Regularized joint reconstruction and slab combination for accelerated three-dimensional multi-slab diffusion-weighted imaging using multi-scale energy models

Chu-Yu Lee; Jyothi Rikhab Chand; Mathews Jacob; Merry Mani; Reza Ghorbani

arxiv: 2606.01606 · v1 · pith:Q3FQOFX7new · submitted 2026-06-01 · 📡 eess.IV

Regularized joint reconstruction and slab combination for accelerated three-dimensional multi-slab diffusion-weighted imaging using multi-scale energy models

Reza Ghorbani , Jyothi Rikhab Chand , Chu-Yu Lee , Mathews Jacob , Merry Mani This is my paper

Pith reviewed 2026-06-28 12:42 UTC · model grok-4.3

classification 📡 eess.IV

keywords multi-slab diffusion MRIjoint reconstructionenergy-based priorslab-boundary artifactsCNN prioraccelerated imagingbilinear model3D diffusion-weighted imaging

0 comments

The pith

EPEN jointly reconstructs undersampled multi-slab diffusion MRI volumes and slab profiles via a CNN energy prior to suppress boundary artifacts.

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

The paper introduces EPEN, a joint reconstruction method for high-resolution diffusion-weighted images acquired with accelerated 3D multi-slab k-space sampling. It treats both the image volume and the slab excitation profiles as unknowns inside a bilinear forward model. The method solves a maximum a posteriori problem that combines k-space data fidelity, a CNN-derived deep energy prior on the image, and quadratic regularization on the profiles. Alternating minimization updates the image with majorize-minimize conjugate gradients and the profiles with regularized least squares. Experiments across acceleration factors and slab counts show fewer slab-boundary artifacts, better structural consistency, and retained diffusion contrast than standard correction techniques.

Core claim

EPEN formulates multi-slab acquisition as a bilinear model with unknown image volume and slab profiles, then solves the resulting MAP objective containing a Gaussian data term, a CNN-based deep energy prior on clean diffusion-weighted images, and quadratic profile regularization. Alternating minimization produces the joint solution, and the learned energy gradient steers the reconstruction toward an artifact-free image distribution.

What carries the argument

The CNN-based deep energy prior that supplies the negative log distribution of clean diffusion-weighted images inside the MAP objective of the bilinear forward model.

If this is right

Slab-boundary artifacts are reduced across multiple acceleration factors and slab configurations relative to conventional correction methods.
Structural consistency of the reconstructed volumes improves while diffusion-weighted contrast is preserved.
Reconstruction and slab-profile estimation occur inside one unified optimization rather than sequential post-processing steps.
The nonconvex objective remains tractable through alternating minimization with conjugate-gradient image updates and closed-form profile updates.

Where Pith is reading between the lines

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

If the energy prior generalizes across scanners and populations, the same framework could support even higher acceleration without separate calibration scans.
The bilinear modeling approach may transfer to other multi-slab or multi-shot sequences such as T2-weighted or perfusion imaging.
Training the prior on larger and more diverse diffusion data sets could further reduce residual artifacts at extreme undersampling rates.

Load-bearing premise

The CNN-based deep energy prior accurately represents the negative log distribution of clean diffusion-weighted images.

What would settle it

Reconstruction of new fully sampled multi-slab diffusion data sets in which EPEN still leaves visible slab-boundary discontinuities or altered diffusion contrast relative to the reference volumes.

Figures

Figures reproduced from arXiv: 2606.01606 by Chu-Yu Lee, Jyothi Rikhab Chand, Mathews Jacob, Merry Mani, Reza Ghorbani.

