arxiv: 2604.06482 · v1 · submitted 2026-04-07 · ⚛️ physics.med-ph · cs.LG

Recognition: 2 theorem links

· Lean Theorem

Spatiotemporal Gaussian representation-based dynamic reconstruction and motion estimation framework for time-resolved volumetric MR imaging (DREME-GSMR)

Jiacheng Xie , Hua-Chieh Shao , Can Wu , Ricardo Otazo , Jie Deng , Mu-Han Lin , Tsuicheng Chiu , Jacob Buatti

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Viktor Iakovenko You Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:59 UTC · model grok-4.3

classification ⚛️ physics.med-ph cs.LG

keywords dynamic MRI reconstructionGaussian representationmotion estimationtime-resolved imagingreal-time trackingradiotherapylow-rank motion modelk-space encoding

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The pith

A spatiotemporal Gaussian representation reconstructs time-resolved 3D MR images from a single pre-treatment scan without anatomical or motion priors.

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

The paper presents a framework that represents both a reference MRI volume and a low-rank motion model using 3D Gaussians. A dual-path encoder estimates motion coefficients directly from k-space signals, enabling reconstruction of dynamic 3D volumes at approximately 400 milliseconds temporal resolution. The same model supports real-time inference of motion coefficients from new k-space data during treatment, with inference times around 10 milliseconds per volume. This approach avoids the need for patient-specific priors or retraining, as demonstrated through evaluations on digital phantoms, physical phantoms, and MR-LINAC data from volunteers and patients showing low center-of-mass errors.

Core claim

DREME-GSMR represents a reference MRI volume and corresponding low-rank motion model as 3D Gaussians, incorporates a dual-path MLP/CNN motion encoder to estimate temporal motion coefficients from raw k-space-derived signals, and uses the solved motion model to infer coefficients from new online k-space data for intra-treatment volumetric MR imaging and motion tracking.

What carries the argument

The spatiotemporal Gaussian representation of anatomy and low-rank motion basis components, together with the dual-path encoder that maps k-space signals to motion coefficients.

If this is right

Dynamic reconstructions achieve approximately 400 ms temporal resolution with 10 ms inference per volume.
Mean target center-of-mass errors remain below 1.5 mm across phantom and clinical datasets for both dynamic and real-time modes.
Motion coefficients can be estimated directly from new k-space data without additional priors or retraining.
A motion-augmentation strategy improves robustness when encountering motion patterns not seen in training.
The method supports cross-evaluation between independent scans from the same patients.

Where Pith is reading between the lines

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

This representation could shorten pre-treatment preparation time by relying on one 3D scan instead of multiple motion-specific acquisitions.
The approach may extend to other time-resolved modalities if k-space signals can be similarly mapped to Gaussian motion coefficients.
Real-time capability at these speeds could support online treatment adjustments on MR-guided systems.
If the low-rank assumption scales, the framework might reduce variability in motion tracking across different scanner models.

Load-bearing premise

The low-rank motion model and trained dual-path encoder sufficiently capture the full range of deformable motions in real-time patient imaging without patient-specific retraining.

What would settle it

A new set of patient scans during real-time imaging that produces mean liver center-of-mass errors above 2 mm would show the claimed generalization does not hold.

Figures

Figures reproduced from arXiv: 2604.06482 by Can Wu, Hua-Chieh Shao, Jacob Buatti, Jiacheng Xie, Jie Deng, Mu-Han Lin, Ricardo Otazo, Tsuicheng Chiu, Viktor Iakovenko, You Zhang.

