Accelerated Physics-guided Diffusion Model for 3D Limited-angle Reconstruction of Cardiac Computed Tomography

Abstract

Cardiovascular diseases (CVDs) remain the leading cause of morbidity and mortality worldwide. Computed tomography (CT) plays an important role in the diagnosis and management of CVDs. However, the temporal resolution of current cardiac CT imaging techniques is still insufficient to freeze the beating heart. One solution to improve temporal resolution is fast data acquisition for a limited-angle range. Unfortunately, this approach introduces significant limited-angle artifacts that compromise image quality. Diffusion models have shown promising results to address this issue by effectively reconstructing missing data, but the process is typically time-consuming. In this paper, we propose a novel method for 3D limited-angle reconstruction using a diffusion model, optimized to significantly reduce computational time. Specifically, our approach integrates the regularized derivative least square (RDLS) algorithm with denoising diffusion implicit models. The numerical simulations and clinical data experiments demonstrate that the proposed model outperforms the existing methods both quantitatively and qualitatively. Compared with DDS, our method delivers a 1.8× speedup on the simulated dataset and a 3.4× speedup on the real clinical dataset.

I. Introduction

CARDIOVASCULAR diseases (CVDs) remain the leading cause of mortality globally, claiming millions of lives each year [1]. The pervasive threat of CVDs requires effective diagnostic tools that enable early detection and timely intervention. Among these, cardiac computed tomography (CT) emerges as a crucial imaging modality [2]. It offers a detailed visualization of the cardiac anatomy, particularly the coronary arteries such as the right coronary artery (RCA), left anterior descending (LAD), and left circumflex (LCX). This capability is crucial for identifying and evaluating vascular diseases within these critical regions, facilitating accurate diagnosis and management of CVDs. However, the efficacy of cardiac CT can be compromised by inherent limitations. Heart beat often results in motion artifacts when imaging time is extended [3]. Furthermore, the associated radiation dose poses a risk of increasing cancer susceptibility. Limited-angle CT reconstruction offers a solution to these challenges by using incomplete projections, which not only enhances the imaging speed but also reduces the radiation dose, thereby increasing temporal resolution and mitigating motion artifacts. However, this technique introduces severe limited-angle artifacts, which can degrade the quality of the reconstructed images.

To address the limited-angle artifacts, several innovative methods have been developed. They can be broadly categorized into two main branches: iterative-based and network-based approaches. Iterative methods, which are based on optimization frameworks, have demonstrated considerable success in reconstructing images from incomplete projection data by incorporating prior knowledge through regularization techniques. A widely adopted regularizer in this context is the minimization of total variation (TV), which employs the sparsity of image gradients to suppress noise and artifacts while preserving edge information [4]. The TV regularization approach assumes that natural images have piecewise smoothness, making it particularly effective for reducing oscillatory noise and stabilizing the reconstruction process. Despite its strengths, TV-regularized iterative reconstruction faces notable limitations, particularly when applied to severely scenarios with limited-angle measurements, such as scanning ranges restricted to 120° or less. At these reduced angular spans, the ill-posedness of the inverse problem is exacerbated, leading to pronounced streaking artifacts and loss of structural fidelity in the reconstructed images. In contrast, network-based methods, leveraging advances in machine learning [5], [6], [7], [8], have shown significant promise in medical imaging[9], [10], [11], [12], [13]. Initially, Convolutional Neural Networks (CNNs) were the primary techniques for image reconstruction tasks. However, more recently, transformer-based approaches demonstrated superior performance due to their ability to capture long-range dependencies and global contextual information [14]. Furthermore, in recent years, deep network-based methods, particularly generative models, have gained popularity. Notably, diffusion models [15] have emerged as the premier technique, distinguished by their capability to generate high-quality samples while ensuring model convergence and diversity [16]. Despite their advantages, diffusion models are sometimes limited by their slow sampling speed, which pose significant challenges in time-sensitive clinical applications.

Diffusion models were originally designed for unconditional image generation tasks. However, when applied to CT reconstruction, unconditional diffusion models are insufficient due to the need for precise control over generated outputs based on limited or degraded inputs. Consequently, conditional diffusion models, which incorporate specific conditions into the generative process, become critical for controlling the output, thereby ensuring that reconstructed images align closely with the real clinical scenarios. The significance of conditional diffusion models lies in their capability to effectively integrate prior knowledge or observed data into the generative process, thus significantly improving reconstruction quality in medical imaging applications. Unlike the unconditional models, conditional diffusion models explicitly guide the generation process using auxiliary information such as partial observations, measurements, or specific image characteristics. This feature is particularly advantageous in challenging tasks like limited-angle CT reconstruction, where incomplete sinogram data demand effective conditioning strategies to recover accurate and clinically relevant images.

