Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport
Authors: Taoran Zheng
Yan Yang
Xing Li
Xiang Gu
Jian Sun
Zongben Xu
Venue: NeurIPS 2025 poster ()
Keywords: prospective medical image reconstruction
dynamic optimal transport
inverse problem
imaging knowledge-informed
Primary area: applications
TL;DR: This paper introduces a novel imaging knowledge-informed dynamic optimal transport framework incorporating imging physics into both transport cost and equation, enabling to learn from unpaired real data for prospective medical image reconstruction.

Abstract: Medical image reconstruction from measurement data is a vital but challenging inverse problem. Deep learning approaches have achieved promising results, but often requires paired measurement and high-quality images, which is typically simulated through a forward model, i.e., retrospective reconstruction. However, training on simulated pairs commonly leads to performance degradation on real prospective data due to the retrospective-to-prospective gap caused by incomplete imaging knowledge in simulation. To address this challenge, this paper introduces imaging Knowledge-Informed Dynamic Optimal Transport (KIDOT), a novel dynamic optimal transport framework with optimality in the sense of preserving consistency with imaging physics in transport, that conceptualizes reconstruction as finding a dynamic transport path. KIDOT learns from unpaired data by modeling reconstruction as a continuous evolution path from measurements to images, guided by an imaging knowledge-informed cost function and transport equation. This dynamic and knowledge-aware approach enhances robustness and better leverages unpaired data while respecting acquisition physics. Theoretically, we demonstrate that KIDOT naturally generalizes dynamic optimal transport, ensuring its mathematical rationale and solution existence. Extensive experiments on MRI and CT reconstruction demonstrate KIDOT's superior performance. Code is available at https://github.com/TaoranZheng717/KIDOT.

Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport
Taoran Zheng', Yan Yang Xing Li', Xiang Gu Jian Sun Zongben Xu'
'School of Mathematics and Statistics, Xi an Jiaotong University, Xi' an, China
State Industry-Educatto Integration Center for Medical Innovations at Xi'an an Jiaotong University taoranzhengdstu.. xjtu. edu. cn; flistar0810, yangyan, xianggu, jiansun, zbxu)xxttuu eenjjuueduu edu.cn
Abstract
Medical image reconstruction from measurement data a vital but challenging inverse problem. Deep learning approaches have achieved promising results, but often requires paired measurement and high-quality images, which is typically simulated through forward model, i.e., retrospective reconstruction. However, training on simulated pairs commonly leads to performance degradation on real prospective data due to the retrospective-to-piie prroopecttee gap caused by incomplete imaging knowledge in simulation. To address this challenge, this paper introduces imaging Showledge-Informee Dynamic Optimal Transport (KIDOT), novel dy- namic optimal transport framework with optimality in the sense preserving consistency with imaging physics transport, that conceptualizes reconstruction as finding a dynamic transport path. KIDOT learns from unpaired data by modeling reconstruction as continuous evolution path from measurements to images, guided by an imaging www.wlege-informee. cost function and transport equation. This dynamic and knowledge-awar approach enhances robustness and better leverages unpaired data while respecting acquisition physics. Theoretically, we demonstrate that KIDOT naturally generalizes dynamic optimal transport, ensuring its mathe- matical rationale and solution existence. Extensive experiments on MRI and CT reconstruction demonstrate KIDOT's s superior performance. Code is available https: /tt::://ithtttt thub. com com/Toorraahheeg111/0000000
1 Introduction
Medical imaging techniques like Magnetic Resonance Imaging (MRT) and Computed Tomography (CT) vital tools for visualizing internal anatomy, yet reconstructing high-fidelity images from the acquired measurements remains significant challenge [1, 2]. This process fundamentally an ill-posed inverse problem inferring complete, clean image from often incomplete and noisy measurement data. Classical reconstruction algorithms, such as filtered back-projection or iterative methods [3-7], typically rely on precise mathematical modeling of the image acquisition physics, incorporating essential imaging knowledge. However, accurately capturing complex factors present in real-world scenarios (e.g., noise, non-ides sampling) within these models remains challenging. Consequently, the performance of traditional methods can degrade significantly when this knowledge incomplete or imperfect, yielding images with noise or residual artifacts.
Deep learning (DI.) has emerged as powerful alternative [8-10], achieving state-of-the-arr re- sults by learning complex mappings directly from data. However, the predominant supervised DL. paradigm| 1-13] faces critical bottleneck: the requirement for large datasets of perfectly registered pairs of low-quality input measurements and their corresponding high-quality ground truth images.
'These authors contributed equally. CCorresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Dynamic optinal transport
Retrospective
Prospective
Imaging knowledge
Guiding
inf B =()(f
y= o S(x)
da(({)
v(t,, x(0) P, a11)
Undersampled k-space data
Resonstruction path
P
Reconstructed
k-space data
Prospoctive degraded domain
High quality domain
X
KIDOT Pramework: leveraging imaging knowledge to guide dynamic process
cecpopppcciive-posoocciee gap reconstruction
Figure 1: (a) Core concept of KIDOT: modeling image reconstruction as continuous evolution from a prospective degraded distribution P to a high quality image distribution Q, guided by dynamic OT. (b) The retrospective curroppetive-ppoopeettee gap MRI: a visual comparison of k-space data from retrospective simulations versus real prospective undersampling (top), and their corresponding k-space reconstruc- tions via supervised learning (bottom).
Obtaining such perfectly aligned data in clinical practice is often difficult, costly, or ethically con- strained. Furthermore, many real-world datasets inherently lack perfect pairing due to patient motion, or differences in acquisition protocols between low-quality and high-quality scans [14, 15].
To circumvent the lack of real paired data, common practice [16, 17) is to generate simulated low-quality data using simplified imaging knowledge (e.g., adding modeled noise, applying idealized undersampling masks) from available high-quality images, creating artificial pairs for supervised training. However, significant gap often exists between these simulations and real-world prospective measurements. Figure 1(b) shows examples of the retrospective- prottee--pspeetiiveee gap in MRI, where k-space data from retrospective simulations (left) differ from that prospective acquisitions (right) in sampling patterns and reconstructed geh-frequency regions. Retrospective simulations often idealize the true data acquisition process, struggling to fully capture the complex noise and genuine artifact characteristics inherent in real-world prospective measurements. Models trained solely on such simulated pairs, therefore, often exhibit a performance drop when deployed in prospective clinical settings, highlighting their sensitivity distribution shifts arising from the disparity between simulated and real-world imaging knowledge. Ideally, a reconstruction method should effectively leverage all available data resources, including potentially large amounts of unpaired real measure- ments and real high-quality images, while robustly incorporating reliable imaging knowledge. This motivates exploring frameworks capable of learning from unpaired distributions manner. Optimal Transport (OT) [18, 19] offers a principled mathematical approach for transporting probability distributions or finding mappings between them, with the potential to learn from unpaired data. However, conventional applications of OT often adopt a static perspective, learning direct map and potentially overlooking the dynamic physical processes inherent in image reconstruction.
physics-aware
To tackle the above challenges, this paper introduces imaging Knowledge-Infoon Informed Dynamic Optimal Transport (KIDOT), novel framework for unpaired medical image reconstruction. KiDOT integrates imaging knowledge, primarily through the forward physical model, directly into a dynamic OT formulation. As conceptually depicted in Fig. ((a), instead of learning a static transformation, KIDOT models reconstruction as continuous evolution path from the measurement distribution (L") to the target image distribution (Q). This evolution is guided by an instantaneous cost function and governed by a transport equation, both incorporating imaging knowledge, ensuring the transport path remains consistent with the physics throughout the transformation. This dynamic, knowledge-infonee approach enables more realistic modeling, suited for leveraging unpaired data while remaining grounded in the physics image acquisition. To translate KIDOT into practice, we introduce a neural network implementation strategy that learns from combination of unpaired real and paired simulated medical images. Furthermore, the theoretical underpinnings of KIDOT, including the rationale and the existence of its learned transport solution, established, validating its mathematical rigor.

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The main contributions of this work are threefold: 1) We propose the KIDOT framework, which uniquely integrates imaging physics into both the cost function and transport equation of dynamic OT for prospective medical image reconstruction. To the best of our knowledge, this is the first dynamic OT framework that integrates medical imaging physics, with optimality in the sense of preserving consistency with imaging knowledge throughout its transport path. 2) We develop a practical implementation algorithm of KIDOT based on neural networks, enabling it to learn to reconstruct from unpaired medical image data. Theoretical analysis for KIDOT is established, showing its rationale and the existence of the learned transport solution. 3) We demonstrate the practical effectiveness and superior performance of KIDOT through extensive experiments challenging MRI and CT reconstruction tasks, particularly showcasing its advantages in handling prospectively acquired and unpaired clinical data.
2 Background
Prospective medical image reconstruction. Medical imaging seeks to reconstruct internal struc- tures from indirect, often noisy measurements. This task is fundamentally an ill-posed inverse problem: determining an underlying image z given observed data y. The relationship is typically modeled as y = A(x) + n. where A represents the forward model dictated by the imaging physics (e.g., the Radon transform in CT [20] or the partial Fourier transform in MRI [21]), and n accounts for measurement noise and errors. Current approaches to solving such inverse problems can be broadly categorized into three types. Classical iterative methods [3, 4] minimize the difference between predicted and measured data, guided by the physical model and incorporating prior assumptions about the image x (e.g., sparsity [3] or low-rank structure [4]) via regularization:
i arg min ||A(x)) - v + XR(x),
(1)
where R(x) is the regularization term, and À is the trade-off parameter. While grounded in physics, these methods may struggle to recover fine details from limited or noisy data. More recently, deep learning techniques [22-26] have shown promise, learning direct mappings from measurements y to images x. These data-driven models can achieve high accuracy and speed but typically require large datasets of paired examples (i.e., measurement y and corresponding ground truth image a:). Hybrid approaches 1, 27] attempt to combine the strengths of both data- and model-driven methods, integrating physical models within neural network architectures. However, like purely data-driven methods, they usually depend on the availability of paired training data. A significant practical challenge arises because acquiring such perfectly aligned, high-quality reference images alongside clinical measurements is often difficult or impossible (e.g., obtaining perfectly matched low-dose and standard-doss CT scans). This common scenario, where only measurements y are readily available without corresponding high-quality æ, defines the challenging prospective inverse problem setting that remains underexplored [16, 17].
Optimal transport and its applications in medical image reconstruction. OT offers robust mathematical framework for comparing probability distributions via the minimal cost required to transform one into another [18, 28, 29]. Static OT finds an optimal coupling between source IP and target @ distributions that minimizes the expected transport cost [18]:
inf xEI((F.)) (P) B(r.)><(((r, v)].
where is the cost function, and II is the set of joint distributions with marginals PP, Q Extending this, dynamic OT models the continuous evolution from P to Q over time [18, 19]. Dynamic OT seeks a velocity field vtt, x())) that optimally transports particles from an initial distribution 20) ~ IP to final distribution (1) ~ Q. This is achieved by finding v that minimizes the action functional:
inf E S
1 2 (l(((,((0))
dx((() = v(t,,(()), x00) ~ P. s(1) Q
s.t.
(3)
dt
The OT principles have proven valuable for inverse problems lacking paired data, e.g., natural image processing [30-34]. In medical imaging, static OT has been explored to address the challenges of unpaired or misaligned clinical data, For example, [35, 36] use static OT to derive a CyeleGAN frame- work for learning from unpaired data. [37] further integrates variational methods with reconstruction
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Imaging Knowledge-Informt Dynamic Optimal Transport (KIDOT) Objective
P
3p
XEg (o(x))
(7+ (P)
T (P)
Pg
T
T fei
I P
Discrete tramsport equation
Prospecrive degraded domain
I T, (4) 7 (A)())))) -4) 14(A,1)
Reconstructed domain
High quality
Figure 2: Illustration of the KIDOT framework. KIDOT employs sequence of transformations To governed by a learned transport equation driven by the medical image reconstruction model, to map measurement y from the prospective degraded domain to the reconstructed domain. The transport process is guided by the KIDOT objective function, which critically incorporates imaging knowledge by enforcing data consistency along the transport path. Simultaneously, it promotes distribution matching between the reconstructed domain and the target high-quality domain using potentials Paa
and Wasserstein-1 distance for unpaired learning in CT. [38, 39] preserve structural consistency [38] or incorporate contextual features [39] in OT-based generative models. Static OT has also been utilized for aligning and synthesizing different MRI sequences to enhance reconstruction quality [40]. More recently, dynamic OT has been explored for semi-supervised problems, FSBM [41] utilizes a objective to build dynamic transport process in Schrödinger bridge framework.
Despite these advances, prior OT-based methods medical imaging often utilize generic costs (e.g.. Euclidean distance) disconnected from the underlying image formation physics, employ static OT which considers only the source and target distributions, without modeling the dynamics of the transformation between them. We introduce novel constraint and objective by requiring the dynamic OT path to be consistent with the physical forward model of the imaging system. This embeds crucial domain knowledge directly into the transport dynamics, offering an imaging-informed informed alternative to standard dynamic OT formulations for inverse problems.
3 Imaging Knowledge-Informed Dynamic Optimal Transport
Given measurement distribution P and high quality image distribution Q in prospective medical image reconstruction, this paper aims to model principled evolution from an initial state y ~ IP to corresponding target state I ~ Q. Dynamic OT provides a framework for modeling the evolution between probability distributions, characterized by a transport cost and governing transport dynamics. Conventional dynamic OT often assumes the transport cost is based on a standard metric distance (e.g., Euclidean distance), optimizing for the shortest path length. However, in inverse problems, e.g., medical image reconstruction, a more physically meaningful cost might relate the evolving estimate to the observed measurements via the forward model, and this cost may not be a standard metric. Meanwhile, the physical knowledge should also be incorporated into the transport dynamics. This necessitates adapting the dynamic OT framework. Towards this goal, we propose an imaging Knowledge-Infonee Informed Dynamic Optimal Transport (KIDOT) approach for prospective medical image reconstruction. As illustrated in Fig. 2, in KIDOT, we design a transport cost reflecting local consistency with the imaging physics at each point along the evolution. Meanwhile, the dynamics of KIDOT, represented by the transport equation, driven by the gradient flow of inverse problems that incorporate imaging knowledge. We also develop practical implementation based neural networks for KIDOT, enabling it to learn from unpaired real measurements for prospective medical image reconstruction. Next, we elaborate on these components in detail.
3.1 Dynamic Transport Cost Informed by Imaging Knowledge
In static OT, the objective is to find coupling и between source P and target Q distributions that minimizes an average transport cost (Eq. (2)). To effectively apply OT to imaging inverse problems, where measurements y are related to an underlying image by forward model y se A(z), the