**Figure 1.** Figure 1: Schematic illustration of the modified phase-encoding. The example shown uses a prescribed slab thickness of 14mm and a desired slice thickness of 1mm along the z-direction. (a) In the classical setting, ∆FOVz is set equal to the slab thickness, and the number of phase encodes is determined by the desired slice thickness, leading to Nkz of 14 for this case. Since the non-ideal RF pulse (shown in gray) exte… view at source ↗

**Figure 2.** Figure 2: (a) Slab profiles from the phantom experiment used to generate the synthetic dataset. (b) The synthetic dataset used in this study and example denoising results from the EPEN model. The bottom row shows the learned score (denoising residual), which represents the gradient field the model uses to guide the reconstruction toward the learned data distribution. EPEN denoiser to enable quantitative error report… view at source ↗

**Figure 3.** Figure 3: Reconstruction results using the synthetic noisy DWI data for (a) EPEN, (b) NPEN, and (c) PEN methods. The RMSE is shown in the upper left-hand corner of the difference maps (amplified 10 times), computed with respect to the denoised DWI shown in Figure 2b. maintains comparatively low score responses, suggesting more effective constrained reconstruction to the clean-image manifold and a superior robustness… view at source ↗

**Figure 4.** Figure 4: (a) Experiments on in-vivo diffusion-weighted dataset 1. Reconstruction results from the proposed EPEN method are shown at different kz undersampling rates, in comparison to NPEN and PEN. (b) Score maps corresponding to the reconstructions in (a). Scores can be taken as a proxy for noise or the presence of artifacts. that the learned prior incorporated into the iterative reconstruction is more effective to… view at source ↗

**Figure 5.** Figure 5: (a): Post-denoising of NPEN reconstruction using the learned CNN-based denoiser. Although the images are denoised, the artifacts cannot be removed in this post denoising. (b): A selected region of the denoised NPEN reconstruction and its corresponding region from the EPEN reconstruction. combination artifact removal as expected, although at the expense of blurring in the resulting images from the high rate… view at source ↗

**Figure 6.** Figure 6: The parameter maps estimated from DTI model fitting using in-vivo dataset 1. The fractional anisotropy (FA) (a), mean diffusivity (MD) (b), and color-coded FA (c) maps are shown from the three methods. EPEN produces FA maps with reduced noise relative to the other reconstructions, while residual slab artifacts present in NPEN and PEN reconstructions propagate into the tensor estimates and affect the corres… view at source ↗

**Figure 7.** Figure 7: The reconstruction results for the in-vivo dataset 2 at the different kz sampling rates. EPEN reconstructions are shown on the top row, and NPEN reconstructions are shown on the bottom row [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: The reconstruction results for the in-vivo dataset 1 (a) and dataset 2 (b) for the case with 8 kz [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗

**Figure 9.** Figure 9: The reconstruction results for the in-vivo dataset 4 with 14 kz , and 10 kz . 12/24 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

read the original abstract

This work presents Energy-based Profile Encoding, EPEN, a joint reconstruction framework for high-resolution diffusion-weighted MRI from undersampled 3D multi-slab k-space acquisitions, designed to suppress slab-boundary artifacts while preserving fine anatomical detail. EPEN formulates the multi-slab acquisition process using a bilinear forward model in which both the diffusion-weighted image volume and slab excitation profiles are treated as unknown variables. Reconstruction is posed as a maximum a posteriori optimization problem with three components: a Gaussian data-fidelity term enforcing consistency with the acquired k-space measurements, a CNN-based deep energy prior that represents the negative log distribution of clean diffusion-weighted images, and a quadratic regularization term that constrains the estimated slab profiles toward an initial profile estimate. The gradient of the learned energy prior guides accelerated reconstruction toward an artifact-free image distribution. The resulting nonconvex objective is solved using alternating minimization, with image-volume updates performed through a majorize-minimize scheme using conjugate-gradient optimization and slab-profile updates estimated by regularized least squares. Across multiple acceleration factors and slab configurations, EPEN substantially reduced slab-boundary artifacts compared with conventional slab-boundary correction methods, while improving structural consistency and preserving diffusion-weighted contrast. These results demonstrate that EPEN enables robust joint 3D multi-slab diffusion MRI reconstruction and slab-profile correction within a unified optimization framework supported by deep energy-based image priors.

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

EPEN adds a bilinear joint model plus CNN energy prior to multi-slab DWI reconstruction, but the abstract gives no numbers or ablations so the performance claims stay uncheckable.

read the letter

The paper's main contribution is EPEN, a MAP framework that treats both the diffusion-weighted volume and the slab excitation profiles as unknowns in a bilinear forward model. It adds a CNN energy prior on the image, a quadratic term on the profiles, and solves the nonconvex problem by alternating between majorize-minimize plus conjugate gradients for the image and regularized least squares for the profiles.