Several approaches have demonstrated the potential of conditional diffusion models. Particularly, Saharia et al. introduced SR3 [17], a pioneering work leveraging diffusion models for conditional generation. SR3 formulates image super-resolution as a conditional iterative refinement process, requiring paired low- and high-resolution image datasets for training. Despite its remarkable performance in resolution enhancement tasks, SR3 employs an additional conditioning channel and requires a significantly large model size. These requirements limit its applicability to resource-constrained or lightweight scenarios commonly encountered in clinical practice, such as real-time CT reconstruction. Another important advancement was proposed by Song et al. [18]. This work was specifically designed for sparse-view CT reconstruction, and it approximates the diffusion process on observed sparse sinogram data. However, this method involves padding the sparse observation data to match the dimensions of a complete sinogram. Although effective for sparse-view scenarios, such padding becomes highly problematic for limited-angle reconstruction tasks. Padding limited-angle sinograms to match a complete angular coverage often introduces artificial artifacts and misleading data, making this approach unsuitable for scenarios involving severe angular limitations.

Recently, the combination of iterative and network-based approaches has gained popularity in the field of limited-angle CT reconstruction. This hybrid strategy leverages the strengths of both methodologies, enhancing the reconstructed image quality while mitigating the limitations inherent to each individual approach. Chung et al. [19] developed DiffusionMBIR to integrate traditional model-based iterative reconstruction with modern diffusion models for pioneering 3D medical image reconstruction from pretrained 2D models. Han et al.[20] proposed the PSDM method for further enhancement, which substitutes the ADMM component in DiffusionMBIR with PDHG and incorporates a Fourier fusion technique, significantly reducing sampling times while maintaining promising results. These results underscore the transformative potential of diffusion models in medical imaging, especially when optimization steps such as ADMM or PDHG are seamlessly integrated to iteratively correct noisy variables. However, both DiffusionMBIR and PSDM directly apply projection steps to the noisy variable, which is suboptimal. Although PSDM demonstrates faster convergence, compared to DiffusionMBIR, to process only one 256 × 256 CT image with 30 iterations per sampling step, PSDM still requires more than ten minutes.

In this paper, we propose an Accelerated Physics-guided Diffusion Model (APDM), a novel framework to integrate iterative methods with diffusion models. By using GPU acceleration for the iterative component and employing DDIM [21] for the diffusion aspect, our method significantly speeds up the sampling process. For a single phase of cardiac images with dimensions of 32 × 256 × 256, APDM requires less than two minutes to complete processing, using 100 iterations per sampling step. Furthermore, unlike typical methods that rely on 2D TV image priors where smoothness and edge preservation are enforced independently on each axial slice, our model incorporates full 3D TV regularization during the sampling phase. By penalizing variations across all three spatial dimensions simultaneously, it not only suppresses noise more effectively, but also preserves fine structural details and anatomical continuity between slices. This volumetric consistency is crucial in 3D CT reconstruction because it reduces inter-slice artifacts, enhances contrast resolution of small features such as vessel walls or micro-calcification, and ultimately yields more accurate, clinically reliable reconstructions than slice-by-slice approaches.

The rest of this paper is organized as follows. Section II reviews iterative reconstruction and diffusion models, surveys existing conditional diffusion methods for CT reconstruction, and presents the fundamental theory and overall workflow for our APDM. In Section III, we present the reconstruction results obtained with our APDM method and the state-of-the-art comparative algorithm. Section IV discusses relevant issues and summarizes the contributions of APDM.

II. METHOD

A. Iterative Reconstruction

In the context of CT image reconstruction, the goal is to estimate the image x from the measured projection data y . This inverse problem is typically formulated as minimizing a cost function of data fidelity. The typical least squares (LS) method aims to find the image x that minimizes the squared Euclidean norm of the difference between the measured data and the forward projections:

$$\underset{x}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{1}{2}\Vert \mathit{A}\mathit{x}-\mathit{y}{\Vert }_2^2,$$ (1)

where A represents the CT system matrix. Because this approach focuses solely on data fidelity, although it attempts to produce an image that best fits the measured data, it can be highly sensitive to noise and may yield unstable solutions, especially when the system is ill-posed. For the limited-angle CT reconstruction problem, the available projection data is incomplete. Such incompleteness can cause severe artifacts in the reconstructed images when traditional analytical methods like filtered backprojection (FBP) are used. To overcome the limitations of the LS method, the regularized least squares (RLS) method incorporates a regularization term R(x),

$$\underset{x}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{1}{2}\Vert \mathit{A}\mathit{x}-\mathit{y}{\Vert }_2^2+\lambda R(\mathit{x}),$$ (2)

where λ is a parameter to trade-off data fidelity and regularization, and R(x) is the regularization term to incorporate prior information about the image. The regularization term imposes prior knowledge or desired properties on the solution, such as smoothness or sparsity, which helps to stabilize the inversion process and reduce the effect of noise. Common choices for R(x) include the ${\ell }_2$-norm of the image gradient and its variants [22], [23] that have shown high performance in reducing artifacts while preserving edges and other important features.

