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Efficient Flow Matching for Sparse-View CT Reconstruction
Jiayang Shi, Lincen Yang, Zhong Li, Tristan van Leeuwen, Daniel M. Pelt, K. Joost Batenburg
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
The paper introduces FMCT and EFMCT, novel frameworks for sparse-view Computed Tomography (CT) reconstruction utilizing Flow Matching (FM) instead of Diffusion Models. FMCT leverages the deterministic nature of Ordinary Differential Equations (ODEs) in Flow Matching to ensure stable data consistency corrections. EFMCT further optimizes efficiency by reusing predicted velocity fields across consecutive sampling steps, significantly reducing Neural Network Function Evaluations (NFEs) and inference time while maintaining competitive reconstruction quality compared to state-of-the-art diffusion-based methods.
Entities (11)
Relation Signals (8)
FMCT → uses → Flow Matching
confidence 95% · we propose an FM-based CT reconstruction framework (FMCT)
EFMCT → uses → Velocity Reuse
confidence 95% · EFMCT that reuses previously predicted velocity fields over consecutive steps
FMCT → evaluatedon → AAPM Dataset
confidence 90% · We conduct experiments on two publicly available CT datasets... AAPM 2016
FMCT → evaluatedon → Medical Segmentation Decathlon
confidence 90% · The second dataset is the Medical Segmentation Decathlon CT dataset
Velocity Reuse → reduces → NFE
confidence 90% · substantially reduce the number of Neural network Function Evaluations (NFEs)
FMCT → comparedwith → DDS
confidence 85% · Decomposed Diffusion Sampler (DDS) [6]
FMCT → comparedwith → DPS
confidence 85% · We compare the proposed method with several representative diffusion-based CT reconstruction approaches, including Diffusion Posterior Sampling (DPS)
FMCT → outperforms → Diffusion Models
confidence 85% · significantly improving computational efficiency compared with diffusion-based methods
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Abstract
Abstract:Generative models, particularly Diffusion Models (DM), have shown strong potential for Computed Tomography (CT) reconstruction serving as expressive priors for solving ill-posed inverse problems. However, diffusion-based reconstruction relies on Stochastic Differential Equations (SDEs) for forward diffusion and reverse denoising, where such stochasticity can interfere with repeated data consistency corrections in CT reconstruction. Since CT reconstruction is often time-critical in clinical and interventional scenarios, improving reconstruction efficiency is essential. In contrast, Flow Matching (FM) models sampling as a deterministic Ordinary Differential Equation (ODE), yielding smooth trajectories without stochastic noise injection. This deterministic formulation is naturally compatible with repeated data consistency operations. Furthermore, we observe that FM-predicted velocity fields exhibit strong correlations across adjacent steps. Motivated by this, we propose an FM-based CT reconstruction framework (FMCT) and an efficient variant (EFMCT) that reuses previously predicted velocity fields over consecutive steps to substantially reduce the number of Neural network Function Evaluations (NFEs), thereby improving inference efficiency. We provide theoretical analysis showing that the error introduced by velocity reuse is bounded when combined with data consistency operations. Extensive experiments demonstrate that FMCT/EFMCT achieve competitive reconstruction quality while significantly improving computational efficiency compared with diffusion-based methods. The codebase is open-sourced at this https URL.
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- Source: https://arxiv.org/abs/2603.00205v2
- Canonical: https://arxiv.org/abs/2603.00205v2