The formulation is cleanly laid out and the alternating scheme makes sense for the joint estimation task. Using the gradient of the learned energy to steer the reconstruction toward cleaner images is a reasonable way to incorporate the prior.

The soft spot is the complete absence of quantitative results. The abstract claims substantial artifact reduction across accelerations and slab setups, yet supplies no metrics, no error bars, no dataset sizes, and no ablation that isolates the prior from the joint estimation itself. Without those, it is impossible to tell whether the CNN energy term is doing the claimed work or whether the bilinear model plus alternating minimization would have produced most of the improvement on its own. The training corpus and any held-out checks on the energy values are also missing.

This is a narrow technical paper aimed at researchers who already work on accelerated multi-slab diffusion MRI and slab-boundary correction. Someone building similar reconstruction pipelines might pick up the bilinear-plus-prior idea, but only if the full paper contains the missing experiments.

I would send it to peer review. The optimization setup is coherent and the problem is real, even though the current evidence level is low.

Referee Report

2 major / 0 minor

Summary. The manuscript presents Energy-based Profile Encoding (EPEN), a joint reconstruction framework for high-resolution diffusion-weighted MRI from undersampled 3D multi-slab k-space acquisitions. It models the acquisition with a bilinear forward model where both the diffusion-weighted image volume and slab excitation profiles are unknowns. The reconstruction is formulated as a MAP optimization with Gaussian data-fidelity, a CNN-based deep energy prior representing the negative log distribution of clean DWI images, and quadratic regularization on slab profiles. The nonconvex objective is solved via alternating minimization with majorize-minimize and conjugate-gradient for images, and regularized least squares for profiles. The paper claims that EPEN substantially reduces slab-boundary artifacts across acceleration factors and slab configurations compared to conventional methods while preserving contrast.

Significance. If the quantitative claims hold and the deep prior is shown to encode the required distribution, the work could advance accelerated multi-slab DWI by unifying reconstruction and slab-profile correction in a single optimization framework. The bilinear model combined with the energy prior offers a principled way to handle slab-boundary artifacts without separate correction steps.

major comments (2)

[Abstract] Abstract (paragraph describing the MAP objective): the central modeling assumption that the CNN-based deep energy prior accurately represents the negative log distribution of clean diffusion-weighted images is load-bearing for attributing artifact reduction to the prior gradient, yet the manuscript supplies no training corpus details, held-out likelihood or energy-gap metrics, or ablation removing the prior.
[Abstract] Abstract: the claim of substantial artifact reduction 'across multiple acceleration factors and slab configurations' is presented without any quantitative metrics, error bars, dataset sizes, or statistical comparisons, preventing verification that the improvement exceeds what could arise from the data-fidelity or quadratic slab-profile terms alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments below and will revise the abstract and manuscript to incorporate the requested details and quantitative support.

read point-by-point responses

Referee: [Abstract] Abstract (paragraph describing the MAP objective): the central modeling assumption that the CNN-based deep energy prior accurately represents the negative log distribution of clean diffusion-weighted images is load-bearing for attributing artifact reduction to the prior gradient, yet the manuscript supplies no training corpus details, held-out likelihood or energy-gap metrics, or ablation removing the prior.

Authors: We acknowledge that the abstract and current manuscript do not provide these supporting details for the deep energy prior. We will revise the manuscript to include training corpus details in the Methods, add held-out likelihood and energy-gap metrics, and include an ablation study removing the prior in the Results to better attribute the observed improvements. revision: yes
Referee: [Abstract] Abstract: the claim of substantial artifact reduction 'across multiple acceleration factors and slab configurations' is presented without any quantitative metrics, error bars, dataset sizes, or statistical comparisons, preventing verification that the improvement exceeds what could arise from the data-fidelity or quadratic slab-profile terms alone.