**Figure 1.** Figure 1: Overview of DREME-GSMR. In the training stage, DREME-GSMR simultaneously reconstructs a dynamic MRI sequence and trains a dual-path motion encoder for real-time motion estimation, all based on a pre-treatment MR scan. After training, a motion-compensated, reference-frame MRI is solved along with the MBCs and the motion encoder capable of estimating corresponding MBC scores to represent the dynamic CBCT seq… view at source ↗

**Figure 3.** Figure 3: XCAT study results visualizations. (a) Comparison of reconstructed referenceframe MRIs from six motion scenarios (X1-X6, from left to right) between DREME-MR and DREME-GSMR. (b) Liver tumor center-of-mass trajectories in the SI direction of the XCAT study. The DREME-MR and DREME-GSMR were trained on the X1 scenario and tested on all six (X1-X6) scenarios. (c) An example of DREME-GSMR’s dynamic reconstruct… view at source ↗

**Figure 4.** Figure 4: Physical phantom study results visualizations. (a) SI center-of-mass trajectories of the spherical tracking target in the physical phantom study for the dynamic reconstruction of Motions 1-4. Trajectories from DREME-GSMR, XD-GRASP, and PCA reconstructions are overlaid with the programmed ‘ground-truth’ curves. Enlarged views highlight representative regions with visible differences among the methods. (b) R… view at source ↗

**Figure 5.** Figure 5: Clinical study result visualizations. (a) Representative liver SI motion trajectories from clinical scans of volunteers (V1-V3) and patients (P1-P3) under cross-scan testing. For each case, the model was trained on one scan and tested on the paired scan, and the dynamic reconstruction of the test scan was used as the pseudo ‘ground truth’. The two columns show the two directional cross-scan testing experim… view at source ↗

**Figure 8.** Figure 8: Comparison between DREME-GSMR with two single encoder variants. (a) Comparison of dynamic reconstruction and cross-scenario real-time SI motion trajectories in the ablation phantom study. DREME-GSMR, DREME-GSMR-MLP, and DREMEGSMR-CNN were trained on Motion 4. The top row shows the training motion and dynamic reconstruction results, and the lower three rows show testing on unseen Motions 1-3. Horizontal da… view at source ↗

**Figure 9.** Figure 9: (a) Dynamic reconstruction and intra-scenario real-time SI motion trajectories for physical phantom motion 4 under different undersampling ratios. The top row shows dynamic reconstruction using the full dataset (100%). The lower four rows show models [PITH_FULL_IMAGE:figures/full_fig_p040_9.png] view at source ↗

read the original abstract

Time-resolved volumetric MR imaging that reconstructs a 3D MRI within sub-seconds to resolve deformable motion is essential for motion-adaptive radiotherapy. Representing patient anatomy and associated motion fields as 3D Gaussians, we developed a spatiotemporal Gaussian representation-based framework (DREME-GSMR), which enables time-resolved dynamic MRI reconstruction from a pre-treatment 3D MR scan without any prior anatomical/motion model. DREME-GSMR represents a reference MRI volume and a corresponding low-rank motion model (as motion-basis components) using 3D Gaussians, and incorporates a dual-path MLP/CNN motion encoder to estimate temporal motion coefficients of the motion model from raw k-space-derived signals. Furthermore, using the solved motion model, DREME-GSMR can infer motion coefficients directly from new online k-space data, allowing subsequent intra-treatment volumetric MR imaging and motion tracking (real-time imaging). A motion-augmentation strategy is further introduced to improve robustness to unseen motion patterns during real-time imaging. DREME-GSMR was evaluated on the XCAT digital phantom, a physical motion phantom, and MR-LINAC datasets acquired from 6 healthy volunteers and 20 patients (with independent sequential scans for cross-evaluation). DREME-GSMR reconstructs MRIs of a ~400ms temporal resolution, with an inference time of ~10ms/volume. In XCAT experiments, DREME-GSMR achieved mean(s.d.) SSIM, tumor center-of-mass-error(COME), and DSC of 0.92(0.01)/0.91(0.02), 0.50(0.15)/0.65(0.19) mm, and 0.92(0.02)/0.92(0.03) for dynamic reconstruction/real-time imaging. For the physical phantom, the mean target COME was 1.19(0.94)/1.40(1.15) mm for dynamic/real-time imaging, while for volunteers and patients, the mean liver COME for real-time imaging was 1.31(0.82) and 0.96(0.64) mm, respectively.