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11institutetext: Centrum Wiskunde en Informatica 22institutetext: LIACS, Leiden University 33institutetext: Great Bay University 44institutetext: Mathematical Institute, Utrecht University 44email: l.yang@liacs.leidenuniv.nl Efficient Flow Matching for Sparse-View CT Reconstruction Jiayang Shi Lincen Yang Zhong Li Tristan van Leeuwen Daniël M. Pelt K. Joost Batenburg Abstract Generative models, particularly Diffusion Models (DM), have shown strong potential for Computed Tomography (CT) reconstruction serving as expressive priors for solving ill-posed inverse problems. However, diffusion-based reconstruction relies on Stochastic Differential Equations (SDEs) for forward diffusion and reverse denoising, where such stochasticity can interfere with repeated data consistency corrections in CT reconstruction. Since CT reconstruction is often time-critical in clinical and interventional scenarios, improving reconstruction efficiency is essential. In contrast, Flow Matching (FM) models sampling as a deterministic Ordinary Differential Equation (ODE), yielding smooth trajectories without stochastic noise injection. This deterministic formulation is naturally compatible with repeated data consistency operations. Furthermore, we observe that FM-predicted velocity fields exhibit strong correlations across adjacent steps. Motivated by this, we propose an FM-based CT reconstruction framework (FMCT) and an efficient variant (EFMCT) that reuses previously predicted velocity fields over consecutive steps to substantially reduce the number of Neural network Function Evaluations (NFEs), thereby improving inference efficiency. We provide theoretical analysis showing that the error introduced by velocity reuse is bounded when combined with data consistency operations. Extensive experiments demonstrate that FMCT/EFMCT achieve competitive reconstruction quality while significantly improving computational efficiency compared with diffusion-based methods. The codebase is open-sourced at https://github.com/EFMCT/EFMCT. 1 Introduction Generative models have recently gained popularity for Computed Tomography (CT) reconstruction due to their strong capability to learn expressive image priors [24, 14, 13, 4]. Among them, Diffusion Models (DM) have achieved state-of-the-art (SOTA) performance in sparse-view CT reconstruction [13, 4, 18] by learning the distribution of full-view reconstructions and leveraging it as a powerful prior for solving ill-posed inverse problems. Despite their strong reconstruction quality, DM methods are inherently stochastic because they model the forward diffusion and reverse denoising processes as Stochastic Differential Equations (SDEs) [21]. When combined with inverse problem solving, the repeated data consistency correction steps required for CT reconstruction can interfere with the stochastic evolution of the SDE [26, 5], leading to a push-and-pull effect between prior-driven denoising and physics-based correction. This interaction often results in unstable or inconsistent reconstruction behavior [18, 5], and necessitates a large number of iterations and Neural network Function Evaluations (NFEs), typically on the order of thousands. Although deterministic schedulers such as DDIM [19] have been proposed to reduce the number of reverse steps, achieving high-quality reconstructions with substantially fewer iterations remains challenging due to the inherently stochastic formulation of diffusion models. In clinical practice, CT reconstruction is frequently time-critical. In emergency and interventional settings, rapid image availability directly impacts diagnosis and treatment decisions, making time to first image a key performance metric [8, 23]. In addition, modern medical CT systems increasingly target higher spatial resolutions, which lead to larger reconstruction volumes and further exacerbate computational demands [11]. While conventional Filtered Backprojection (FBP) provides fast reconstructions, its image quality degrades substantially under sparse-view or low-dose acquisition, where advanced prior-based methods become necessary. As a result, the excessive computational cost of diffusion-based reconstruction methods poses a significant challenge for their practical deployment in real-world clinical workflows [18]. Flow Matching (FM) offers a fundamentally different modeling perspective that is particularly well suited to this setting. Instead of stochastic sampling, FM models