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cost function c(T, y) should encode this physical relationship. Therefore, instead of using generic distances, we propose an -ngeomeed cost based on data fidelity: c(z, y) = ly A(x) This cost directly quantifies the discrepancy between an observed measurement y and the measurement predicted by the forward model A on a candidate image x.
Yet, how to define the transport cost in the dynamic setting is difficult. A key challenge arises because our informedd cost c(x, y) |yy A(x))] is generally not a metric. Consequently, the standard dynamic OT formulation based on minimizing path length defined based on a metric is un- suitable. As an alternative, we propose minimize the expected value of integrated instantaneous cost over potential paths:
inf T8 (( To) where c(æ,v) A(x))11
(4)
Here, the dynamics di ctt, Ii) are governed by the velocity field and It represents the path evolving from an initial measurement In- This objective seeks dynamics v such that, on average, the evolving path I maintains consistency with its specific originating measurement In (via the forward model A) throughout the entire time interval [0, I]. We next analyze the rationale of this cost.
de
Rationale analysis. To evaluate the suitability of minimizing such an integrated cost, we analyze its behavior in the Euclidean space. Considering paths connecting a starting point to an endpoint æ, examine which path minimizes the integral of the instantaneous Euclidean distance between the point on the path st) and the fixed endpoint æ. We have the following theorem.
Theorem 3.1 (Existence and Geometry of Minimizer for L Distance Integral). Let z, E RR" be distinct points and let M | yll2 be constant. Define the feasible set of paths XM as Xs = {s E AC([O, 1J, I2) ] s(0) y, s(1) = I, ||s'Iloc I MM where AC([0, I)) is the space of absolutely continuous paths. Then, an optimal solution
Xy to problem
inf [ [|s(t) yyl2dt SEXM
(5)
exists and its geometric trajectory (t) I [0, 1]} is the straight line segment connecting y and
The proof is provided in Appendix A.2. This theorem indicates that when the static cost is taken the squared Euclidean distance, minimizing the proposed dynamic transport cost recovers the standard straight-line path, consistent with the conventional dynamic OT. This suggests that minimizing this integral formulation could be a reasonable way to define transport dynamics based on general cost, including non-metric, imaging knowledge-informe data-fidelity cost.
3.2 Transport Equation Guided by Imaging Knowledge
We now specify the dynamics that governs the transformation path 4e from measurements to re- constructed images, guided by imaging knowledge. Our approach draws inspiration from inverse problems, which commonly minimize an objective that combines data consistency with a regu- larization term R[I) encoding prior knowledge about the desired image. The continuous-timg gradient flows of the inverse problems naturally provide a principled evolution dynamic. Specifically, minimizing Eq. (1) via gradient flow leads to the following ordinary differential equation (ODE):
dI dt = - (AA - Io) + VR(L)),
(6)
where Ln represents the initial measurement y, AA the adjoint of the forward operator A, and (R(())) is the gradient of the regularization term evaluated at the current state I,. This ODE describes a path moving away from simple measurements towards reconstructions that better satisfy both data fidelity and prior constraints. We adopt this gradient flow to define the velocity field v(t, It) that drives the transport our KIDOT framework. Note that the regularization is often learned based on paired data in a supervised manner in previous methods. By taking Eq. (6) as the transport equation, our KIDOT will provide an approach to learn the regularization/prior from prospective unpaired data.
difficult to define and
5

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Combining the proposed integrated cost (Eq. (4)) with the imaging-knowledge dynamics (Eq. (6)), the KIDOT formulation seeks the optimal regularization by solving:
inf EOO J// I - A(Till de
dl, (AA((AA( - Io) + VR(I)). 1 P,, dt
s.t.
In problem (7), the objective function minimizes the expected cost along the trajectory, while the constraints enforce that the evolution follows the defined dynamics that starts from the distribution of input measurements P. and ultimately transforms it into the target distribution of high-quality images Q at time 1. This formulation integrates imaging knowledge into both the transport cost and equation, forming an imaging seowledge-nnformee dynamic OT model. We note that while the gradient flow in Eq. (6) serves as our main example, the KIDOT framework inherently flexible and can readily accommodate more advanced optimization flows, e.g., proximal gradient flows, to incorporate the imaging knowledge into the transport equation in more different ways.
3.3 Practical Implementation Based Neural Networks
This section focuses on the practical solution to KIDOT problem (7). Since the optimization variable function in the KIDOT problem, we parameterize it using neural networks ease implementation.
Parametrization and relaxation. For convenience, we directly parameterize VR using a neural network Hr with parameters c. To solve problem (7), another challenge is the ODE-based constraint with initial and ending distribution constraints, making optimization difficult. To tackle this challenge, apply a relaxation to the ending distribution to seek approximation of the optimal solution. Specifically, given initial state Io P, produce using ODE by
(To) Io
- Io) + Ho (L+))dr.
By denoting the distribution of Tg as The (P), the constraint 4 Q becomes Tay (P) = Q equivalent to W1 (T Q) 0 where W1 is the 1-Wasserstein distance. By introducing the Lagrange relaxation, problem (7) becomes
inf J(dAA, & C(o) AW (T) &
(8)
where C(o) = EI,F Jo' c(T To) dt} Leveraging the Kantorovich-Rubinste duality [18] for the W1 distance, we further transform the optimization problem
inf ((0)+) +1 sup (Ee~ole(x)) ((o=el((((((o)))))
11
where || -IILip is the Lipschitz norm.
Discretization and training. To solve the above problem, we also parameterize using neural network fo with parameters A. The integral to produce (Ta) and the cost are approximated numerically via discrete sum over N time steps to <
< iy 1, with step size f++1 - t; 1/N. Using forward Euler discretization for the ODE in Eq. (7), the evolution at each discrete step becomes 4 = 4t N (AA,,) 1o) (L,,), where Lo = lo. Let T (Io) denote the state Iu.,, obtained after + such steps. Substituting these into the dual objective Eq. (9) and discretizing the time integral, we arrive the following loss function for training:
N-1
(oioor 0) Ee
Σ A(T (v)) + (T NN---- (w)) i=0
-Egolo(()))
(10)
LKIDOT can be implemented on unpaired mini-batcl samples of the unpaired prospective medical images. In medical image reconstruction, the simulation measurements produced by the forward model A on high-quality medical images are also available for training, serving as partially paired data P Prair. For these data, we apply supervised training loss
(o) = E (v ) Precr [F(T (1p'" xp)].
(11)