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

The paper's Gaussian-plus-low-rank pipeline for model-free dynamic MR reconstruction and 10ms tracking is a concrete step forward, but the generalization claim rests on unquantified motion diversity in the patient data.

read the letter

The core advance is representing both the reference anatomy and a low-rank motion model as 3D Gaussians, then training a dual-path encoder to map raw k-space signals directly to the motion coefficients. This lets them start from one pre-treatment 3D scan, reconstruct at roughly 400 ms temporal resolution, and do real-time inference in about 10 ms per volume without building a patient-specific model in advance. The motion-augmentation step is a sensible addition to stretch the basis a bit further. They evaluate on XCAT, a physical phantom, and 26 human subjects with cross-evaluation on sequential scans, reporting liver center-of-mass errors around 1.3 mm in volunteers and 0.96 mm in patients for the real-time path, plus decent SSIM and DSC numbers on the phantoms. That is usable data for the radiotherapy motion-management setting they target. The low-rank basis and Gaussian representation are not brand new on their own, but the end-to-end k-space-to-coefficient pipeline with the dual-path network is a fresh framing for this application. The main soft spot is exactly the one the stress-test note flags: the low-rank motion model is derived from a single pre-treatment scan, and the encoder is trained on augmented versions of those same patterns. The paper does not quantify how much motion diversity exists across the patient scans or test against clearly out-of-distribution deformations such as irregular breathing or bulk shifts. If those cases fall outside the learned span, both reconstruction fidelity and the claimed inference speed will degrade without retraining. The abstract and reported metrics do not include enough detail on rank selection, Gaussian count, or training splits to judge how brittle the numbers are. This work is aimed at the MR-Linac and adaptive radiotherapy community. It has enough concrete implementation and human data to merit a serious referee, even though the validation would benefit from tighter motion-diversity analysis and more independent test cases. I would send it to review.

Referee Report

3 major / 3 minor

Summary. The paper introduces DREME-GSMR, a framework representing anatomy and motion via 3D Gaussians and a low-rank motion basis derived from a single pre-treatment 3D MR scan. A dual-path MLP/CNN encoder estimates temporal motion coefficients from k-space signals for dynamic reconstruction (~400 ms temporal resolution) and real-time inference (~10 ms/volume). Motion augmentation is used for robustness. Evaluation spans XCAT digital phantom, physical phantom, and MR-LINAC data from 6 volunteers plus 20 patients (cross-evaluation on sequential scans), reporting SSIM ~0.92, tumor COME ~0.5-0.65 mm, DSC ~0.92 on phantoms, and mean liver COME of 1.31 mm (volunteers) / 0.96 mm (patients) for real-time imaging.

Significance. If the generalization to unseen motions holds, the work offers a potentially impactful advance for motion-adaptive radiotherapy by enabling fast volumetric MR without patient-specific priors or models. Strengths include multi-dataset validation (digital/physical/human with cross-evaluation), quantitative reporting of reconstruction fidelity and motion accuracy (COME, SSIM, DSC), and emphasis on inference speed. The Gaussian-plus-low-rank approach with neural encoding is a coherent technical contribution.

major comments (3)

[Methods (low-rank motion model and dual-path encoder)] Methods (motion model and encoder): The central claim that a low-rank motion basis from one pre-treatment scan plus augmentation suffices for arbitrary unseen intra-treatment deformations (including irregular breathing or bulk shifts) is load-bearing for the 'no prior model' and real-time applicability assertions, yet the manuscript provides no explicit analysis or residual-error quantification of motion components outside the learned basis.
[Results (human subjects evaluation)] Results (human subjects cross-evaluation): The reported patient liver COME of 0.96(0.64) mm rests on sequential scans whose motion diversity is not quantified (e.g., no metrics on periodicity, amplitude range, or out-of-basis components), so it is unclear whether the test set actually probes the generalization regime required by the real-time imaging claim.
[Methods (motion encoder training and augmentation)] Methods (training details): The motion encoder is trained on patterns derived from the same low-rank basis used for the reference volume; this creates dependence that must be mitigated by augmentation, but no ablation or sensitivity analysis on augmentation strength versus basis rank is provided to support the independence of the reported test metrics.

minor comments (3)

