of the Condat-Vu scheme and validity of the hypothesis test.
  2. [Numerical experiments] The experimental assessment on the two MRI and one CT problems supplies no quantitative comparison of test decisions or convergence behavior against the original BUQO formulation, nor any diagnostic checking that the NN preserves the null-distribution properties of the test statistic. Without such evidence the claim that the procedure remains a valid uncertainty-quantification tool is unsupported.
minor comments (2)
  1. Notation distinguishing the NN-based inpainting operator from the original mathematical operator should be introduced explicitly to avoid ambiguity in the algorithm statement.
  2. The abstract states that the approach is assessed on the cited problems but provides no numerical results, error metrics, or figures; the main text should ensure all quantitative claims are accompanied by the corresponding tables or plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable comments on our manuscript. We address each major comment below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [Method / algorithm description] The central substitution of the NN for the mathematical inpainting operator (described in the method section following the BUQO recall) is presented without any derivation or stated conditions showing that the trained network satisfies firm nonexpansiveness, monotonicity, or the same fixed-point characterization as the original hand-crafted operator. This equivalence is load-bearing for both convergence of the Condat-Vu scheme and validity of the hypothesis test.

    Authors: We acknowledge that the manuscript provides no derivation establishing that the trained inpainting network satisfies firm nonexpansiveness, monotonicity, or the fixed-point properties of the original hand-crafted operator. The presentation treats the network as a plug-and-play module whose output is inserted into the Condat-Vu scheme. In the revised manuscript we will add an explicit discussion paragraph after the algorithm description that states the assumptions inherited from BUQO, notes that these properties are not verified for the trained network, and clarifies that convergence and test validity are supported only by the observed numerical behavior rather than by the original theoretical guarantees. revision: partial

  2. Referee: [Numerical experiments] The experimental assessment on the two MRI and one CT problems supplies no quantitative comparison of test decisions or convergence behavior against the original BUQO formulation, nor any diagnostic checking that the NN preserves the null-distribution properties of the test statistic. Without such evidence the claim that the procedure remains a valid uncertainty-quantification tool is unsupported.

    Authors: The reported experiments demonstrate feasibility on discrete and non-uniform Fourier undersampling (MRI) and Radon (CT) problems but contain no side-by-side quantitative comparison of test decisions or iteration counts with the original hand-crafted BUQO, nor any Monte-Carlo verification that the test statistic retains its null distribution under the NN inpainter. In the revision we will add (i) a table comparing test outcomes and convergence metrics on identical instances and (ii) empirical null-distribution diagnostics obtained by simulating under the null hypothesis, thereby providing direct evidence for the retained validity of the uncertainty-quantification procedure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension of prior BUQO framework with external NN component

full rationale

The paper extends the existing BUQO hypothesis-testing minimization (solved via Condat-Vu) by substituting its hand-crafted inpainting operator with a trained convolutional NN. No equations, derivations, or self-citations are shown that reduce the test statistic, convergence properties, or validity claim to a fitted quantity defined by the same data, a self-referential definition, or a load-bearing self-citation chain. The central premise remains an independent modeling choice whose correctness rests on the (unproven here) equivalence assumption rather than on any circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that a learned inpainting network can stand in for the mathematical inpainting operator without altering the statistical validity of the hypothesis test; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption A convolutional inpainting network can be trained to approximate the mathematical definition of local artefact inpainting used in BUQO
    Invoked when the paper states that the NN performs the inpainting procedure previously done by hand-crafted techniques.

pith-pipeline@v0.9.0 · 5787 in / 1158 out tokens · 19449 ms · 2026-05-24T09:26:21.121953+00:00 · methodology

discussion (0)

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