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where F(.,)) is standard supervised loss function (e.g., L1 or Le distance), encouraging the predicted endpoint T (Up) match the known ground truth Tp. The final training objective is
inf sup LXIDOT
*Lsup((),
(12)
where y hyperparameter. Eq. (12) is optimized by alternately updating and through gradient descent and ascent, respectively. We provide the detailed training algorithm in Appendix A.I.
Theoretical analysis. We now study the existence of an optimal solution to the formulation presented in Eq. (8). We make the assumptions: A1) Measures P and Q have compact support, and the parameter set Rd is non-empty and compact. A2) T% T pointwise whenever on + Φ and there exists h E ((0, 1]) such that (e(T), (v)- y)1 - h(t) for all n and t E [0, 1]. A3)
De lT S0. A4) For fixed the cost function cly. -) is continuous in its second argument.
Theorem 3.2 (Existence of Minimizer for KIDOT Objective). Suppose the assumptions Al-A4 hold, then there exists a minimizer for the optimization problem inf (( Q)
The proof is provided in Appendix A.3. This theorem confirms that the KIDOT objective is well- defined and that set of optimal parameters o² guiding the transport path exists.
Experiments
We conduct experiments encompassed simulated MRI data, prospectively acquired real-world MRI data, and prospectively acquired clinical CT data. Full experimental details and supplementary results are available in Appendix due to space limit.
Simulated MRI data. Our experiments on simulated MRI data utilized the publicly accessible fastMRI multi-coil knee dataset [42]. We selected 2500 fully sampled MRI slices, partitioning them into 1000 for training, 500 for validation, and 1000 for testing. Undersampled measurements were simulated using k-space masks corresponding to acceleration factor. A key aspect this evaluation was to assess KIDOT's robustness to scenarios where test data characteristics differ from training data, a common challenge prospective clinical deployment. achieve this, we generated two distinct sets of undersampled inputs: one set, with specific sampling patterns, was used for training supervised baselines, while a different set of sampling patterns was employed for testing to mimic prospective acquisition conditions. For this dataset, the core transport network To within KIDOT was implemented using an unfolding architecture based on E2E- VarNet [12].
Quantitative results on the simulated MRI dataset are presented in Table 1. KIDOT demonstrates superior performance across all evaluated metrics, encompassing both image fidelity measures (PSNR and SSIM) and perceptual quality indicators (FID [43] and KID [44]). These scores outperform conventional methods like CS-wavelet [45, 46), supervised learning models like 22E--arNet [12], contemporary unpaired learning techniques such as OT-CycleGAN [35], UAR [37] and FSBM [41], as well as strong diffusion- based models like DDS [47]. This highlights KIDOT's effectiveness in producing reconstructions that achieve both high fidelity to the ground truth and strong perceptual realism, even under these challenging simulated prospective conditions. Visual comparisons, further illustrating these improvements, provided in Appendix B.8.
Table 1: Quantitative comparison the simulated MRI dataset (4x acceleration). Metrics include PSNR. SSIM (higher better t), FID, and KIDx x 100 (lower is better J). Best results are bolded.
Method
PSNR 1
SSIM 1
FID +
KIDx 100
CS-wavelet [45]
29.17
0.7981
95.76
5.77
VarNet [12]
32.56
0.8372
24.35
1.29
OT-CycleGAN [35]
26.45
0.7751
156.43
7.25
UAR [37]
33.01
0.8425
19.56
1.05
FSBM [41]
26.52
0.7634
113.62
6.03
DDS [47]
31.27
0.8136
29.46
1.65
KIDOT (ours)
33.31
0.8518
10.68
0.83

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Real prospective MRI data. To further assess KIDOT's performance in realistic clinical scenarios, we utilized a multi-contrast brain MRI dataset acquired prospectively using 3.OT United Imaging scanner equipped with 32-channel head coil. Our experiments centered on the T2-FLAIR weighted sequence, which was undersampled with a 10x acceleration factor. The dataset, comprising images of size 256 x 256 x 176, was divided into 3077 training, 1860 validation, and 3077 test slices. A key challenge with this dataset is the separate acquisition of undersampled data and their corresponding fully sampled references, common practice that often introduces inherent spatial misalignments between them. Consequently, standard pixel-wise fidelity metrics such as PSNR or SSIM are less reliable for evaluating performance on the test set.
For this dataset, the transport network To in KIDOT was parameterized using an unfolding architecture based on PromptMR [13]. Given the inherent spa- tial misalignments, which render pixel-wise metrics unreliable, our evaluation on this dataset focused on distribution-based metrics: FID and KID. Further- more, we conducted a rigorous benchmark against state-of-the-art DDS model under three practical strategies: using a pre-trained model, fine-tuning it on our data, and training it from scratch. As detailed in Table 2, KIDOT achieved the leading FID and KID scores, outperforming both supervised and un- supervised baselines. KIDOT surpasses all DDS vari- ants, which highlights a key advantage: its physics- informed dynamics provide powerful inductive bias, making it more data-efficient and robust, especially when real training data is limited. Visual results pre-
Table 2: Quantitative comparison on the Real Prospective MRI Data (10x accelcration). Metrics include FID and KID x 100 (lower is better 4). Best results are bolded.
Method
FID
KIDx 100 J
CS-wavelet [45]
67.75
3.34
E2E-VarNee [12]
65.27
2.83
PromptMR [13]
29.52
1.46
OTT-CylleGAN [35]
221.83
10.94
UAR [37]
29.09
1.23
FSBM [41]
147.62
7.13
DDS (Pre-trained) [47]
62.94
2.71
DDS (From scratch) [47]
58.32
2.53
DDS (Finc-tuncd) [47]
39.78
1.84
KIDOT (ours)
28.26
1.12
sented in Figure 3 corroborates these quantitative findings. On representative test slices, KIDOT reconstructions exhibit clear anatomical detail, effective noise suppression. and reduced Gibbs arti- facts. Particularly in challenging areas (e.g.. highlighted region in red). KIDOT demonstrates superior preservation of fine structures compared to other approaches. Notably, as FSBM is not tailored for medical imaging and its visual results were poor, we omitted its visualizations for brevity.
C VINEEE MARN
Reference
CS-wavelet
E2E E2E-vamet
PromptMR
01-CycKGAN
UAR
DDS (Fine-tuned)
KIDOT
Figure 3: Qualitative comparison of MR reconstruction methods on the prospectively acquired United Imaging brain dataset (T2-FLAIR 10x acceleration). KIDOT demonstrates enhanced detail recovery (e.g.. red box).
Clinical prospective CT data. We finally evaluated KIDOT on a challenging clinical low-dose CT (LDCT) task, utilizing a dataset of prospectively acquired abdominal scans from 30 consented patients under Institutional Review Board approval. The LDCT protocol involved an approximate 10-fold reduction in radiation dose compared the corresponding normal-dose CT (NDCT) scans. Crucially, differences in scan parameters and acquisition directions between the LDCT and NDCT protocols
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resulted in significant spatial misalignments between the image pairs, common complication in prospective clinical studies. From 28,251 available slices, 2,420 were selected for this evaluation. To benchmark KIDOT against methods reliant aligned data, we generated simulated low-dose CT (SLDCT) images from the NDCT seans, injecting compound Gaussian and Poisson noise into the uncorrupted projections by using the method in [48, 49]. The transport network Tp within KIDOT for this task was based on the ISTA-Net architecture [50].
Due to the substantial spatial misalignments inherent in the dataset, performance was also primarily evalu- ated using the distribution-ban -assed metrics FID and KID. Table presents the quantitative comparison. KIDOT achieves the best scores among all evaluated methods, outperforming conventional approaches like BM3D, su- pervised deep learning models such as REDCNN and the baseline -et, and unpaired methods includ- ing OT-CycleGAN and UAR. This demonstrates KI- DOT's strong capability to learn effective mappings for dynamic image reconstruction even from severely mis- aligned input distributions, reflecting its potential utility in practical clinical scenarios. Figure 4 provides qualita- tive results LDCT data processed and the reconstruc- tion achieved by various methods, including KIDOT, underscoring the challenges posed by noise and the need for robust handling of real-world data complexities like misalignment. Among these, KIDOT's reconstruction (bottom-right) particularly stands out, demonstrating superior noise suppression and enhanced preservation fine anatomical details compared other approaches.
Table Quantitative comparison on Clin- ical Prospective CT Data (10-fold reduc- tion). FID/KID are reported (lower is het- ter J). Best results are highlighted in bold.
Method
FID|
KID x 100 J
BM3D [51]
100.82
6.12
REDCNN [52]
46.43
1.93
ISTA-Net [50]
24.86
1.59
OT OT-CycleGAN [35]
63.54
3.28
UAR [37]
23.82
1.45
FSBM [41]
68.42
3.61
KIOTT (ours)
22.25
1.32
Low Dose
Reference
BM-3D
NeT
REDCNN
OT-CycleGAN
CAR
KIDOT
Figure 4:
Qualitative results for LDCT reconstruction clinical prospective data. Comparison illustrates noise reduction and detail enhancement achieved by KIDOT.
Ablation Study. We conduct an ablation study on the fastMRI knee dataset (4x acceleration) evaluate the effectiveness of each component in KIDOT. We compared four different configurations: (1) training
Table 4: Ablation study on the fastMRI
knee dataset (4x accelcration).
Method
PSNR
SSIM
with only supervised loss CsUP in Eq. (11), (2) training with the loss associated with P8 Eq. (10) and LSUP, denoted as LsUP + Pp. (3) adding the transport cost Cfinal |ly A(T IN
CSUP CSUP + +:0
32.56
0.8372
32.92
0.8422
CSUP + (e + CRual
33.01
0.8425
& (v))ll for the final output in Eq. (10) to CSUP + Po. denoted as LSUP + +o Cinal- (4) using whole loss (KIDOT). Table (upper two rows)
KIDOT
33.31
0.8518
shows that adding the potential network ypu to the supervised baseline resulted in performance improvements. These gains indicate that so successfully guides learning process towards better alignment with the target data distribution. It can be observed from Table that incorporating the

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bongiing-nowledgee informed transport cost to the final output can further improve performance. Furthermore, our full KIDOT model that enforces consistency to imaging knowledge throughout the transport process, further improves the results, demonstrating the effectiveness of our imaging snowledge-informet dynamic OT for medical image reconstruction. The key difference between (3) and (4) lies in when and how the imaging- knowledge-informe cost is applied. The third setting represents an ablation where the physics-consistems cost is applied only to the final output of the N-step transport path. In contrast, our full KIDOT model enforces this physics consistency throughout the entire dynamic transport process, integrating the cost over intermediate steps of the trajectory.
Conclusion and Limitations
We presented KIDOT, a novel framework for prospective medical image reconstruction. Our key innovation is framing reconstruction as a dynamic transport process whose learned path is explicitly constrained by imaging physics. Theoretical analysis and extensive experiments show the rationale and effectiveness of KIDOT. While KIDOT demonstrates promising results, it relies on the availability of the physical forward model (i.c., A),, which may limit its application when the information of A is missing. Meanwhile, the computational cost associated with simulating the dynamic transport path via numerical ODE integration can be higher than static methods. We will investigate more techniques in our framework tackle these limitations in the future.
Acknowledgement
This work was supported by National Key R&D Program(2022YTA 1004201), NSFC (12090021, 12125104, 12401671, 12426313, 12501708, 12501709, 623B2084), Postdoctoral Research Fund- ing Project of Shaanxi Innovative Talents (BX20240276), China Fundamental Research Funds for the Central Universities (xzy022025047), China Postdoctoral Science Foundation (2025M773058) and the Key Laboratory of Biomedical Imaging Science and System, Chinese Academy of Sciences.
China National Postdoctotal Program for
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Does the paper describe the usage of LLMs if it is important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: The core methodology of KIDOT does involve the use of Large Language Models (LLMs) as an important, original, or non-standard component.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer to LLM policy (https: / hctps//ecceooeeeeee//////// /LLM) for what should or should not be described.
20