deterministic probability transport through an Ordinary Differential Equation (ODE) [12]. Consequently, FM-based sampling follows a constant trajectory without stochastic noise injection. This deterministic formulation enables more coherent interaction with repeated data consistency corrections in inverse problems [25, 17], thereby mitigating the push-and-pull effect observed in diffusion-based methods and allowing for more efficient sampling. Furthermore, the velocity fields learned by FM are observed to evolve smoothly along the ODE trajectory, with strong correlations between adjacent time steps, as demonstrated in [2] and empirically confirmed in our experiments (see Fig. 1). This observation suggests that repeatedly re-evaluating the neural network at every integration step may be redundant. Motivated by these properties, we propose an efficient FM-based CT reconstruction framework that reuses previously predicted velocity fields to substantially reduce the number of NFEs. The reuse of velocity fields introduces only a controlled integration error, which we show is of the same order as the Euler discretization error and remains bounded through the explicit data consistency corrections. As a result, the proposed strategy achieves significant efficiency gains while maintaining competitive reconstruction quality. Our main contributions are summarized as follows: 1) We propose, to the best of our knowledge, the first flow-matching-based framework for CT reconstruction. 2) We introduce a velocity field reuse strategy that significantly improves the efficiency of FM-based CT reconstruction by reducing the required NFEs. 3) We provably show that the single-step reuse error is of the same order as Euler discretization, and does not alter the overall convergence behavior under bounded consecutive reuse steps. 4) We conduct extensive experiments to demonstrate the reconstruction performance and efficiency of the proposed FM-based CT reconstruction method. Figure 1: Left: Cosine similarity of predicted velocity fields at consecutive iterations in FMCT, with and without data consistency correction, showing strong correlation across adjacent steps. Right: Reconstruction performance vs. NFE for DM methods and FMCT/EFMCT, demonstrating that FMCT/EFMCT achieve competitive quality at substantially lower NFEs. DM methods here use the deterministic DDIM sampler for fairness; DPS/MCG results with the original 1000-step DDPM sampler are shown as isolated points. Results are averaged over the same randomly selected 21 reconstructions from the AAPM dataset across all iteration settings; shaded regions indicate standard deviation. 2 Method Problem Formulation. CT reconstruction aims to recover an unknown object ∈ℝm x ^m from a set of projection measurements ∈ℝn y ^n. The measurement process can be mathematically modeled as a linear system =, y= A x, (1) where ∈ℝn×m A ^n× m is the system matrix determined by the acquisition geometry. In sparse-view CT, the number of measurements is insufficient (n<mn<m), making the inverse problem underdetermined and ill-posed. In addition, measurement noise further degrades reconstruction quality. In this work, we focus on the sparse-view aspect of CT reconstruction. Flow Maching. Given a source distribution p(1)p( x_1) and a target distribution p(0)p( x_0), FM defines a time-dependent velocity field t:ℝd→ℝd v_t:R^d ^d that transports samples from p(1)p( x_1) to p(0)p( x_0) via the ODE [12] dtdt=t(t),t∈[0,1]. d x_tdt= v_t( x_t), t∈[0,1]. (2) The idea of FM is to train to approximate t v_t by a neural network t v_t θ, which allows sampling from the target distribution by numerically integrating the ODE. The optimization objective of FM then becomes ℒFM=t∼(0,1),t∼p(t)‖t(t)−t(t)‖2.L_FM=E_t (0,1),\> x_t p( x_t)\| v_t( x_t)- v_t θ( x_t)\|^2. (3) However, the marginal velocity field t(t) v_t( x_t) is generally intractable, as it requires integration over all possible 0 x_0. The common way is to replace the marginal velocity with a conditional velocity field t(t|0) v_t( x_t| x_0), which is proven in [12] to yield gradients equivalent to those of Equation 3. In this work, we