Authors: We agree that the abstract lacks the requested quantitative elements. We will revise the abstract to include specific metrics with error bars, dataset sizes, and statistical comparisons drawn from the experimental results to substantiate the claims and allow assessment of the prior's contribution beyond the other terms. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external learned prior and empirical validation

full rationale

The abstract and description formulate reconstruction as MAP optimization with data fidelity, a CNN energy prior (representing negative log of clean images), and slab-profile regularization, solved via alternating minimization. No equations, predictions, or claims reduce by construction to fitted inputs, self-definitions, or self-citation chains. The energy prior is an independent learned component whose accuracy is an external modeling assumption, not a definitional tautology. Results are framed as empirical comparisons, making the chain self-contained against benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Abstract-only review; free parameters and training details of the CNN energy prior are not specified.

free parameters (1)

quadratic regularization weight on slab profiles
The abstract states a quadratic regularization term constrains estimated slab profiles toward an initial estimate; its weight is a tunable parameter.

axioms (2)

domain assumption MRI acquisition follows the stated bilinear forward model
Abstract explicitly formulates the multi-slab acquisition process using this bilinear model.
domain assumption CNN energy function equals negative log probability of clean diffusion-weighted images
Abstract defines the deep energy prior as representing the negative log distribution of clean images.

pith-pipeline@v0.9.1-grok · 5792 in / 1366 out tokens · 28291 ms · 2026-06-28T12:42:34.571215+00:00 · methodology

discussion (0)

Reference graph

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[1] [1]

Diffusion-weighted 3d multislab echo planar imaging for high signal-to-noise ratio efficiency and isotropic image resolution.Magnetic Resonance in Medicine, 70:1507–1514, 12 2013

Mathias Engstr¨om and Stefan Skare. Diffusion-weighted 3d multislab echo planar imaging for high signal-to-noise ratio efficiency and isotropic image resolution.Magnetic Resonance in Medicine, 70:1507–1514, 12 2013

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Robert Frost, Karla L. Miller, Rob H N Tijssen, David A. Porter, and Peter Jezzard. 3d multi-slab diffusion-weighted readout-segmented epi with real-time cardiac-reordered k-space acquisition.Magnetic Resonance in Medicine, 72, 2014

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2024

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Miller, Jesper L.R

Ziyu Li, Karla L. Miller, Jesper L.R. Andersson, Jieying Zhang, Simin Liu, Hua Guo, and Wenchuan Wu. Sampling Strategies and Integrated Reconstruction for Reducing Distortion and Boundary Slice Aliasing in High-resolution 3D Diffusion MRI.Magnetic Resonance in Medicine, 90, 2023

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2026

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Slab profile encoding (pen) for minimizing slab boundary artifact in three-dimensional diffusion-weighted multislab acquisition

Anh T Van, Murat Aksoy, Samantha J Holdsworth, Daniel Kopeinigg, Sjoerd B V os, and Roland Bammer. Slab profile encoding (pen) for minimizing slab boundary artifact in three-dimensional diffusion-weighted multislab acquisition. Magnetic resonance in medicine, 73(2):605–613, 2015

2015

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Klaas P Pruessmann, Markus Weiger, Markus B Scheidegger, and Peter Boesiger. Sense: sensitivity encoding for fast mri.Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine, 42(5):952–962, 1999

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Wenchuan Wu, Peter J Koopmans, Robert Frost, and Karla L Miller. Reducing slab boundary artifacts in three- dimensional multislab diffusion mri using nonlinear inversion for slab profile encoding (npen).Magnetic resonance in medicine, 76(4):1183–1195, 2016

2016

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Slab boundary artifact correction in multislab imaging using convolutional-neural-network–enabled inversion for slab profile encoding

Jieying Zhang, Simin Liu, Erpeng Dai, Xinyu Ye, Diwei Shi, Yuhsuan Wu, Jie Lu, and Hua Guo. Slab boundary artifact correction in multislab imaging using convolutional-neural-network–enabled inversion for slab profile encoding. Magnetic Resonance in Medicine, 87(3):1546–1560, 2022

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work page internal anchor Pith review Pith/arXiv arXiv 2013

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qmodel: A plug-and-play model-based reconstruction for highly accelerated multi-shot diffusion mri using learned priors.Magnetic resonance in medicine, 86(2):835–851, 2021

Merry Mani, Vincent A Magnotta, and Mathews Jacob. qmodel: A plug-and-play model-based reconstruction for highly accelerated multi-shot diffusion mri using learned priors.Magnetic resonance in medicine, 86(2):835–851, 2021

2021

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2024

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