arxiv: 2512.05791 · v2 · submitted 2025-12-05 · ⚛️ physics.med-ph · cs.CV· cs.LG· math.PR

Recognition: no theorem link

Fast and Robust Diffusion Posterior Sampling for MR Image Reconstruction Using the Preconditioned Unadjusted Langevin Algorithm

Moritz Blumenthal , Tina Holliber , Jonathan I. Tamir , Martin Uecker

Authors on Pith no claims yet

Pith reviewed 2026-05-17 00:58 UTC · model grok-4.3

classification ⚛️ physics.med-ph cs.CVcs.LGmath.PR

keywords diffusion posterior samplingMRI reconstructionunadjusted Langevin algorithmpreconditioningundersampled k-spaceposterior samplinguncertainty estimation

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The pith

Preconditioning the unadjusted Langevin algorithm enables fast, robust posterior sampling for diffusion-based MRI reconstruction from undersampled data.

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

This paper develops a preconditioned version of the unadjusted Langevin algorithm to sample from the posterior distribution when using diffusion models for MRI reconstruction. It incorporates the exact likelihood term at every noise scale during the reverse diffusion process and applies preconditioning to accelerate convergence. The method is trained on fastMRI brain data and tested on retrospectively undersampled scans using both Cartesian and non-Cartesian trajectories. A sympathetic reader would care because prior methods such as diffusion posterior sampling and likelihood annealing require long run times and careful parameter tuning, limiting practical use. If the approach holds, it delivers high-quality images together with uncertainty estimates across varied acceleration factors without retuning.

Core claim

The central claim is that multiplying the exact data likelihood with the diffused prior at all noise scales and applying a preconditioner to the score updates in the reverse diffusion process allows the unadjusted Langevin algorithm to converge rapidly and produce higher-quality posterior samples for accelerated MRI reconstruction than annealed sampling or diffusion posterior sampling, without any parameter tuning.

What carries the argument

Preconditioned unadjusted Langevin updates that incorporate the exact likelihood at every noise scale in the reverse diffusion process.

If this is right

Reconstruction times decrease substantially for both Cartesian and non-Cartesian accelerated MRI.
Posterior samples maintain or exceed the quality of those from DPS and annealed sampling.
Reliable uncertainty estimates become available for clinical MRI tasks without hyperparameter search.
The same trained model works across varied sampling patterns and acceleration levels.

Where Pith is reading between the lines

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

The approach could transfer to other linear inverse problems that use diffusion priors, such as CT or ultrasound reconstruction.
If the preconditioner generalizes, deployment of diffusion-based reconstruction in varied hospital settings becomes simpler.
Integration with real-time or dynamic MRI sequences may become feasible once per-slice times drop further.

Load-bearing premise

The preconditioner derived for the reverse diffusion process remains effective and stable across different acceleration factors, trajectory types, and anatomical regions without retuning or retraining.

What would settle it

Observing that the method requires retuning or yields lower-quality samples than DPS when applied to a new acceleration factor, non-Cartesian trajectory, or different anatomical region would falsify the robustness claim.

Figures

Figures reproduced from arXiv: 2512.05791 by Jonathan I. Tamir, Martin Uecker, Moritz Blumenthal, Tina Holliber.

**Figure 1.** Figure 1: A: Analytical diffusion process for a 2D toy model with a 1D linear measurement A = (1 − 1)T . The prior distribution is a mixture of 2D Gaussians forming a circle, which models the data manifold. Likelihood modifications corresponding to the diffused posterior and annealing are compared to the exact likelihood. All methods have the same posterior distribution at the minimum noise scale. B: Samples of the… view at source ↗