[Abstract and Results] The abstract and results report mean(s.d.) values but omit details on statistical testing, sample-size justification, or data-exclusion criteria for the 26 human subjects.
[Methods] Hyperparameter choices (number/scale of Gaussians, motion-basis rank, network architecture) are listed as free parameters but lack explicit selection procedure or sensitivity results.
[Discussion] No direct comparison to existing low-rank or Gaussian-based dynamic MRI methods is included, which would help situate the quantitative gains.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We have addressed each major comment below with targeted revisions to strengthen the presentation of our methods, results, and claims regarding generalization. All requested analyses will be incorporated into the revised version.

read point-by-point responses

Referee: Methods (motion model and encoder): The central claim that a low-rank motion basis from one pre-treatment scan plus augmentation suffices for arbitrary unseen intra-treatment deformations (including irregular breathing or bulk shifts) is load-bearing for the 'no prior model' and real-time applicability assertions, yet the manuscript provides no explicit analysis or residual-error quantification of motion components outside the learned basis.

Authors: We agree that explicit quantification of out-of-basis residuals would better support the generalization claim. In the revised manuscript we will add a dedicated analysis subsection that computes the residual motion variance after projecting intra-treatment deformations onto the pre-treatment low-rank basis (using the same rank as in the main experiments). We will report the captured variance fraction and show that augmentation (random scaling and combination of basis coefficients) reduces effective out-of-basis error on held-out sequences. This addition will clarify the practical limits of the low-rank-plus-augmentation approach without requiring a full patient-specific motion model. revision: yes
Referee: Results (human subjects cross-evaluation): The reported patient liver COME of 0.96(0.64) mm rests on sequential scans whose motion diversity is not quantified (e.g., no metrics on periodicity, amplitude range, or out-of-basis components), so it is unclear whether the test set actually probes the generalization regime required by the real-time imaging claim.

Authors: We acknowledge that motion diversity metrics for the sequential patient scans were not reported. In the revision we will add quantitative descriptors for both volunteer and patient test sets, including breathing amplitude ranges (extracted from diaphragm tracking), periodicity via dominant frequency analysis, and the norm of out-of-basis residuals relative to the pre-treatment basis. These metrics will be presented alongside the existing COME values to demonstrate that the cross-evaluation spans a range of motion patterns distinct from the training basis, thereby supporting the real-time generalization claim. revision: yes
Referee: Methods (training details): The motion encoder is trained on patterns derived from the same low-rank basis used for the reference volume; this creates dependence that must be mitigated by augmentation, but no ablation or sensitivity analysis on augmentation strength versus basis rank is provided to support the independence of the reported test metrics.

Authors: We agree that an ablation study on augmentation strength and basis rank is needed to substantiate robustness. We will perform and report new experiments that systematically vary (i) the augmentation scaling factor applied to motion coefficients and (ii) the motion-basis rank (e.g., 3–12 components). For each combination we will report SSIM, COME, and DSC on held-out phantom and human data, thereby showing that the reported test metrics remain stable across a reasonable range of these hyperparameters and are not artifacts of a single augmentation setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on independent cross-evaluation rather than construction from inputs

full rationale

The paper constructs a patient-specific low-rank motion basis and 3D Gaussian representation from a single pre-treatment 3D MR scan, then trains a dual-path encoder on augmented motions derived from that basis to map k-space signals to motion coefficients. Real-time inference applies the trained encoder to new k-space data. Evaluation uses held-out sequential scans from volunteers and patients (independent of the pre-treatment scan used for basis construction), reporting metrics such as liver COME that are not inputs to the fitting process. While the encoder training distribution overlaps with the low-rank basis generation, this creates only moderate statistical dependence rather than a reduction of the reported performance numbers to the training inputs by construction. No self-citations, uniqueness theorems, or ansatzes are invoked in a load-bearing way that collapses the derivation chain.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The framework rests on domain assumptions about motion structure and representation capacity rather than new physical axioms; several modeling choices are fitted during training.

free parameters (3)