focus on one specific conditional linear flow model, also known as Rectified Flow [15]. It models the transportation along straight-line trajectories through interpolation between 0 x_0 and 1 x_1 t=(1−t)0+t1,t(t|0)=1−0. x_t=(1-t) x_0+t x_1, v_t( x_t| x_0)= x_1- x_0. (4) This leads to the conditional flow matching loss ℒCFM=t∼(0,1),0∼p(0),1∼p(1)‖t((1−t)0+t1)−(1−0)‖2.L_CFM=E_t (0,1),\> x_0 p( x_0),\> x_1 p( x_1)\| v θ_t((1-t) x_0+t x_1)-( x_1- x_0)\|^2. (5) In practice, we sample 1 x_1 from a standard Gaussian distribution and use full-view CT images as 0 x_0. Once trained, the model enables deterministic sampling by integrating the learned velocity field backward from t=1t=1 to t=0t=0. Figure 2: Overview of our proposed FMCT(left) and EFMCT(right) method. Flow Matching for CT Reconstruction. Given a pretrained FM model, we incorporate physics-based data consistency into the sampling process to enable CT reconstruction. Each integration step consists of three components. First, a flow transport step is performed using an explicit Euler update t−Δt′=t−Δtt(t). x_t- t = x_t- t v_t θ( x_t). (6) Second, we estimate the terminal sample at t=0t=0 via linear extrapolation, and apply the data consistency operation (⋅)DC(·) to obtain the correction direction v_DC from measurement aware refinement ^0=t−ttθ(t),=(^0,,). x_0= x_t-t v_t^θ( x_t), v_DC=DC( x_0, y, A). (7) In this work, we implement (⋅)DC(·) using conjugate gradient [9, 6], though the framework is compatible with other data consistency schemes. Finally, the data-consistent update is applied as t−Δt=t−Δt′+. x_t- t= x_t- t + v_DC. (8) These steps are repeated iteratively until t=0t=0 to obtain the final reconstruction. An overview of the proposed framework is illustrated in Fig. 2. Velocity Reuse for Efficient Sampling. Empirically, we observe that the predicted velocities t v_t θ are highly correlated in consecutive steps (Fig. 1), resulting in near-straight trajectories toward the reconstructed image. This behavior is consistent with the smooth evolution of velocity fields along the underlying ODE in flow matching. These observations suggest that repeatedly re-evaluating the neural network at every integration step may be redundant. Moreover, when combined with explicit data consistency operations, we argue that the small deviations introduced by reusing a previously predicted velocity can be effectively compensated by physics-based corrections, as conceptually illustrated in Fig.2. Specifically, after computing t=t(t) v_t= v_t θ( x_t) at time t, we reuse this velocity for up to M consecutive steps: t−(j+1)Δt′=t−jΔt−Δtt,j∈[1,M]. x_t-(j+1) t = x_t-j t- t v_t, j∈[1,M]. (9) Since t v_t is reused without additional network evaluations, this strategy directly reduces the number of NFEs. To ensure stability and data fidelity, we perform an adaptive refinement check at each reuse step ‖t−(j+1)Δt′−‖2≤η‖t−jΔt′−‖2,\| A x_t-(j+1) t - y\|^2≤η\| A x_t-j t - y\|^2, (10) where η>1η>1 is a relaxation factor. If the condition is violated, velocity reuse is terminated and the velocity is recomputed using the neural network. This adaptive mechanism balances efficiency and reconstruction accuracy. η is emprically set at 1.051.05. The complete algorithm in summarized in the appendix. Proposition 1 Assume that v is locally Lipschitz in both x and t, i.e., ‖t()−t(′)‖≤Lx‖−′‖\| v_t( x)- v_t( x )\|≤ L_x\| x- x \| and ‖t()−t′()‖≤Lt|t−t′|\| v_t( x)- v_t ( x)\|≤ L_t|t-t | for all relevant ,′,t,t′ x, x ,t,t . Then, for a single reuse step, the local deviation between the reuse update ~k+1 x_k+1 and the standard Euler update k+1 x_k+1 satisfies ‖~k+1−k+1‖=O(Δt2)\| x_k+1- x_k+1\|=O( t^2), i.e., velocity reuse introduces a local error of the same order as the Euler discretization itself. Moreover, if the same velocity is reused for at most M consecutive steps (independent of Δt t), and the data-consistency correction is non-expansive with respect to its image argument, then the deviation remains controlled. The accumulated deviation after one reuse block is O(M2Δt)O(M^2 t). Implication. The proposition implies that velocity reuse introduces a local error of the same order as the Euler discretization itself. When the reuse length is bounded, the accumulated error remains controlled over consecutive reuse steps. Proofs are in Appendix. 