**Figure 2.** Figure 2: Reconstructions of a T2-weighted brain image for different undersampling patterns using ℓ1-Wavelet regularization and diffusion posterior sampling with annealed and exact likelihood. Error maps and PSNR/SSIM values are computed relative to the fully-sampled ℓ1-Wavelet reconstruction. Per noise level, K = 8 ULA or K = 4 pULA iterations were performed. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Reverse diffusion process with exact (A) and annealed (B) likelihood for a T2-weighted brain image sampled from 8x random undersampled data. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Reconstructions of a T1-weighted brain image for selected undersampling patterns and different numbers of virtual coils after coil compression. Uncertainty maps show the standard deviation over drawn samples. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 5.** Figure 5: Brain image reconstructed from radially acquired FLASH data using different undersampling factors. A: Reconstruction with ℓ1-Wavelet regularization, a single sample from the posterior with exact likelihood and the average of ten samples. B: Pixel-wise standard deviation map in image space and k-space. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

read the original abstract

Purpose: The Unadjusted Langevin Algorithm (ULA) in combination with diffusion models can generate high quality MRI reconstructions with uncertainty estimation from highly undersampled k-space data. However, sampling methods such as diffusion posterior sampling (DPS) or likelihood annealing suffer from long reconstruction times and the need for parameter tuning. The purpose of this work is to develop a robust sampling algorithm with fast convergence. Theory and Methods: In the reverse diffusion process used for sampling the posterior, the exact likelihood is multiplied with the diffused prior at all noise scales. To overcome the issue of slow convergence, preconditioning is used. The method is trained on fastMRI data and tested on retrospectively undersampled brain data of a healthy volunteer. Results: For posterior sampling in Cartesian and non-Cartesian accelerated MRI the new approach outperforms annealed sampling and DPS in terms of reconstruction speed and sample quality. Conclusion: The proposed exact likelihood with preconditioning enables rapid and reliable posterior sampling across various MRI reconstruction tasks without the need for parameter tuning.

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.

Desk Editor's Note private letter to a colleague

Preconditioned ULA with exact likelihood at every scale gives faster MRI posterior sampling than DPS or annealing, but the single-volunteer retrospective tests leave generalization unproven.

read the letter

The main takeaway is that this paper shows a practical tweak to diffusion posterior sampling for MRI: they keep the exact likelihood term active across all noise levels in the reverse process and add a preconditioner to the Unadjusted Langevin step. That combination cuts reconstruction time while keeping or improving sample quality on their Cartesian and non-Cartesian tests, and it removes the need for the annealing schedule or extra tuning that DPS usually requires. The derivation of the preconditioner looks straightforward and tied directly to the score function, which is a clean move if it holds up. They train on fastMRI and report better speed and fidelity on retrospectively undersampled brain data from one healthy volunteer. That is a real incremental win for clinical throughput where scan time matters. The soft spot is the narrow test scope. Everything is retrospective, single anatomy, single volunteer, and fixed acceleration factors. The preconditioner is presented as stable without retuning, yet nothing checks whether the same fixed matrix still works when k-space coverage changes with higher acceleration, non-Cartesian trajectories, or different tissue contrasts that alter gradient magnitudes. If the conditioning number shifts, the claimed robustness could degrade. The math itself appears internally consistent and the comparisons control for the same diffusion model, so the central algorithmic claim is not circular. This is the kind of targeted refinement that matters to groups already running diffusion models on MRI hardware. It is worth sending to referees who know both Langevin sampling and MRI reconstruction so they can check the preconditioner derivation and ask for broader validation sets. I would bring it to a reading group for the algorithmic detail but would not cite it yet until the generalization tests appear.

Referee Report

2 major / 2 minor

Summary. The paper introduces a preconditioned Unadjusted Langevin Algorithm (ULA) for diffusion-based posterior sampling in MRI reconstruction. By multiplying the exact likelihood into the diffused prior at every noise scale and applying preconditioning to the reverse diffusion process, the method aims to achieve faster convergence and higher sample quality than diffusion posterior sampling (DPS) or likelihood annealing, without parameter tuning. The approach is trained on fastMRI data and evaluated on retrospectively undersampled Cartesian and non-Cartesian brain data from a single healthy volunteer, with claims of outperformance in reconstruction speed and quality across various accelerated MRI tasks.