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A Proofs and Algorithm
A.1 KIDOT Training Algorithm
Algorithm 1 KIDOT Training Algorithm
Input:
Real prospective degraded dataset (samples y ~ Y); high-quality dataset (samples
X); unfolding transport network Ta and potential network yoo; learning rates aø, og; number of critic updates per generator update No.
1: while ø has not converged do
2: for k ===1,, 1, N₂ do
3: % Training potential network soo ]==1 4: Sample mini-batch { from and (a()) from X.
5: Evolve (y6²) (Te '(y(0)) using discrete transport equation for N steps.
6:
Compute critic loss: Co Ei= (o(x
13 (T N (y(
7:
Update critic parameters: 0 + 0 + ag VaLa. end for
8:
9:
% Training unfolding transport network T
10:
Sample mini-batch { from Y.
11:
Evolve { along paths (y())) N and to endpoint
(
12:
Compute generator loss:
Lo Σィ= (Ei-o NV (1y A(T (y()))) - Appo(TT ;(y())))
14:
if paired data is used then
Sample mini-batch {wp ()) )1 د from Paair-
15:
Ap
16:
L. Lo + E, ESn F(T
xx-1 (yr ), 1p (2)).
17:
end if
Update generator parameters: Φ + Φ ag V LL..
19:
end while
A.2 Proof of Theorem 3.1
Theorem A.1 (Existence and Geometry of Minimizer for I,² Distance Integral). Let R be distinct points and let M Z || yll2 be constant. Define the feasible set of paths X'M as
XM {s E AC((O, R") s(0) y, s() x. |s S MM,
where AC( 1], IR)) denotes the space of absolutely continouss paths. Consider the optimization problem
inf [[s], where [[s] [[s]=] -f ||s(t) (bst))-līdtt. N
(13)
Then, an optimal solution s* E XM to problem (13) exists, and its geometric trajectory {8 (1) [0, 1}} coincides with the straight line segment connecting y and x.
|.E
Proof:
The proof proceeds in two parts. First, we establish the existence of an optimal solution. Second, we prove that the geometric trajectory of any such optimal solution must be the straight line segment connecting y and x.
Part 1: Existence of Optimal Solution
We equip the space of continuous functions C((O, R") with the supremum norm || Iloo. where ll supre(o.1) [t)]l) We next show that the feasible set Xxs is nonempty and compact in this space and that the functional T[s] is continuous, thus obtaining the existence of an optimal solution.
Non-emptiness of Xas- Consider the straight line path so(!) = (1 - l)y + lr. It is absolutely continuous, (0) = y, and s0(1) ))- = Its derivative is (() = y a.e., with 1| (4) | yll
Since M 2 || y yl22. we have ||=6 I|x - vlI2 S M. Thus, 8o E XMs, and the feasible set nonempty.
21

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2) Compactness of Xas (in C(O0, 11. We apply the Arzelà-Ascoli theorem by verifying that MM closed, uniformly bounded. and equicontinuous.
Uniform Lipschitz Bound: The condition I| S M implies that every path s E AM is Lipschitz
continuous with uniform Lipschitz constant M, ie., [|s(t)) s(t2) I M tzl for all 1,,12 E 10.
Closedness: Let { (5)) C AM be a sequence converging uniformly to 8 E C( 1]. R"). i.e. ||8n 0|100 - 0. The limit function is continuous. Pointwise convergence implies s(0) lim 8n (0) = and s(1) = lim,- 8y (1) I. Taking the limit in the Lipschitz condition [[s,(ti)- s,, (t2)ll2 S M1t - t2l yields [|s(ta) (o(ti)=((ta)))) s(t2)ll2 S M| -t t2 for all t1, 12 E [0,1]. This shows s is M-Lipschitz continuous. By Rademacher's 's theorem, a Lipschitz continuous function is differentiable almost everywhere, and at points where it is differentiable, its derivative st) satisfies (s)(l) I M. This implies that is absolutely continuous and its supremum norm satisfies Il'll S M. Thus, s E XM. proving that Xas is closed in C(OO, C( 1).
Uniform Boundedness: For any XM and [0. [|s(t]]| I ||s(0)]l2 + [|s(t) s(0) s l|yl|2 + +Mtt 0| & llvlla + M. Hence, | I llllla + M for all E Xs, making the set uniformly bounded.
Equicontinuiry: Since all functions in Xas share the same Lipschitz constant M, the family Nas is equicontinuous. Specifically, for any e > O, choose d = = (M > 0). Then for all 8 E Xas. 141 /2] < 5 implies [|=(4) a(12)]l2 I M14 [2] < M6 =6
By the Aizelå-Ascoll theorem, Xa is compact in C((O, 1], R"). 3) Continuity of the Functional [s}] Let 8n + 8 uniformly in C((0, 1], R"). Then
[[sp]] [[s]] S
[1s)(()) - u1l - - y122ddt
[. (1)(1) s(t), (t) + s(t) 2y)[dt
$ / [|),(t) s(t) (t) (t) + s(t) 2yll2dt.
Since } is uniformly bounded, [|s,.()) + s(t) - 2ylla is bounded by some constant C², Therefore,
[[sm.] [[s] C² S/ [lsn(1) s(1]l1dd C² S, ||sm sI C" || - -lloc-
As [|s, - s sloc + 0, we have [[ss]] [[s]] + 0. Hence, [[s] is continuous on C((0, 1], R") with the supremum norm.
4) Existence Conclusion. By the Weierstrass Extreme Value Theorem, a continuous real-valued functional (7(s) defined on non-empty compact set (Xx)) attains its minimum value that set. Therefore, there exists at least one optimal solution 8 E X'M for problem (13).
Part 2: Geometry of the Optimal Trajectory
Let Xas be an optimal solution, which exists by Part 1. We show that its geometric trajectory must be the straight line segment connecting and x. Let I denote this line segment:
{2 R
Es E [0. 1]. 2 (1 s)y + sx).
For any t E [0, 1J. let (t) = proj (s* (t)) be the orthogonal projection of s" s*(() onto 1.
Assume, for the sake of contradiction, that the geometric trajectory of s* is not the line segment I. This implies that the set T = {t E [0. 1] st) & 1} has a positive Lebesgue measure. For t E T, s* (1) o (l). By the property of orthogonal projection (Pythagorean theorem), for any E [0, 1]:
(t) v13 | (t) q() + llq(t) -
Since (t) - qBt)lB 2 0, we have (t) yl z (]q(t) - y1l3 for all t E [o. 1]. Crucially, for t T, since s" (t) A qt),, we have [|s((t) (t) q(t)ll 0, which implies the striet inequality:
(t) -11 > lla(t) -11.. E.
22

---

Now, we integrate over [0, 1] obtaining
1|s²| = / (t) allaet
1 [ts - -yllact + / p.NVT [ts - -ylldd
/ lla(t) vlğdt + f 10.133 lla(t) y aliat
f. (la(t) wll2dt = [[g].
where the inequality is because 7 has a positive Lebesgue measure. Thus, we have [[s*] > [[].-
Next, verify that the projected path (t) is feasible, i.e., 9 XM-
Boundary Conditions: Since s" (0) E 7 and s* (1)
I, their projections are themselves:
q(0) = (y)j = y and q(1) = proj / (z) x.
Absolute Continuity and Speed Bound: The orthogonal projection onto a convex set (like the line segment or the line containing it) is I-Lipschitz (non-expansivee, i.e., (Iproi,(a) (b)i( S ||a b1|2- Therefore,
(l((t)) ((t2)ll2 = (s - (5 & (t1)
(t2)ll2-
Since s* E XM is M-Lipschitz, q(t) ) is also M-Lipschitz. This implies g(t) is absolutely continuous and almost everywhere differentiable, its derivative satisfies |/ (t) M almost everywhere, hence, 11 M.
Thus, ((t) belongs to the feasible XM.. We have now found feasible path q E XM such that T[g] <[s] This contradicts the assumption that s* is an optimal solution (minimizer) for problem (13). Therefore, the geometric trajectory of any optimal solution s" must be the straight line segment connecting and x.
A.3 Proof of Theorem A.2
We make the assumptions: Al) Measures IP and Q have compact support, and the parameter set
K IRd non-empty and compact. A2) T - To pointwise whenever on. - á and there exists h E L L'((0, 1l) such that le(T%,, (z). v)1 S htt) for all n and t E [0, 1]. A3) supØek T co.
A4) For fixed y, the cost function c(y. -) is continuous in its second argument.
Theorem A.2 (Existence of Minimizer for KIDOT Objective). Suppose the assumptions Al-A4 hold, then there exists a minimizer for the optimization problem
inf J( Q) = EiNe V." (TT) dt + AW1(To) (P).Q).
(14)
Øek
Proof: Let (To.) } be minimizing sequence such that
lim J(To.. 1A, =nf A,P.) n-000
(15)
Since on E K, where K is compact and finite- dimensionnnii there exists a convergent subsequence, up to sub-sequences, (still indexed by n) such that én, - 6² for some o" E K. By assumption, this implies pointwise convergence:
To (v)
(x-())
(v) for all E. [0
We next show that T.. is minimizer of (14), By the continuity of c(y,-) -) in its second argument, have
[ c(,, ccy, Th. (v)) + S'' cv. (v))
23

---

Moreover, since cv. To (v)) is uniformly bounded by an integrable function h(t), the Dominated Convergence Theorem yields
lim Er V c(v, Tan To.
(v))at) = Exr Bo clu. (o,TT(()))t
(16)
1-000
Next, for any up (R*).,,, we observe
If (()) p. ) / d(x)) 4PP & / (v)) ((T (v)) dP(y) - 0,
which implies that (Tg, #P converges narrowly to (Tg))
Additionally, since supØek To < co, we apply Dominated Convergence Theorem again to obtain
/ | ((T 2 / d( )1PP + O.
By [53, Theorem 5,1 narrow convergence and convergence of first moments together imply that W2 ) (Tx.) 0.
By the triangle inequality, follows that
lim Wi =) w1 PT) 1-+00
(17)
Combining Eq. (15), (16), (17), we obtain
J[T P) Ep U%: clu Ty- (v))dd) + (Q))))
lim Ep [ c(u. To. (w)) + +AW, (Q, (T.))))
(16217)
n-
lim J(T 0.. P) n-+00
(15)
inf (T 1P)
This completes the proof that To minimizer of (14).
B Additional Experiments
B.1 Implementation Details and Baselines
This section introduces the compared methods and detailed settings in experiments.
Implementation details. We trained distinct KIDOT models tailored each specific datasets (simulated MRI data, real prospective MRI data and clinical prospective CT data). Optimization utilized the RMSProp algorithm with differential learning rates: x 10 for the transport network (T6) and 2 x 10 for the potential network (4po). The inner iteration parameter N, was consistently set to 1. A learning rate decay schedule applied factor of 10 reduction after each block of 30 training epochs. The underlying architecture for T was adapted based on the dataset: using the E2E- VanNet backbone [12] for simulated MRI data, the PromptMR backbone [13] for real prospective MRI data, and the MC-CDic backbone [50] for clinical prospective CT data. Experiments were performed using Py Torch on an NVIDIA 4090 GPU. The source code will publicly released following possible publication.
24