3 Results Comparison Methods. We compare the proposed method with several representative diffusion-based CT reconstruction approaches, including Diffusion Posterior Sampling (DPS) [5], Manifold Constrained Gradient (MCG) [7], Pseudoinverse-Guided Diffusion Models (PGDM) [20], and Decomposed Diffusion Sampler (DDS) [6]. These methods represent a diverse set of strategies for incorporating measurement information into diffusion-based reconstruction and serve as strong SOTA baselines. In addition to learning-based methods, we include two classical reconstruction approaches, FBP and Model-Based Iterative Reconstruction (MBIR) method, Alternating Direction Method of Multipliers with Split-Bregman Total Variation (ADMM-TV) [3, 10]. Table 1: Quantitative comparison of reconstruction performance across different methods. Mean and standard deviation are reported for PSNR, SSIM, and data fidelity. The best average result for each metric is shown in bold, and the second best is underlined. NFE∗ counts only neural network forward evaluations. ↓ indicates efficiency improvement in NFE and computation time (on RTX4090) of FMCT/EFMCT compared with the most efficient diffusion-based method. Dataset Method 40 views 20 views NFE∗ Time/s PSNR SSIM Data Fit PSNR SSIM Data Fit AAPM FBP 26.98±0.4526.98± 0.45 0.691±0.0270.691± 0.027 2783.64±233.302783.64± 233.30 24.01±0.4924.01± 0.49 0.547±0.0290.547± 0.029 1798.77±171.001798.77± 171.00 - 0.02 ADMM-TV 31.94±0.4231.94± 0.42 0.836±0.0270.836± 0.027 22.11¯±2.29 22.11± 2.29 29.45±0.4629.45± 0.46 0.798±0.0290.798± 0.029 17.53¯±2.00 17.53± 2.00 - 9.85 DPS 30.05±0.6930.05± 0.69 0.795±0.0330.795± 0.033 261.35±20.58261.35± 20.58 28.78±0.8128.78± 0.81 0.764±0.0320.764± 0.032 142.93±35.99142.93± 35.99 1000 142.16 MCG 30.00±0.7930.00± 0.79 0.791±0.0320.791± 0.032 198.07±14.20198.07± 14.20 27.57±0.4227.57± 0.42 0.736±0.0310.736± 0.031 180.83±11.87180.83± 11.87 1000 143.44 PGDM 30.26±0.6130.26± 0.61 0.803±0.0300.803± 0.030 208.98±13.72208.98± 13.72 29.43±0.6029.43± 0.60 0.775±0.0310.775± 0.031 106.51±5.30106.51± 5.30 100 29.79 DDS 31.06±0.4031.06± 0.40 0.834¯±0.030 0.834± 0.030 19.09±1.1419.09± 1.14 29.21±0.3629.21± 0.36 0.785±0.0310.785± 0.031 9.87±0.619.87± 0.61 50 3.83 FMCT 31.63¯±0.39 31.63± 0.39 0.810±0.0310.810± 0.031 60.57¯±18.52 60.57± 18.52 29.98±0.2329.98± 0.23 0.769±0.0300.769± 0.030 46.46±14.8646.46± 14.86 25¯(↓50%) 25( 50\%) 1.92¯(↓50%) 1.92( 50\%) EFMCT 31.52±0.2831.52± 0.28 0.832±0.0270.832± 0.027 182.80±104.39182.80± 104.39 29.52¯±0.23 29.52± 0.23 0.795¯±0.027 0.795± 0.027 111.41±79.80111.41± 79.80 7(↓75%)7( 75\%) 0.83(↓78%)0.83( 78\%) Decathlon FBP 28.01±0.7628.01± 0.76 0.763±0.0370.763± 0.037 2772.79±434.362772.79± 434.36 24.85±0.9624.85± 0.96 0.618±0.0390.618± 0.039 1731.97±305.771731.97± 305.77 - 0.02 ADMM-TV 34.40±1.2334.40± 1.23 0.900±0.0210.900± 0.021 15.73¯±0.55 15.73± 0.55 30.26±1.0330.26± 1.03 0.851±0.0280.851± 0.028 14.76¯±0.50 14.76± 0.50 - 9.85 DPS 33.85±1.2533.85± 1.25 0.870±0.0180.870± 0.018 250.11±18.61250.11± 18.61 31.69±1.4431.69± 1.44 0.842±0.0250.842± 0.025 121.25±18.57121.25± 18.57 1000 142.16 MCG 30.03±1.2030.03± 1.20 0.824±0.0300.824± 0.030 301.17±35.25301.17± 35.25 28.67±1.2628.67± 1.26 0.798±0.0380.798± 0.038 198.22±22.68198.22± 22.68 1000 143.44 PGDM 34.78±1.2334.78± 1.23 0.899±0.0140.899± 0.014 147.10±15.20147.10± 15.20 31.82±1.1931.82± 1.19 0.860±0.0180.860± 0.018 132.96±13.69132.96± 13.69 100 29.79 DDS 37.66±1.2037.66± 1.20 0.923¯±0.009 0.923± 0.009 15.10±9.2915.10± 9.29 33.00±1.2933.00± 1.29 0.873¯±0.018 0.873± 0.018 5.01±0.365.01± 0.36 100 7.65 FMCT 37.65¯±1.53 37.65± 1.53 0.912±0.0200.912± 0.020 334.44±74.21334.44± 74.21 33.41±1.5733.41± 1.57 0.841±0.0360.841± 0.036 294.97±58.87294.97± 58.87 50¯(↓50%) 50( 50\%) 4.09¯(↓47%) 4.09( 47\%) EFMCT 37.56±1.3737.56± 1.37 0.940±0.0140.940± 0.014 621.99±48.99621.99± 48.99 33.21¯±1.29 33.21± 1.29 0.888±0.0240.888± 0.024 470.58±33.62470.58± 33.62 11(↓89%)11( 89\%) 1.72(↓78%)1.72( 78\%) Figure 3: Visual