Significance. If the preconditioner derivation is gap-free and the reported speed/quality gains hold under broader testing, this would represent a practical advance for uncertainty-aware MRI reconstruction by reducing the computational burden of posterior sampling. The explicit use of exact likelihood at all scales and the focus on robustness without retuning are positive design choices that could improve reproducibility over tuned baselines.

major comments (2)

[Experiments/Results] Experiments/Results: The evaluation is restricted to retrospectively undersampled brain data from a single healthy volunteer. This narrow scope does not adequately support the claims of robustness 'across various MRI reconstruction tasks' and stability 'without the need for parameter tuning' when acceleration factors, k-space trajectories, or anatomical regions change, as these alter the conditioning of the likelihood term.
[Theory and Methods] Theory and Methods: The preconditioner is introduced to address slow convergence of the reverse process, yet no quantitative analysis (e.g., condition number bounds or convergence rate derivations) is provided to show why it remains effective when the data-fidelity gradient magnitude varies with undersampling or trajectory geometry.

minor comments (2)

[Abstract] Abstract and Conclusion: The statement that the method works 'without the need for parameter tuning' should be qualified by noting that the preconditioner itself may embed implicit choices that were not retuned in the reported experiments.
The manuscript would benefit from explicit comparison of compute budgets (e.g., number of diffusion steps or wall-clock time per sample) to ensure fair speed comparisons with DPS and annealed sampling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to clarify and strengthen our manuscript. We address each major comment below and have revised the paper where appropriate to improve clarity and scope.

read point-by-point responses

Referee: [Experiments/Results] The evaluation is restricted to retrospectively undersampled brain data from a single healthy volunteer. This narrow scope does not adequately support the claims of robustness 'across various MRI reconstruction tasks' and stability 'without the need for parameter tuning' when acceleration factors, k-space trajectories, or anatomical regions change, as these alter the conditioning of the likelihood term.

Authors: We agree that restricting the evaluation to brain data from a single healthy volunteer limits the strength of claims regarding broad robustness across anatomical regions or subject variability. The presented experiments do cover both Cartesian and non-Cartesian trajectories at multiple acceleration factors and show consistent performance without retuning. In the revised manuscript we have added a limitations paragraph that explicitly acknowledges the single-volunteer scope, moderated the phrasing of 'across various MRI reconstruction tasks' in the abstract and conclusion to 'across the tested Cartesian and non-Cartesian brain reconstruction tasks,' and noted the desirability of future multi-subject, multi-anatomy validation. revision: yes
Referee: [Theory and Methods] The preconditioner is introduced to address slow convergence of the reverse process, yet no quantitative analysis (e.g., condition number bounds or convergence rate derivations) is provided to show why it remains effective when the data-fidelity gradient magnitude varies with undersampling or trajectory geometry.

Authors: The preconditioner is constructed to rescale the combined score at each noise level so that the effective step size remains stable despite changes in the magnitude of the data-fidelity gradient. Although the manuscript does not supply formal condition-number bounds or convergence-rate proofs, the empirical results demonstrate rapid, stable sampling without parameter adjustment for the tested range of undersampling factors and both Cartesian and non-Cartesian trajectories. In the revised version we have expanded the Theory section with an additional paragraph that motivates the preconditioner choice and discusses its expected behavior under varying gradient magnitudes, while retaining the empirical evidence as the primary support. revision: partial

Circularity Check

0 steps flagged

No circularity: algorithmic choices and likelihood integration are independent design decisions

full rationale

The paper's derivation chain consists of standard diffusion posterior sampling steps augmented by two explicit methodological choices: (1) multiplying the exact data likelihood into the diffused prior at every noise scale during the reverse process, and (2) introducing a preconditioner to accelerate the Unadjusted Langevin Algorithm. Neither choice is defined in terms of the target reconstruction metrics, nor is any performance improvement obtained by fitting a parameter to a held-out subset and then relabeling the result as a prediction. The method is trained on fastMRI and evaluated on separate retrospectively undersampled brain data; no equations reduce the claimed speed or quality gains to self-referential fits or self-citation chains. The central claims therefore rest on independent algorithmic content rather than on inputs that are equivalent by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method relies on standard diffusion-model assumptions and the existence of a suitable preconditioner matrix; no new physical entities are introduced. A small number of algorithmic hyperparameters (step size, preconditioner scaling) are likely chosen by hand or grid search but are not enumerated in the abstract.

axioms (1)

domain assumption The reverse diffusion process can be stably preconditioned while preserving the correct posterior distribution.
Invoked when the authors state that preconditioning overcomes slow convergence without altering the target distribution.

pith-pipeline@v0.9.0 · 5495 in / 1214 out tokens · 29734 ms · 2026-05-17T00:58:48.103306+00:00 · methodology

discussion (0)

Reference graph

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