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Representative compared methods. In MRI reconstruction experiments, we conducted compar- isons against several existing methods, including the traditional CS- CS-wavelet [45], two representative deep learning unfolding networks, F22E-VarNe [12] and PromptMR [13], diffusion-based method (DDS [47]) and three comparative methods based on OT: OT-CycleGAN [35], UAR [37] and FSBM [41].
For CT reconstruction, our set of comparison methods consists of the traditional BM3D [51], two representative deep learning methods (ISTA-Net [50], an unfolding network, and REDCNN [52]), and three OT-based approaches, OT-CycleGAN [35], UAR [37] and FSBM [41].
Motivation for Transport Network Implementation Our motivation for the implementation of the transport network, Ta is guided by three core principles: leveraging task-specific knowledge, ensuring fair comparisons, and demonstrating the broad applicability of our KIDOT framework.
First, we utilize backbones incorporated with task-specific knowledge. In medical image reconstruc- tion, different imaging modalities, such as MRI and CT, possess unique physical properties and data characteristics. For example, MRI heavily relies on multi-coil acquisition, where effectively leveraging the physical information from coil sensitivity maps is often critical for optimal reconstruc- tion. Meanwhile, the imaging mechanism in CT and MRI are different. CT images are produced by Radon Transform and MR images are produced by Fourier Transform. This necessity to incorporate modality-speccifc knowledge has led the research community to develop specialized, state-of-the-an architectures for each task. To evaluating the performance of our approach for each task, we adopt the state-of-the ttthhe-rrt art architectures for each task. Note that, for MRI, 12EV-arNet more light- weight than PromptMR, and we use Var-et for simulated MRI for the sake of efficient computation.
Second, we ensure fair comparison in each experiment. For any given task, the transport network in our KIDOT framework uses the same backbone architecture as the corresponding supervised baseline it is compared against. For example, when comparing with VarNee the simulated MRI dataset, our KIDOT model also uses an E2E- VarNet-based architecture for its transport network T
Finally, this approach allows us demonstrate the versatile usage of KIDOT on different backbones. By deliberately choosing backbones that are identical or highly similar to the supervised basclines, we effectively isolate the contribution of our framework. This ensures that the observed performance gain is attributable to our novel knowledge-inforet rather than simply employing a more powerful (and different) network architecture.
dynamic optimal transport formulation itself,
B.2 Datasets Details
Simulated MRI data. All images were center-cropped to 320x320 pixels. We simulated undersam- pled measurements by applying k-space masks equivalent to a 4x acceleration factor. A key aspect of this evaluation was to assess KIDOT's robustness to scenarios where test data characteristics differ from training data, a common challenge in prospective clinical deployment. To achieve this, generated two distinct sets of undersampled inputs: one set, with specific sampling patterns, was used for training supervised baselines, while different set of sampling patterns was employed for testing to mimic prospective acquisition conditions. Specifically, the test mask was generated by randomly flipping 3% of the entries in the training mask. Examples of these distinct masks are shown in Figure. 5.
Real prospective MRI data. For real prospective MRI data experiments, we used a multi-contrast MRI dataset acquired on a 3.OT United Imaging scanner equipped with a 32-channel receiver head coil. We selected the T2-FLAIR weighted sequence from this dataset. The T2-FLAIR images were acquired using a 2D Inversion Recovery (IR) sequence with the following parameters: TR=6000 ms, TE-396.44 ms, and volume size 256x256x 176. The dataset was partitioned into training, validation, and test sets consisting of 3077, 1860, and 3077 slices, respectively.
Clinical prospective CT data. For our clinical prospective CT reconstruction evaluation, we utilized dataset of real Low-Dose CT (LDCT) and Normal-Dose -Doee CT (NDCT) images. Data acquisi- tion was approved by the Institutional Review Board of the xxx hospital Each subject underwent triple-phase CT acteiaar portal venous, and delayed-on a United Imaging Healthcare (UIH) uCT960+ 320-slice helical scanner (rotation time 0.5 s/rotation, pitch 0.9937, collimation 80 mm; reconstructed via UIH ulnnovation-CT Explorer R001). Scans exhibited similar anatomical structures
25

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Fully sampled image
Retrospective image
Prospective image
image
Rerropective
Prospective
Visualization of simulated imagres
[b] Visualizationn ion of sampling mask
Figure 5: (a) Visualization of simulated images: the first column shows fully sampled images, the second column is used for supervised learning. the third column represents the undersampled prospective data, and the fourth column shows the residuals (difference between supervised and prospective images). (b) Visualization of simulated images:the first row the retrospective mask, and the second row is the prospective mask.
and consistent slice counts but differed in contrast enhancement. The dataset involved 30 consented patients with breathing training. Bolus tracking was used (abdominal aorta, 150 HU threshold). NDCT scans (head-to-foot, 100 kVp, 213 mAs DL 2) were triggered 16s, 50s, and 120s post- threshold. LDCT scans (foot-to-head, 80 kVp, 47 mAs, 1/10th NDCT dose) followed immediately. Slice counts varied (98-645), yielding 28,251 512x512 abdominal LDCT/NCCT images. We used a subset of 2420 pairs.
Given the potential for anatomical misalignment real prospective acquisitions (due to different scan directions/timings)) we also generated perfectly-pairet Simulated Low-Dose CT (SLDCT) images by adding noise in the projection domain. We inject compound Gaussian and Poisson noise into the uncorrupted projections by using the method in [48, 49]. The incident scale photon flux is 40000. We adopt a fan-beam geometry to simulate the uncorrupted projections from the CT images. The geometrical parameters for projection were: nourceto-issoeener distance 570.0 mm, source- to-detector distance 962.9 mm, image size 512 x 512 (with pixel spacing 0.6934 0.6934 mn?), detector bin number 864, detector bin width 1.0336 mm, and 1200 projection views acquired over 360-degrec orbit.
To visualize the characteristics of these different CT data types and the discrepancy between our simulated low-dose data and real clinical low-dose data, we provide examples in Figure From left to right, the figure shows: the normal dose image, the generated simulated low dose image, corresponding real low dose image from our prospective dataset, and the residual image calculated as the difference between the SLDCT and LDCT images (SLDCT- LDCT). This residual highlights aspects of real low-dose noise and artifacts not fully captured by our simulation model.
B.3
Additional Ablation Studies
B.3.1 Effect of Different Number of Discretization Steps N
We conduct an ablation study to investigate the sensitivity of our method to the number discretiza- tion steps N, which is a key hyperparameter in our framework. The choice of N represents a critical trade-off between the fidelity the ODE approximation and the associated computational cost. A smaller N can lead to coarse and inaccurate discretization of the continuous transport path, resulting in suboptimal reconstruction quality. Conversely, a large N increases the computational burden, as inference time and memory grow linearly with N. and also introduce training instability due to the risk of gradient exploding or vanishing in deeper backpropagation paths.
This analysis is performed the real prospective MRI data with 10x acceleration factor to find the optimal balance. The results are reported in Table As shown, the reconstruction quality, measured by both FID and KID, consistently improves N increases from 6 to 12, achieving the
26

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Norm dose
Simulated low dose
Real low dose
Residual image
-0.
0.3
0.3
0.0
-01
Figure 6: Comparison of Normal dose, Simulated low dose, and Real low dose CT Images with Residual images calculated as the difference between the SLDCT and LDCT images.
optimal results at N = 12. However, when we further increase the number of steps to N = 15, the performance begins degrade. This degradation is likely attributable the increased possibility of gradient soolodin/vvaiihingg during training. Based on this analysis, we fix N 12 for our experiments.
Table Comparison of different number of discretization steps N on the Real Prospective MRI Data (10x acceleration). Best results are bolded.
discretization steps N
FID 4
KIDx 100 4
6
35.31
2.01
9
30.06
1.62
12
28.26
1.12
15
29.23
1.31
B.3.2 Sensitivity to the Supervised Trade-off Parameter
We analyze the sensitivity of KIDOT the supervised trade-off parameter using simulated MRI data (4x acceleration). As shown in Table 6, the choice of x impacts KIDOT's performance metrics. We observe general trend where increasing from 1 leads to consistent improvements in reconstruction quality (higher PSNR and SSIM) and fidelity (lower FID and KID). The best results across all evaluated metrics achieved when 10a. Further increasing the trade-off parameter results in a slight degradation across all metrics compared to 10a. Based on this result, we fix 7 10a in experiments.
10a
Table 6: Comparison of different trade-off off parameter y on the Simulated MRI data (4x acceleration).
Best results are bolded.
tra parameter y
PSNR t
SSIM +
FID
KIDx 100 +
30.87
0.7815
54.82
2.47
10
31.71
0.7987
37.74
1.52
102
32.28
0.8204
29.52
1.38
103
33.04
0.8408
17.23
1.01
104
33.31
0.8518
10.68
0.83
105
33.10
0.8490
15.39
0.97
27

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B.3.3 Effect of Different Lagrange Multiplier À
We investigate the effect of the Lagrange multiplier À KIDOT's performance using simulated MRI data (4x acceleration). Table summarizes the results for different values of A. We observe that the choice of À has impact on both reconstruction quality and fidelity metries. Notably, the optimal performance across all evaluated metries (highest PSNR/SSIM and lowest FID/KID) is achieved with a Lagrange multiplier of À 1. Increasing À beyond this value leads to consistent and substantial degradation in performance. As increases to 102, PSNR and SSIM decrease, while FID and KID> 100 values increase drastically, indicating considerable loss of image fidelity and introduces more artifacts. Based on these results, we fix À = 1 for our experiments.
Table 7: Comparison of different Lagrange multiplier À on the Simulated MRI data. (4x accelera- tion). Best results are bolded.
Lagrange multiplier
PSNR t
SSIM f
FID 4
KIDx 4
0.5
33.07
0.8469
18.43
1.01
1
33.31
0.8518
10.68
0.83
2
32.94
0.8457
20.14
1.07
5
32.87
0.8436
26.69
1.25
27.62
0.8132
113.43
6,21
10a
26.15
0.7518
178.59
7.33
B.3.4 Effect of the Physical Forward Operator AA in Transport Equation
We conducted an ablation study to evaluate the importance of explicitly incorporating the physical forward operator A within our KIDOT framework. This comparison was performed on the simulated MRI data with 4x acceleration factor. The setting "w/o A" indicates that the physical forward operator was removed from the transport equation, the network architecture Ta- As summarized in Table 8, including the physical forward operator A leads to substantial improvement in reconstruc- tion quality and fidelity. Comparing the setting "w AA to "w/o A", we observe that incorporating A results in notable increase in PSNR (33.31 vs 30.88) and SSIM (0.8518 vs 0.8 0.8174). The fidelity metrics are improved, with FID decreasing from 36.53 to 10.68, and KID x 100 decreasing from 1.48 to 0.83, These results clearly demonstrate the crucial role of the physical forward operator in guiding the reconstruction process and highlight its contribution to KIDOT's superior performance.
Table Results of the method w/wo physical forward operator on the Simulated MRI data (4x acceleration). Best results are bolded.
Setting
PSNR f
SSIM f
FID 4
KIDx 100 +
W A w/o A
33.31
0.8518
10.68
0.83
30.88
0.8174
36.53
1.48
B.4 Statistical Significance Analysis
We perform statistical significance tests compare baseline methods against our proposed KIDOT method on the simulated MRT dataset with 4x acceleration. Table 9 summarizes the p-values from these tests. Each test compares the results of baseline method directly against KIDOT.
For image-level quality metrics (PSNR and SSIM), p-values are computed via paired t-tests on the per-image metric values. Specifically, for each image, obtain the metric values from all runs for both methods, then conduct paired t-tests across the set of images, thus accounting for within-image variability and dependencies between paired samples.
For distribution- -basd fidelity metrics (FID and KIDx 100), which are traditionally computed once dataset, we obtain multiple samples for statistical testing using bootstrap resampling procedure over the test images within each run. The FID and KID scores computed on these bootstrap samples form the basis for independent two-sample t-tests comparing baseline methods with KIDOT.
28