comparison of reconstructions across different methods, datasets, and views. PSNR, SSIM, and reconstruction time are shown in the lower-left, lower-right, and upper-right corners of each image, respectively. We visualize only the best three diffusion-based methods for better visibility. Zoom in for more details. Datasets. We conduct experiments on two publicly available CT datasets to evaluate the reconstruction performance of the proposed method. The first dataset is the AAPM 2016 Low Dose CT Grand Challenge dataset [16]. From this dataset, CT images (512×512512× 512 pixels) from nine patients are used to train the diffusion and flow matching models, while images from patient L506 are reserved for testing. The second dataset is the Medical Segmentation Decathlon CT dataset [1]. From the Task_06 Lung subset (512×512512× 512 pixels), we randomly select ten patients for training and evaluate reconstruction performance on images from patient 017. To obtain sparse-view measurements, we simulate 20-view and 40-view projection data using the ASTRA Toolbox [22]. All experiments are conducted using a parallel-beam geometry. This choice is made deliberately to provide a controlled and widely used experimental setting that minimizes confounding effects introduced by geometric complexity and allows us to isolate the impact of the reconstruction methodology itself. The proposed FMCT and EFMCT with velocity reuse strategy do not rely on geometry-specific assumptions and can be readily extended to other acquisition geometries by modifying the forward operator. Reconstruction Performance. Table 1 reports the quantitative reconstruction results. FMCT/EFMCT are often among the highest PSNR and SSIM across most datasets and view configurations, while requiring substantially fewer NFEs and lower computation time than diffusion-based baselines. ADMM-TV and DDS achieve the best and second-best data fidelity, reflecting their strong emphasis on measurement consistency. Compared with FMCT, EFMCT exhibits a slight reduction in PSNR and data fidelity but attains comparable or improved SSIM, accompanied by a significant efficiency improvement. This demonstrates that the proposed velocity reuse strategy effectively reduces computational cost while preserving reconstruction quality. Figure 3 provides a visual comparison. Although ADMM-TV achieves high PSNR/SSIM, its reconstructions are overly smooth and lack fine details. Diffusion-based methods and FMCT/EFMCT produce visually sharper and more detailed reconstructions, though they do not always exactly match the reference. This behavior is expected in sparse-view CT, where the problem is severely ill-posed and multiple reconstructions can satisfy the measurements. Generative methods balance data consistency with adherence to the learned prior, which may lead to small deviations in fine-scale details. Figure 4: Ablation of the velocity reuse strategy. Left: Reuse enabled after different iteration indices. Right: Varying the maximum number of consecutive reuse steps. Shaded regions indicate standard deviation. Impact of Velocity Reuse Strategy. We analyze the effect of the proposed velocity reuse strategy through an ablation study shown in Fig. 4. The experiments are conducted on 21 randomly selected test CT images, with the total number of sampling iterations fixed to 50. In the first ablation, we vary the iteration at which velocity reuse is enabled while fixing the maximum number of consecutive reuse steps to 10. Enabling reuse too early leads to an initial drop in reconstruction performance, whereas delaying reuse to later iterations gradually improves PSNR and SSIM. This behavior suggests that early-stage sampling is more sensitive to approximation errors, while reuse becomes safer once the reconstruction has approached a more stable trajectory. Consequently, velocity reuse is better suited to mid- or late-stage sampling. In the second ablation, velocity reuse is enabled from the first iteration, and the maximum number of allowed consecutive reuse steps is varied. Increasing the reuse limit initially improves PSNR and SSIM, indicating that moderate reuse does not adversely affect reconstruction quality. However, allowing excessive consecutive reuse results in a general performance degradation, revealing a trade-off between reconstruction quality and computational efficiency. Aggressive reuse substantially reduces computational cost but may compromise reconstruction accuracy. For all EFMCT results, we adopt a reuse strategy that enables velocity reuse after the first iteration and allows up to 10 consecutive reuse steps in all experiments. 