Number and scale of 3D Gaussians
Core representation parameters chosen to model reference volume and motion fields.
Rank of the motion basis
Dimensionality of the low-rank motion model fitted to training data.
Motion encoder network weights
Parameters of the MLP/CNN trained to map k-space signals to motion coefficients.

axioms (2)

domain assumption Patient anatomy and deformable motion can be compactly represented by a finite set of 3D Gaussians and a low-rank basis
Central modeling choice stated in the abstract.
domain assumption K-space-derived signals contain sufficient information to recover motion coefficients via the trained encoder
Required for both reconstruction and real-time inference steps.

pith-pipeline@v0.9.0 · 5755 in / 1666 out tokens · 48188 ms · 2026-05-10T17:59:54.895982+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

we used three MBCs (i.e. i=1,2,3) for each Cartesian direction to model complex breathing motion
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Representing patient anatomy and associated motion fields as 3D Gaussians... low-rank motion model (as motion-basis components)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 1 canonical work pages

[1]

The CNN-based motion encoder was designed to enable image-domain motion estimation

followed by rectified linear unit (ReLU) activations, while the final layer remains linear to yield a scalar output corresponding to a specific motion component. The CNN-based motion encoder was designed to enable image-domain motion estimation. For each SOS stack, a central k-space patch of size 3×6 was retained, while the remaining k-space samples were ...

2020
[2]

of the low-rank motion model, we incorporate a normalization loss to promote the normality of MBCs 𝒆&(𝒙) : 𝐿4AB=1955kl||𝑒&,C||??−1l?n% &'(C',,<,= , (10) where ||⋅||? is the L2 norm. Secondly, a zero-mean regularization is applied to the MBC scores 𝒘&(𝑡): 𝐿D4E=1955o1𝑁15𝑤&,C(𝑡)1o?,% &'(C',,<,= (11) This loss penalizes any time-independent baseline offsets i...

2025
[3]

An ROI mask was generated from the reference image using intensity thresholding to separate the low-intensity air/background and focus the Jacobian regularization on the anatomy

by constraining the local deformation induced by the dynamic DVFs: 𝐿F@G=1∑|Ω1|155k𝑑𝑒𝑡k𝐽∅%(𝒙)n−1n?𝒙∈K%1 , (12) where 𝐽∅%(𝒙)=∇∅1(𝒙) represents the Jacobian matrix of the transformation ∅1(𝒙)=𝒙+𝒅(𝒙,𝑡) at point 𝒙, while Ω1 denote the region of interest (ROI) at frame 𝑡. An ROI mask was generated from the reference image using intensity thresholding to separat...

2025
[4]

Both physical phantom and clinical data are acquired with the stack-of-stars trajectory from a 1.5T MR-LINAC

simulations, physical phantom measurements, and clinical data. Both physical phantom and clinical data are acquired with the stack-of-stars trajectory from a 1.5T MR-LINAC. The XCAT simulation study provided ‘ground-truth’ anatomy and motion, enabling quantitative evaluation of both image reconstruction quality and motion estimation accuracy. The physical...

2004
[5]

Data were acquired on a 1.5T Elekta Unity MR-LINAC (Elekta AB, Stockholm, Sweden) at the UT Southwestern Medical Center with a TR of 4.5 ms and 8 receive coils

were sinusoidal, with a period of 4 s and an amplitude of 30 mm, and a period of 3 s and an amplitude of 24 mm, respectively. Data were acquired on a 1.5T Elekta Unity MR-LINAC (Elekta AB, Stockholm, Sweden) at the UT Southwestern Medical Center with a TR of 4.5 ms and 8 receive coils. A total of 673 stacks were continuously acquired using SOS golden‐angl...

2019
[6]

Flashgs: Efficient 3d gaussian splatting for large-scale and high-resolution rendering.arXiv preprint arXiv:2408.07967, 2024

Conclusion In this study, we present DREME-GSMR, a novel framework for time-resolved dynamic MRI reconstruction and real-time motion management based on 3D Gaussian representations. Leveraging the strong representation power of Gaussians, DREME-GSMR enables ‘one-shot’ dynamic MRI reconstruction directly from raw k-space data, eliminating the need for prio...

work page arXiv 2025