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As shown in Table 9, all p-values fall below the conventional significance threshold of 0.05. Notably, comparisons against CS-wavelet, O2E-VarNet, and OT- C..ClelGANN yield p-values < 0.001 across all metrics, indicating strong statistical significance. For UAR, p-values range from 0.009 (KIDx 100) to 0.024 (PSNR), still demonstrating significant improvements by KIDOT, These results collectively confirm that KIDOT outperforms all tested baselines on the simulated MRI dataset with high statistical confidence.
Table 9: Statistical significance (p-values) of baseline methods compared to KIDOT on the Simulated MRI data (4x acceleration). Lower p-values indicate more significant differences. P-values are from independent t-tests.
Method
PSNR
SSIM
FID
KID 100
CS-wavelet
< 0.001
0.001
<.0001
< 0.001
E2E- VarNet
< 0..01
< 0.001
< <.0001
< 0.001
OT-CycleGAN UAR
< 0.001
< 0.001
< 0.001
< 0.001
0.024
0.001
< 0.001
0.009
B.5 Training Cost Curves for Three Datasets
In Figure 7, we display the cost curves over three datasets (i.e., simulated MRI data, real prospective MRI data and clinical prospective CT data) of To and (p in the training process. The To cost curve is normalized to [0, 1]. Se cost is scaled [0, 1] and then take the negative.
Simulated MRI data
Real prospective MRI data
Clinical prospective CT data
1.
04
52
0.4
30
Figure 7: The training cost curves for datasets. The cost of To is scaled to [0, 1]. The cost of yoo scaled to [0, 1] and then takes the negative.
B.6 Performance in Low-data Regimes
To evaluate the robustness of our method under data scarcity, we conduct an additional ablation study in low-data regime. We intentionally restrict the amount of available training data by using reduced subsets of the simulated MRI dataset. Specifically, we train our model and the baselines on only 20% and 10% of the original paired data.
The results are presented in Table 10. both settings, our KIDOT framework consistently outper- forms the E2E- VarNet and UAR baselines across all metrics. This improved performance under data scarcity can be directly attributed to the regularization imposed by KIDOT's physics-informed constraints. Unlike purely data-driven methods that are prone to overfitting small datasets, our framework's adherence to the physical model provides robust inductive bias, significantly reducing its dependency on the volume of training data and enhancing its generalization capability.
29

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Table 10: Performance comparison in low-data regimes on the Simulated MRI data. We train all methods using 20% and 10% of the original training paired data. Best results for each setting are bolded.
Data
Method
PSNR
SSIM
FID 4
KIDx 100
E2E-VarNes
30.97
0.7978
72.43
4.02
20%
UAR
31.30
0.8054
40.32
2.95
KIDOT (Ours)
31.75
0.8131
31.61
2.32
29.92
0.7681
92.41
6.07
10%
UAR
30.64
0.7857
87.07
5.68
KIDOT (Ours)
30.88
0.7952
73.14
4.11
B.7
Generalization to More Aggressive Out-of- sit-of-DDstriibtion (OOD) or Heterogeneous Data
To further investigate the robustness and generalization capabilities of our model, we conducted two challenging cross-domain experiments on the fastMRI dataset, evaluating performance on both cross-anatomy and cross-contrast tasks. These experiments designed to assess how well the models learn the fundamental principles MRI reconstruction, rather than simply memorizing features specific to the training data distribution.
Cross-Anatone (Knee - Brain). In this setup, the models are tasked with reconstructing brain supervised baseline (E2E-VarNet), we used paired
images after being trained on knee data. knee scans for training. For our unpaired method, KIDOT, we trained it using unpaired data consisting of high-quality knee scans and low-quality brain seans. Both models were then evaluated on their ability to reconstruct brain scans. As shown in Table 11, KIDOT significantly outperforms E2E- VarNet, demonstrating its ability to generalize anatomical knowledge.
Table 11: Generalization performance on the Cross-Anatomy (Knee bolded.
Brain) task. Best results are
Method
PSNR f
SSIM
FID 1
KIDx 100 4
E2E-VarNet
33.82
0.7880
80.21
4.24
KIDOT (Ours)
34.62
0.8145
26.87
1.53
Cross-Contrast (T1 - T2). Similarly, this experiment tests generalization across different MRI contrasts. The models were trained dataset of unpaired high-quality T1 T'l-weigtted brain scans and low-quality T2-weighted brain scans. The evaluation was then performed on the task of reconstructing T2-weighted brain scans. The results presented in Table 12. Again, KIDOT shows superior performance, indicating its robustness to variations in image contrast.
Table 12: Generalization performance on the Cross-Contrast (T1 - T2) task. Best results are bolded.
Method
PSNR
SSIM
FID |
KIDx 1DDx1001 4
D2E-VarNet
32.06
0.8229
31.29
1.42
KIDOT (Ours)
32.48
0.8348
29.74
1.33
The superior performance of KIDOT in both of these challenging OOD scenarios provides strong evidence for the benefit of its thysics--uided inductive bias. This bias encourages the model to learn generalizable reconstruction process grounded
principles of MR imaging, rather than overfitting to the specific anatomical or contrast features of the training domain. While standard supervised baseline performs well on in-domain data, it struggles when faced with a significant domain shift, its learned prior is purely data-driven and less generalizable.
30

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B.8
Additional Visual Results
B.8.1
Additional Visual Comparison on Simulated MRT Data
Figure 8 presents additional visual comparisons on simulated MRI data, highlighting that our method reconstructs clearer structural details with fewer artifacts in the regions of interest (red boxes).
Reference
CS-wavelet
OT-CyyllGGAN
UAR
KIDOT
Figure 8: Visual comparison on Simulated MRI data.
B.8.2 Additional Visual Comparison on Real Prospective MRI data
Figures 9 and 10 display more visual results of the compared methods.
Zero Filling
Reference
CS-wavelet
E2E-varnet
PromptMR
OT-CycleGAN
UAR
KIDOT
Figure 9: Visual comparison on Real prospective MRI data.
31

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Zero Filling
Refercace
CS-wavelet
E2E-vamet
PromptMR
OT-CycleGAN
UAR
KIDOT
Figure 10: Visual comparison on Real prospective MRI data.
B.8.3 Additional Visual Comparison on Clinical Prospective CT Data
Figures 11 and 12 displays additional visual results of the compared methods the clinical prospec- tive CT data. As shown in the regions of interest, our method (KIDOT) achieves a better trade-off' between noise suppression and structural detail preservation.
Low Dose
Reference
BM-3D
ISTA-Net
REDCNN
OT-CycleGAN
UAR
KIDOT
Figure 11: Visual comparison on Clinical prospective CT data.
32

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Low Dose
Reterence
BM-3D
ISTA-Net
REDCNN
OT-CycleGAN
UAR
KIXXT
Figure 12: Visual comparison on Clinical prospective CT data.
33