4 Conclusion In this work, we proposed FMCT/EFMCT, representing the first application of flow matching to CT reconstruction. By exploiting the deterministic transport structure of flow matching, we introduced a velocity reuse strategy that substantially reduces NFEs. We provided theoretical analysis showing that the error introduced by velocity reuse is of the same order as Euler discretization and remains bounded when combined with explicit data consistency operations. Extensive experiments on multiple datasets demonstrate that FMCT/EFMCT achieve competitive reconstruction quality while significantly improving computational efficiency compared with diffusion-based methods. We believe that the proposed framework offers a practical and principled approach toward efficient generative CT reconstruction, with the potential to facilitate the deployment of advanced reconstruction methodologies in real-world clinical settings. Ackownledgements. The authors are supported by European Union H2020-MSCA-ITN-2020 under grant agreement no. 956172 and Dutch Research Council under grant no. ENWSS.2018.003, NWA.1160.18.316. Disclosure of Interests. The authors have no competing interests to declare that are relevant to the content of this article. 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(11) Define the velocity-reuse update for M consecutive steps using a previously evaluated velocity computed at (tk0,k0)(t_k_0, x_k_0): ~k+1=k−Δttk0(k0),k=k0,…,k0+M−1. x_k+1= x_k- t\, v_t_k_0( x_k_0), k=k_0,...,k_0+M-1. (12) Assume Lipschitz continuity Lx,Lt≥0L_x,L_t≥ 0 such that for all relevant ,′,t,t′ x, x ,t,t , ‖t()−t(′)‖≤Lx‖−′‖,‖t()−t′()‖≤Lt|t−t′|.\| v_t( x)- v_t( x )\|≤ L_x\| x- x \|, \| v_t( x)- v_t ( x)\|≤ L_t|t-t |. (13) Single Step Error. Assume reuse the velocity at the first step k0=k−1k_0=k-1 gives error eke_k, ek e_k =‖tk(k)−tk−1(k−1)‖ =\| v_t_k( x_k)- v_t_k-1( x_k-1)\| (14) ≤‖tk(k)−tk−1(k)‖+‖tk−1(k)−tk−1(k−1)‖ ≤\| v_t_k( x_k)- v_t_k-1( x_k)\|+\| v_t_k-1( x_k)- v_t_k-1( x_k-1)\| ≤Lt|tk−tk−1|+Lx‖k−k−1‖ ≤ L_t|t_k-t_k-1|+L_x\| x_k- x_k-1\| ≤LtΔt+LxΔt‖tk−1(k−1)‖. ≤ L_t t+L_x t\| v_t_k-1( x_k-1)\|. Then the error in ~k+1 x_k+1 and k+1 x_k+1 becomes ‖k+1−~k+1‖ \| x_k+1- x_k+1\| =Δt‖tk(k)−tk−1(k−1)‖ = t\| v_t_k( x_k)- v_t_k-1( x_k-1)\| (15) ≤Δt2(Lt+Lx‖tk−1(k−1)‖). ≤ t^2(L_t+L_x\| v_t_k-1( x_k-1)\|). It shows that the reusing the velocity gives single step error of O(Δt2)O( t^2), which matches the error of standard Euler x(t−Δt)=x(t)−Δtx˙+O(Δt2)x(t- t)=x(t)- t\, x+O( t^2). Accumulated Error. According to our proposed algorithm, we define standard single-step state update as k+1=k−Δttk(k)+(k−tktk(k),,), x_k+1= x_k- t\, v_t_k( x_k)+DC\! ( x_k-t_k v_t_k( x_k), y, A ), (16) and the corresponding state update with velocity-reuse steps as ~k+1=~k−Δttk0(k0)+(~k−tktk0(k0),,),k=k0,…,k0+M−1, x_k+1= x_k- t\, v_t_k_0( x_k_0)+DC\! ( x_k-t_k v_t_k_0( x_k_0), y, A ), k=k_0,…,k_0+M-1, (17) where (,,)DC( x, y, A) is the data consistency operation given measurement y and forward operator A. Here the same velocity tk0(k0) v_t_k_0( x_k_0) is reused for M consecutive steps. Assume (,,)DC( x, y, A) is κ-non-expansive with respect to its image argument, ‖(,,)−(′,,)‖≤κ‖−′‖,0≤κ≤1.\|DC( x, y, A)-DC( x , y, A)\|≤κ\| x- x \|, 0≤κ≤ 1. (18) Let k=~k−k δ_k= x_k- x_k. Then ‖k+1‖ \| δ_k+1\| ≤‖k‖+Δt‖tk(k)−tk0(k0)‖ ≤\| δ_k\|+ t\| v_t_k( x_k)- v_t_k_0( x_k_0)\| (19) +κ‖k−tk(tk0(k0)−tk(k))‖ +κ \| δ_k-t_k ( v_t_k_0( x_k_0)- v_t_k( x_k) ) \| ≤(1+κ)‖k‖+(Δt+κtk)‖tk(k)−tk0(k0)‖⏟velocity mismatch. ≤(1+κ)\| δ_k\|+( t+κ t_k) \| v_t_k( x_k)- v_t_k_0( x_k_0)\|_velocity mismatch. Using Lipschitz continuity, we have ‖tk(k)−tk0(k0)‖⏟velocity mismatch \| v_t_k( x_k)- v_t_k_0( x_k_0)\|_velocity mismatch ≤Lx‖k−k0‖⏟state drift+Lt|tk−tk0|. ≤ L_x \| x_k- x_k_0\|_state drift+L_t|t_k-t_k_0|. (20) Assume the practical updates have bounded increments, ‖j+1−j‖≤ΔtUmax,\| x_j+1- x_j\|≤ t\,U_ , (21) which holds when both the velocity field and the DC correction remain bounded along the trajectory. Then ‖k−k0‖⏟state drift \| x_k- x_k_0\|_state drift ≤∑j=k0k−1‖j+1−j‖ ≤ _j=k_0^k-1\| x_j+1- x_j\| (22) ≤(k−k0)ΔtUmax. ≤(k-k_0) t\,U_ . Since |tk−tk0|=(k−k0)Δt|t_k-t_k_0|=(k-k_0) t, we obtain ‖tk(k)−tk0(k0)‖⏟velocity mismatch≤(k−k0)Δt(Lt+LxUmax). \| v_t_k( x_k)- v_t_k_0( x_k_0)\|_velocity mismatch≤(k-k_0) t\,(L_t+L_xU_ ). (23) Let C=Lt+LxVmaxC=L_t+L_xV_ and tmax=maxktkt_ = _kt_k. Then ‖k+1‖≤(1+κ)‖k‖+(Δt+κtmax)(k−k0)ΔtC.\| δ_k+1\|≤(1+κ)\| δ_k\|+( t+κ t_ )(k-k_0) t\,C. (24) Unrolling over M consecutive reuse steps with k0=0 δ_k_0=0 gives ‖k0+M‖⏟accumlated deviation \| δ_k_0+M\|_accumlated deviation ≤(Δt+κtmax)CΔt∑j=0M−1(1+κ)M−1−jj ≤( t+κ t_ )C t _j=0^M-1(1+κ)^M-1-jj (25) ≤(Δt+κtmax)CΔt(1+κ)M−1M(M−1)2. ≤( t+κ t_ )C t(1+κ)^M-1 M(M-1)2. Therefore for bounded M, ‖k0+M‖=O(M2Δt),\| δ_k_0+M\|=O(M^2 t), (26) i.e. the accumulated error remains controlled for bounded consecutive reuse steps. 6 Appendix: Algorithm We summarize the complete EFMCT algorithm in 1. Algorithm 1 EFMCT: Flow Matching with Velocity Reuse 1:Forward operator A, measurements y, FM model t(⋅) v θ_t(·), number of steps N, max reuse steps M, relaxation factor η 2:Initialize t←1t← 1, t∼(,) x_t ( 0, I), Δt←1/N t← 1/N ⊳ Start from Gaussian 3:prev←∅ v_prev← , r←‖t−‖2r←\| A x_t- y\|^2, m←0m← 0 ⊳ Residual r and reuse counter m 4:for i=1i=1 to N do 5:// Attempt velocity reuse if it improves data consistency 6: if prev≠∅ v_prev≠ and m<Mm<M then 7: t−Δt′←t−Δtprev x_t- t ← x_t- t\, v_prev, ^0←t−Δt′−tprev x_0\ ← x_t- t -t v_prev, r~←‖t−Δt′−‖2 r←\| A x_t- t - y\|^2 ⊳ Reuse velocity 8: if r~≤ηr r≤η\,r then 9: t−Δt←t−Δt′+(^0,,) x_t- t← x_t- t + v_DC( x_0, y, A), r←r~r← r, m←m+1m← m+1 ⊳ DC update 10: else 11: prev←∅ v_prev← , m←0m← 0 12: continue 13: end if 14: else 15:// Evaluate FM network at current time step 16: t−Δt′←t−Δtt(t) x_t- t ← x_t- t\, v θ_t( x_t), ^0←t−Δt′−tt(t) x_0← x_t- t -t v θ_t( x_t) 17: t−Δt←t−Δt′+(^0,,) x_t- t← x_t- t + v_DC( x_0, y, A), r←‖t−Δt−‖2r←\| A x_t- t- y\|^2 18: prev←t(t) v_prev← v θ_t( x_t), m←0m← 0 19: end if 20: t←t−Δt← t- t, t←t−Δt x_t← x_t- t 21:end for 22:return 0 x_0 7 Appendix: Extra Results Table 2: Quantitative comparison of reconstruction perceptual quality across different methods. Mean and standard deviation are reported for LPIPS, where lower values indicate better perceptual similarity. The best average result for each setting is shown in bold, and the second best is underlined. NFE∗ counts only neural network forward evaluations. ↓ indicates efficiency improvement in NFE and computation time (on RTX4090) of FMCT/EFMCT compared with the most efficient diffusion-based method. Dataset Method 40 views 20 views NFE∗ Time/s LPIPS ↓ LPIPS ↓ AAPM FBP 0.391±0.0140.391± 0.014 0.554±0.0210.554± 0.021 - 0.02 ADMM-TV 0.272±0.0270.272± 0.027 0.380±0.0280.380± 0.028 - 9.85 DPS 0.128±0.0180.128± 0.018 0.166¯±0.025 0.166± 0.025 1000 142.16 MCG 0.145±0.0170.145± 0.017 0.208±0.0280.208± 0.028 1000 143.44 PGDM 0.159±0.0220.159± 0.022 0.187±0.0250.187± 0.025 100 29.79 DDS 0.114¯±0.013 0.114± 0.013 0.181±0.0210.181± 0.021 50 3.83 FMCT 0.101±0.0100.101± 0.010 0.148±0.0150.148± 0.015 25¯(↓50%) 25( 50\%) 1.92¯(↓50%) 1.92( 50\%) EFMCT 0.129±0.0140.129± 0.014 0.194±0.0210.194± 0.021 7(↓75%)7( 75\%) 0.83(↓78%)0.83( 78\%) Decathlon FBP 0.348±0.0180.348± 0.018 0.512±0.0260.512± 0.026 - 0.02 ADMM-TV 0.195±0.0250.195± 0.025 0.319±0.0210.319± 0.021 - 9.85 DPS 0.128±0.0370.128± 0.037 0.140±0.0380.140± 0.038 1000 142.16 MCG 0.164±0.0340.164± 0.034 0.186±0.0260.186± 0.026 1000 143.44 PGDM 0.100±0.1290.100± 0.129 0.124±0.0320.124± 0.032 100 29.79 DDS 0.072¯±0.015 0.072± 0.015 0.110±0.0170.110± 0.017 100 7.65 FMCT 0.109±0.0210.109± 0.021 0.180±0.0260.180± 0.026 50¯(↓50%) 50( 50\%) 4.09¯(↓47%) 4.09( 47\%) EFMCT 0.070±0.0110.070± 0.011 0.120¯±0.024 0.120± 0.024 11(↓89%)11( 89\%) 1.72(↓78%)1.72( 78\%)