Reviews:
[{"review_id": "f1iY1517Wq", "reviewer": "Reviewer_YYCZ", "summary": "This paper introduces KIDOT, a dynamic optimal transport framework for medical image reconstruction that integrates imaging physics directly into both the cost function and transport dynamics. The method is designed to work with unpaired data and aims to address the retrospective-to-prospective domain gap, with experiments on MRI and CT showing improved perceptual and fidelity metrics.", "questions": "1.\t*Regarding weakness 1:* Can the authors provide proof for KIDOT’s superior performance in low-data regimes?\n2.\t*Regarding weakness 2:* Prospective image for simulated data was generated by flipping 3% of the entries in the training mask, essentially just changing the undersampling pattern slightly. Diffusion-based reconstruction methods, which are typically trained independently of the sampling mask, are known to be inherently robust to such variations. Would KIDOT still provide an advantage over these models under a more meaningful shift, and why are they not included in the comparison?\n3.\t*Regarding weakness 2:* Would a diffusion model trained on a large existing dataset, such as DDS [1] trained on fastMRI brain data, perform competitively on this prospective dataset at test time? If so, this would suggest that we can still collect prospective data in an unpaired fashion and reconstruct it using strong pre-trained priors, without the need for a method like KIDOT.  Exploring this would help clarify whether KIDOT provides a real advantage in clinical settings.\n4.\t*Regarding weakness 3:* Unpaired training is often motivated by the ability to use heterogeneous data sources, so how would KIDOT perform when the source and target domains differ more significantly, for example in contrast type, anatomy, or acquisition settings?\n\n[1] Chung et al., “Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems”, ICLR, 2024.", "limitations": "Yes, the authors have addressed the limitations in Section 5.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n\n1.\tThe supervision assumption is less restrictive than fully supervised approaches, as it does not require paired measurement-image data. This aligns well with real-world clinical scenarios, where perfectly aligned scans are often unavailable or impractical to obtain.\n2.\tTheoretical analysis is provided to support the formulation, including proof of existence of solutions under mild assumptions.\n3.\tThe method was tested in various settings, focusing on both simulated and prospective MRI data, and clinical prospective CT data.\n\n**Weaknesses:**\n\n1.\tKIDOT is claimed to work without paired data, but it still requires thousands of training slices based on the numbers reported in the paper. There is no evaluation of performance in low-data regimes, which are often the reality in clinical settings. This weakens the motivation for unpaired learning, especially since large paired datasets (such as fastMRI, mridata.org and others) already exist and are widely used.\n2.\tWhile KIDOT is proposed as a solution for unpaired and prospective data, the paper does not compare it to diffusion-based reconstruction methods that use strong priors trained on large datasets like fastMRI. Even though these models are trained with paired data, they are often robust to distribution shifts and can perform well on prospective data if the prior matches the anatomy or modality, which is commonly the case in practice.\n3.\tAlthough KIDOT is trained on unpaired data, the paper does not explore how well the model generalizes to more aggressive out-of-distribution (OOD) examples, such as different contrasts, anatomies, or acquisition protocols. Evaluating under such conditions would strengthen the paper’s premise, since the core motivation of KIDOT is to address the retrospective-to-prospective gap, which is itself a form of distribution shift.", "was_revised": true, "final_justification": "The paper presents strengthened empirical evidence and addresses key concerns, and contingent on the inclusion of the new comparisons and experiments in the final version, I find it suitable for acceptance.", "rebuttal": "We thank the reviewer for the insightful comments and suggestions on our paper. We address the concerns and questions in detail below.\n### **W1 & Q1: Clarification on motivation for unpaired learning and experiments in low-data regimes**\n\nWe clarify the \"paired data\" in medical image reconstruction datasets and the motivation of our method, along with the experiments in low-data regimes, respectively.  \n\n**1. Clarification of \"paired\" data in public datasets**\n\nThough there exist large paired public datasets like fastMRI, the \"pairing\" in these datasets is mainly synthetic. For example, in fastMRI, the low-quality (under-sampled) k-space data is generated by retrospectively simulating a sampling mask on fully-sampled k-space data. Similarly, in low-dose CT datasets like the Mayo Clinic Challenge, the low-dose images are created by simulating noise onto clean, standard-dose scans. These simulation are hard to fully replicate the complex, non-ideal characteristics of real-world prospective data acquisition, such as subtle patient motion between scans, system hardware instabilities, and specific noise distributions. Deep models trained on these \"idealized\" simulated data often suffer a significant performance drop when deployed in real clinical workflows.\n\n**2. KIDOT's motivation: leveraging abundant real-world data**\n\nWe clarify that the motivation for our KIDOT stems from a pragmatic clinical reality. Specifically, in clinical scenario, acquiring **perfectly registered, truly prospective paired data** (e.g., a low-dose scan and a standard-dose scan of the same patient at the exact same moment and position) is extremely difficult, often ethically questionable, and sometimes physically impossible. In contrast, it is **relatively easy and common** to collect large volumes of **unpaired** clinical data: a vast pool of low-quality scans (e.g., under-sampled MRI, low-dose CT) from some patients, and a separate, large pool of high-quality, fully-sampled scans from other patients.\n\nKIDOT is designed to leverage these two abundant but unpaired real-world data streams, bypassing the need for perfect, one-to-one pairing. KIDOT directly tackles the distribution shift problem by learning from the true data distributions encountered in clinical practice.\n\n**3. Performance in low-data regimes**\n\nWe first clarify that the paired training data in simulated datasets, such as fastMRI, is generated by retrospectively applying sampling masks to fully-sampled data. This means that, in principle, the supply of such paired data is abundant, limited only by the availability of fully-sampled scans. As suggested, we also evaluate our method when the source of simulated paired data is intentionally restricted. We conduct a new ablation study on the simulated MRI dataset, in which we train methods on reduced data subsets, 20% and 10% of the original training paired data. The results are presented in Table r1-1 and Table r1-2 below.\n\nTable r1-1: Performance comparison in a low-data regime, using 20% of the original training paired data.\n|Method|PSNR (↑)|SSIM(↑)|FID(↓)|KID*100(↓)|\n|:-|:-:|:-:|:-:|:-:|\n|E2E-VarNet|30.97|0.7978|72.43|4.02|\n|UAR|31.30|0.8054|40.32|2.95|\n| **KIDOT (Ours)**|**31.75**|**0.8131**|**31.61**|**2.32**|\n\nTable r1-2: Performance comparison in a low-data regime, using 10% of the original training paired data.\n|Method|PSNR(↑)|SSIM(↑)|FID(↓)|KID*100(↓)|\n|:-|:-:|:-:|:-:|:-:|\n|E2E-VarNet|29.92|0.7681|92.41|6.07|\n|UAR|30.64|0.7857|87.07|5.68|\n|**KIDOT (Ours)**|**30.88**|**0.7952**|**73.14**|**4.11**|\n\nTables r1-1/2 implies improved performance of KIDOT over baselines. This performance under data scarcity could be directly attributable to the regularization imposed by KIDOT's physics-informed constraints. Unlike purely data-driven methods that are prone to overfitting on small datasets, our framework's adherence to the physical model provides a robust inductive bias, reducing its dependency on the volume of training data.\n\nWe will add this new low-data ablation study to the Appendix. We will also include the discussions into the introduction and motivation sections in the paper.\n\n### **W2 & Q2 & Q3: Comparison with diffusion-based models** \n\nWe are sorry for missing the diffusion-based methods. As suggested, we have conducted experiments on both simulated and real datasets to compare with the diffusion method of Denoising Diffusion Sampler (DDS (Chung et al., ICLR 24)). \n\n**Experiment 1: comparison on simulated data**\n\nTo compare with diffusion-based reconstruction method, we first performed an experiment on **simulated fastMRI dataset**. The authors of DDS provide a publicly available model pre-trained on fastMRI, enabling a direct comparison between the two methods under the same setting. Table r1-3 shows KIDOT outperforms the pre-trained DDS model.\n\nTable r1-3: Quantitative comparison against the SOTA baseline DDS on the simulated fastMRI dataset.\n| Method | PSNR (↑) | SSIM (↑) | FID (↓) | KID*100 (↓) |\n|:- |:-:|:-:|:-:|:-:|\n| DDS | 31.27 | 0.8136 | 29.46 | 1.65 |\n| **KIDOT (Ours)** | **33.31** | **0.8518** | **10.68** | **0.83** |\n\n**Experiment 2: comparing on real prospective MRI data**\n\nFor the real prospective MRI data, a practical challenge arises when applying the DDS model pre-trained on fastMRI, to our clinical dataset due to a mismatch in image dimensions (square size（320x320）for fastMRI v.s. non-squre size (256x176) for our data). We tackle this by padding zeros. Another challenge is the inherent **distribution gap** between the fastMRI data and our real prospective MRI data, which necessitates a more nuanced evaluation. The existence of this gap makes it uncertain whether a pre-trained prior will be effective at all.  Therefore, to ensure a fair and physically valid evaluation, we adopted three practical strategies for comparison:\n\n1.  **DDS (Pre-trained):** The pre-trained DDS model was used directly. Our `256x176` images were zero-padded to `256x256`, processed, and then cropped back.\n2.  **DDS (Fine-tuned):** Starting with the pre-trained model used in (1), we then performed fine-tuning on our clinical dataset.\n3.  **DDS (Trained from scratch):** The DDS architecture was trained from the ground up on our clinical dataset.\n\nTable r1-4: Performance comparison of KIDOT against different application strategies for DDS on the real prospective MRI dataset.\n|Method|FID(↓)|KID*100(↓)|\n|:---|:---:|:---:|\n|DDS(Pre-trained)|62.94|2.71|\n|DDS(Fine-tuned)|39.78|1.84|\n|DDS(Trained from scratch)|58.32|2.53|\n|**KIDOT (Ours)**|**28.26**|**1.12**|\n\nTable r1-4 shows the comparison of KIDOT against different application strategies for DDS on the real prospective MRI dataset. The performance of the DDS (Pre-trained) confirms the existence of the gap between the fastMRI data and our real prospective MRI data. DDS (Fine-tuned) improves performance, but it still lags behind KIDOT. Interestingly, DDS (Trained from scratch) on this dataset performs even worse, suggesting that our authentic clinical dataset may be insufficient for a data-hungry model like DDS to learn a robust prior from scratch. KIDOT achieves superior performance. Its physics-guided dynamics provide a powerful inductive bias that may be both more data-efficient and better aligned with the reconstruction task.\n\nWe will add these experiments into Tables 1-2 and the detailed analysis into the Experiments section of the paper. \n\n### **W3 & Q4: Generalization to more aggressive OOD or  heterogeneous data**\n\nAs suggested, we investigate the performance on cross-anatomy and cross-contrast experiments on the fastMRI dataset.\n\n*   **Cross-Anatomy (Knee → Brain):** In this setup, models are trained on the knee anatomy domain. Paired baselines utilize paired knee scans for training. Our unpaired method is trained using unpaired high-quality **knee scans** and low-quality **brain scans**. The evaluation is then performed on the task of reconstructing **brain scans**.\n*   **Cross-Contrast (T1 → T2):** Similarly, for this task, models are trained on a dataset of unpaired high-quality **T1W brain scans** and low-quality **T2W brain scans**. We then evaluate their generalization performance on **T2W brain scans**.\n\nThe results from these experiments, presented in Table r1-5 and Table r1-6, provide quantitative evidence for KIDOT's physics-guided inductive bias. This bias encourages the model to learn a generalizable reconstruction process grounded in the principles of MR imaging, rather than simply memorizing the anatomical features of the training domain. We anticipated that a standard supervised baseline, while performant on in-domain data, would struggle when faced with a shift in anatomy, as its learned prior is purely data-driven.\n\nTable r1-5: Generalization performance on the Cross-Anatomy (Knee → Brain) task.\n| Method | PSNR (↑) | SSIM (↑) | FID (↓) | KID*100 (↓) |\n|:- |:-:|:-:|:-:|:-:|\n| E2E-VarNet | 33.82 | 0.7880 | 80.21 | 4.24 |\n| **KIDOT (Ours)** | **34.62** | **0.8145** | **26.87** | **1.53** |\n\nTable r1-6: Generalization performance on the Cross-Contrast (T1 → T2) task.\n| Method | PSNR (↑) | SSIM (↑) | FID (↓) | KID*100 (↓) |\n|:- |:-:|:-:|:-:|:-:|\n| E2E-VarNet | 32.06 | 0.8229 | 31.29 | 1.42 |\n| **KIDOT (Ours)** | **32.48** | **0.8348** | **29.74** | **1.33** |\n\nDue to space limit, we will include this experiment to Appendix."}, {"review_id": "63HMH8FNko", "reviewer": "Reviewer_kNxi", "summary": "This paper introduces KIDOT, a novel framework for prospective medical image reconstruction. The core idea is to frame reconstruction as a dynamic optimal transport (OT) problem, where the evolution from a degraded measurement to a high-quality image is modeled as a continuous path. Crucially, this transport path is guided by imaging physics, incorporated into both a novel instantaneous cost function and the transport dynamics equation itself. Experiments demonstrate its superiority on simulated and real-world prospective MRI and CT data.", "questions": "1. The authors performed experiments on real-world CT datasets and described their data collection process. Will these datasets be made publicly available? If not, are there any existing public datasets that can be used to further evaluate the method to improve reproducibility?", "limitations": "Yes.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. This paper is well-motivated and tackles a critical and practical problem in prospective medical image reconstruction. The proposed method, Knowledge-Informed Dynamic Optimal Transport (KIDOT), is novel and logically sound. It offers a more principled, physics-grounded approach.\n2. This paper clearly introduces and validates the proposed theory, grounding the method in the theory of dynamic optimal transport. The formulation of the imaging-informed cost function and the physics-guided transport equation is elegant.\n3. Extensive experiments are conducted, and the proposed method demonstrates state-of-the-art performance. The ablation studies are comprehensive and effectively validate the contribution of their designs.\n4. This paper validates the proposed method on both simulated and real, prospectively acquired MRI and CT datasets, which demonstrates the robustness and practical relevance of the method.\n5.  This paper is very well-written. The motivation is clearly articulated, and the complex methodology is broken down into understandable components. \n\nWeaknesses:\n1. The authors do not explain how the number of discrete time steps is determined for the experiments. And what is the impact on the performance when N increases? Is there a trade-off between the number of steps and reconstruction quality?\n2. Although the proposed method achieves the best performance among the compared baselines, most of the compared methods are relatively outdated. It would be beneficial to include comparisons with recent optimal transport-based methods, such as Feedback Schrödinger Bridge Matching.\n3. A discussion comparing the strengths and weaknesses of the proposed method relative to recent optimal transport-based methods would further clarify its advantages and limitations, such as the high relevant FSBM.", "was_revised": true, "final_justification": "The paper proposes a novel and well-motivated approach, Knowledge-Informed Dynamic Optimal Transport (KIDOT), for prospective medical image reconstruction, which addresses an important practical problem with a principled, physics-grounded method. Extensive experiments, including real clinical data, support the effectiveness of the approach. The authors’ rebuttal satisfactorily addressed my concerns regarding experimental validation and provided further comparative analysis. While a few minor issues remain, they do not affect the overall strength of the contributions. Based on these points, I have increased my score and recommend acceptance.", "rebuttal": "We thank the reviewer for the insightful comments and suggestions on our paper. We address the concerns and questions in detail below.\n\n### **W1: Determination and impact of the number of discrete time steps ($N$)**\n\nIn our work, the number of steps $N$ was treated as a hyperparameter. It was determined through a search on a validation set, with the objective of finding the optimal balance between the fidelity of the ODE approximation and the associated computational cost. \n\nThe choice of $N$ represents a critical trade-off. Smaller $N$ can lead to a coarse and inaccurate discretization of the continuous transport path, resulting in suboptimal reconstruction quality. Too large $N$ may increase the computational burden (inference time and memory grow linearly with $N$). Meanwhile, larger $N$ can also potentially lead to instability of training, because the possibility of gradient exploding/vanishing in backpropagation increases for larger $N$. We report the results on the real prospective MRI data (10× acceleration) with varying $N$ as follows:\n\n\nTable r2-1: Ablation study on the number of discretization steps *N*.\n\n| discretization steps *N* | FID ↓ | KID×100 ↓ |\n|:------------------------:|:-----:|:---------:|\n| 6                        | 35.31 | 2.01      |\n| 9                        | 30.06 | 1.62      |\n| **12**                   | **28.26** | **1.12**  |\n| 15                       | 29.23 | 1.31      |\n\nAs shown in Table r2-1, the reconstruction quality, measured by both FID and KID, consistently improves as $N$ increases from 6 to 12. Our method achieves the best results when $N=12$. When we further increase the number of steps to 15, the performance begins to degrade, which may be due to the increased possibility of gradient exploding/vanishing in training.\n\n\nWe will add this ablation study and the accompanying discussion to the Appendix.\n\n\n### **W2 & W3: Comparison with recent optimal transport-based methods of FSBM**\n\nWe first compare the methodology of FSBM and KIDOT, and then perform experimental comparison of them. \n\n**1. Methodology comparison of FSBM and KIDOT**\n\n**Summary:**\nFSBM tackles the general semi-supervised generative modeling where datasets are partially aligned. It leverages a small subset of pre-aligned \"key-points\" provide feedback and guide the transport path for unpaired data. FSBM utilizes a distance-preserving objective to build a dynamic transport process in Schrödinger bridge framework. \n\nKIDOT aims at tackling the prospective medical image reconstruction problem. KIDOT models the medical image reconstruction problem as an dynamic transport problem from the low-quality images to the high-quality images. KIDOT explicitly embeds the medical imaging knowledge into the transport cost and equation, forming an new knowledge-informed dynamic optimal transport framework. \n\n**Difference:**\nThough both FSBM and our proposed KIDOT tackles data distribution transform using dynamic optimal transport, they are different in both framework and techniques. The core of FSBM is to guide the translation of unpaired data by the paired key-points using an developed feedback model. While our KIDOT aims to incorporate the imaging knowledge into the transport model by designing the knowledge-informed transport cost and equation, for prospective medical image reconstruction. \n\n**Advantages and limitations of KIDOT:**\nAs analyzed above, when the imaging knowledge is clear, our KIDOT has advantage for the task (as verified in experiments), because the imaging knowledge provides inductive bias that can be utilized in KIDOT. However, when the knowledge is not clear or mismatched, the performance of KIDOT could be limited.\n\n**2. Experimental comparison**\n\nTo provide a comparison, we implement FSBM and evaluate it across all three of our experimental datasets. Following the setup in the FSBM paper for high-dimensional data, we perform the transport in the latent space of an autoencoder. For MRI experiments, we utilize the publicly available microsoft/mri-autoencoder-v0.1 model on Hugging Face. For CT experiments, we train a new autoencoder from scratch. This model is trained on our CT dataset, ensuring the learned latent representations are optimally tailored to the target domain. To provide FSBM with the necessary 'key-point' pairs for its semi-supervised guidance, we use our simulated paired data as the ground truth. The trajectories for these key-points were generated via linear interpolation between the paired samples, providing the guidance FSBM requires.\n\nTable r2-2: Performance comparison on simulated MRI data\n| Method | PSNR (↑) | SSIM (↑) | FID (↓) | KID*100 (↓) |\n|:--- |:---:|:---:|:---:|:---:|\n| FSBM | 26.52 | 0.7634 | 113.62 | 6.03 |\n| **KIDOT (Ours)** | **33.31** | **0.8518** | **10.68** | **0.83** |\n\nTable r2-3: Performance on real prospective MRI data\n| Method | FID (↓) | KID*100 (↓) |\n|:--- |:---:|:---:|\n| FSBM | 147.62 | 7.13 |\n| **KIDOT (Ours)** | **28.26** | **1.12** |\n\nTable r2-4: Performance on clinical prospective CT data\n| Method | FID (↓) | KID*100 (↓) |\n|:--- |:---:|:---:|\n| FSBM | 68.42 | 3.61 |\n| **KIDOT (Ours)** | **22.25** | **1.32** |\n\nTable r2-2, Table r2-3 and Table r2-4 show the results. The observed difference in performance can be attributed to KIDOT's explicit integration of the physical model. This provides a strong, task-specific inductive bias that directly constrains the reconstruction to be consistent with the measurement $y$. In contrast, FSBM, as a general-purpose generative model, does not have an inherent mechanism to enforce this specific physical constraint, making it less suited for the high-fidelity demands of this inverse problem.\n\nWe will incorporate this comprehensive comparison—including the conceptual distinctions and the new experimental results—into the **Related Work** and **Experiments** sections of our paper.\n\n### **Q1: Data and code availability for reproducibility**\n\nThe clinical prospective CT dataset was acquired under a strict Institutional Review Board (IRB) protocol, which places necessary restrictions on the public release of the full patient cohort to protect privacy. However, we recognize the immense value of providing real-world data to the community. Therefore, to the fullest extent permitted by our IRB, **we will release a representative, fully anonymized subset of this prospective CT dataset.** This subset will contain low-dose and standard-dose scans and will enable other researchers to directly evaluate their methods on the same challenging, real-world data distribution we addressed.\n\nWe will also release our source code when the paper is public to ensure reproducibility. We will also add a statement to our paper clarifying these points."}, {"review_id": "lcib4rtzjW", "reviewer": "Reviewer_JZ2H", "summary": "This paper proposes a new framework for prospective medical image reconstruction. It frames reconstruction as a dynamic transport process, and its learned path is explicitly constrained by imaging physics. Theoretical analysis is performed to validate the existence and optimality of the solution. Experiments on simulated and real MRI/CT datasets demonstrate the effectiveness of the proposed method.", "questions": "In Table 4, what is the difference between the last two rows, namely (3) and (4). This is a little confusing. Please make it more clear.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. The formulation of prospective medical image reconstruction based on dynamic transport process is reasonable and intriguing.\n2. The proposed framework is supported by sufficient theoretical analysis, which builds a solid basis for following development.\n3. The proposed method is compared with existing methods on simulated and real MRI/CT datasets, showing the effectiveness of the proposed method.\n4. The authors promise the open access to the real prospective datasets, which can benefit the medical imaging community.\n5. The paper organization is clear.\n\nWeaknesses:\nMy concern is mainly about the transport network implementation. The authors use different implementations for the transport network on different tasks, which might be unfair to other comparing methods to some extent. What if using a fixed implementation of the transport network?", "was_revised": true, "final_justification": "The proposed medical image reconstruction framework is justified by sufficient theoretical analysis and experimental evaluation. The rebuttal also addressed my concerns. Hence I recommend to accept this manuscript.", "rebuttal": "We thank the reviewer for the insightful comments and suggestions on our paper. We address the concerns and questions in detail below.\n\n### **W1: Motivation and fairness of transport network implementation**\n\nThanks for this suggestive question. We clarify that our motivation for network implementation is to use the task-specific and state-of-the-art backbones for each task (e.g., E2E-VarNet for simulated MRI, PromptMR for prospective MRI, ISTA-Net for CT), meanwhile ensuring fairness. We elaborate the motivation as follows.\n\n**Using backbones incorporated with task-specific knowledge:**  In medical image reconstruction, different imaging modalities, such as MRI and CT, possess unique physical properties and data characteristics. For example, MRI heavily relies on multi-coil acquisition, where effectively leveraging the physical information from coil sensitivity maps is often critical for optimal reconstruction. Meanwhile, the imaging mechanism in CT and MRI are different. CT images are produced by Radon Transform and MR images are produced by Fourier Transform. This necessity to incorporate modality-specific knowledge has led the research community to develop specialized, state-of-the-art architectures for each task. To evaluating the performance of our approach for each task, we adopt the state-of-the-art architectures for each task. Note that, for MRI, E2E-VarNet is more light-weight than PromptMR, and we use E2E-VarNet for simulated MRI for the sake of efficient computation. \n\n**Fair comparison in each experiment:** We clarify that our KIDOT and the baselines use the same backbones in each individual experiment. To ensure the comparison is as fair as possible.\n\n**Aiming to demonstrate universal usage of KIDOT on different backbones:** Our another goal is to demonstrate the effectiveness of the proposed KIDOT framework on different network architectures. To isolate the contribution of our framework, we deliberately chose backbones that are identical or highly similar to the state-of-the-art supervised baselines for each respective task. For instance, when comparing with E2E-VarNet on the simulated MRI dataset, our KIDOT model also uses an E2E-VarNet-based architecture as its transport network $T\\_\\varphi$. This ensures that the performance gain is attributable to our novel knowledge-informed dynamic optimal transport formulation, but the more powerful backbone. The experiments in Tables 1-3 indeed shows the effectiveness of our method on different backbones.\n\nWe do believe it is interesting to investigate the universal network architecture for different the medical image reconstruction tasks along with the implementation of KIDOT. Such a universal network architecture cloud be possible if introducing in the attention mechanism, gating mechanism, and mixture of experts model for dealing task-specific and common knowledge, similar to large language models. Due to the limited time of the rebuttal, we left this study as future work. \n\nWe will add a detailed explanation in the **\"Implementation Details\" section (Sec B.1)** of the appendix, explicitly stating why different backbones were chosen and how this contributes to a fair comparison.\n\n### **Q1: Clarification of Table 4**\n\nWe apologize for the confusion and will revise the text to make the distinction unambiguous. We denote $C\\_{\\rm inter} = \\sum\\_{i=0}^{N-2} \\|y - \\mathcal{A}(\\mathcal{T}\\_\\phi^{t_i}(y))\\|\\_1$, $C\\_{\\rm final} = \\|y - \\mathcal{A}(\\mathcal{T}\\_\\phi^{t\\_{N-1}}(y))\\|\\_1$.\n\nThen the key difference between row (3) and row (4) lies in **when and how the imaging-knowledge-informed cost is applied**.\n \n*   **Row (3)  ( $\\mathcal{L}\\_{\\rm SUP}+\\varphi\\_\\theta+ C\\_{\\rm final}$ ):** This setting represents an ablation where the physics-consistency cost, $\\|y - \\mathcal{A}(\\mathcal{T}\\_\\phi^{t\\_{N-1}}(y))\\|\\_1$, is applied only to the final output of the N-step transport path. It checks if simply adding a physics-based penalty at the end is beneficial.\n*   **Row (4) (KIDOT= $\\mathcal{L}\\_{\\rm SUP}+\\varphi\\_\\theta+ C\\_{\\rm inter} + C\\_{\\rm final}$ ):** This is our full, proposed model. It enforces physics consistency throughout the entire dynamic transport process. As defined in our objective function (Eq. 10), the cost is integrated over all intermediate steps of the path ($\\sum\\_{i=0}^{N-1} \\|y - \\mathcal{A}(\\mathcal{T}\\_\\phi^{t_i}(y))\\|\\_1$).\n\nThis comparison is crucial as it demonstrates that enforcing consistency along the entire trajectory (our full KIDOT model) is superior to only constraining the final endpoint, which is a core tenet of our dynamic approach.\n\nWe will revise the **\"Ablation Study\" section (Sec. 4)** and the **method descriptions in Table 4** to explicitly and clearly state this difference."}]

Author rebuttal:

Meta-review: The paper introduces a novel framework for prospective medical image reconstruction that leverages optimal transport and imagining physics. The method is validated on simulated and real-world prospective MRI and CT data. 

Initially, the reviewers agreed on the importance of the tackled problem, the novelty of the approach and praised the good writing of the paper. Most of the initial reviewer’s questions revolved around experimental setups and additional comparisons. In the rebuttal, the authors clarified the experimental setup and provided additional evaluations. In the discussion period all reviewers were satisfied with the authors’ responses and recommended the acceptance of the paper. AC agrees with the reviewers and recommends to accept.
Decision: Accept (poster)