Many CT images can be modeled as a combination of low-attenuation background with relatively sparse and high-attenuation foreground structures. Motivated by this observation, the regularized derivative least square (RDLS) method [24] has been proposed to address the reconstruction problem. The key idea is to first reconstruct the foreground of the CT image by exploiting its sparsity. Due to this sparsity, fewer image voxels need to be considered, leading to lower computational costs and faster reconstruction speed. RDLS is tailored to exploit the inherent sparsity of CT data and focus on the computational effort in localized subregions. Its semilocal cost function enables piecewise refinement in areas of interest, while classical CT methods apply global updates. This targeted refinement is especially advantageous when projection data are incomplete. The RDLS method is defined as:

$$\underset{\mathit{x}}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{1}{2}(\nabla (\mathit{A}\mathit{x}-\mathit{y}){)}^T(\nabla (\mathit{A}\mathit{x}-\mathit{y}))+\lambda R(\mathit{x}),$$ (3)

where ∇ represents the gradient operator. The semi-local nature of the RDLS cost function allows for concentrated reconstruction effort in selected subregions, in contrast to classical CT reconstruction methods that are inherently global. As a result, RDLS enables piecewise refinement of the reconstructed image. Although the RDLS method can be used as a stand-alone reconstruction algorithm, the convergence of low-frequency image components can be slow. To improve the convergence of these low-frequency features, one can either apply another iterative reconstruction technique before or after RDLS, or employ a low-pass filter as a preconditioner. We used a simple Gaussian filter in this paper and found it works effectively in this role.

B. Diffusion Model

Denoising diffusion probabilistic models (DDPM) are generative models to approximate a target data distribution $q\left({\mathit{x}}_0\right)$ by defining a parameterized Markov chain of latent variables ${\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_T\in {\mathbb{R}}^d$ with learnable parameters $\mathit{\theta }\in {\mathbb{R}}^d$. The joint distribution modeled by DDPM is given by:

$$p(x_0;\theta )=\int p(x_T;\theta )\prod _{t=1}^{T}p(x_{t-1}\mid x_t;\theta )\text{d}{x}_{1:T},$$ (4)

where $p\left({\mathit{x}}_T\right)=𝒩\left({\mathit{x}}_T;\mathbf{0},{\mathbf{I}}_{d_x}\right)$ is a standard Gaussian prior at time T, and $p\left({\mathit{x}}_{t-1}\mid {\mathit{x}}_t;\mathit{\theta }\right)$ corresponds to the learned reverse conditional probabilities for transitioning between timesteps, typically modeled by neural networks.

The forward diffusion process gradually adds Gaussian noise to the data. This process can be written as a variance-preserving stochastic differential equation (VP-SDE) [25]:

$$d\mathit{x}=-\frac{{\beta }_t}{2}\mathit{x}dt+\sqrt{{\beta }_t}dw,$$ (5)

where the variance schedule ${\beta }_t$ satisfies $0<{\beta }_1<\cdots <$ ${\beta }_T<1$ for T time steps, and dw represents the derivative of Wiener Process. By applying the Anderson’s theorem to Eq.(5), we have the reverse SDE:

$$d\mathit{x}=\left(-\frac{1}{2}\beta (t)\mathit{x}-\beta (t){\nabla }_{\mathit{x}}\mathrm{l}\mathrm{o}\mathrm{g}{p}_t(\mathit{x})\right)dt+\sqrt{\beta (t)}d\bar{w},$$ (6)

where $\bar{w}$ is a Wiener process running backward in time, and ${\nabla }_{\mathit{x}}\mathrm{l}\mathrm{o}\mathrm{g}{p}_t(\mathit{x})$ defines the score function. Hence, the problem of reverse VP-SDE is transformed into a problem of score estimation. In practice, the score function can be approximated using a neural network ${\mathit{s}}_{\mathit{\theta }}$ parameterized by θ, which can be trained via denoising score matching [26].

Because the DDPM sampling process usually requires over 2,000 reverse time steps to achieve high-quality generation, it is computationally expensive and slow. To address this limitation, Song et al. proposed Denoising Diffusion Implicit Models (DDIM) [21]. It defines the diffusion process as a non-Markovian process.

$${\mathit{x}}_{t-1}=\sqrt{{\bar{\alpha }}_{t-1}}{\stackrel{\mathit{ˆ}}{\mathit{x}}}_0\left({\mathit{x}}_t\right)+\sqrt{1-{\bar{\alpha }}_{t-1}-\eta {\delta }_t^2}{s}_{\theta }\left({\mathit{x}}_t,t\right)+\eta {\delta }_t\mathit{ϵ},$$ (7)

where ${\alpha }_t=1-{\beta }_t$, ${\bar{\alpha }}_t=\prod _{i=1}^t{\alpha }_t$, η is a hyperparameter controlling the level of stochasticity, ${\delta }_t$ determines the time-dependent contribution of the noise at each step, and ${\stackrel{\mathit{ˆ}}{\mathit{x}}}_0\left({\mathit{x}}_t\right)$ is the predicted clean data that can be estimated from ${\mathit{x}}_t$ by using Tweedie formula [27].

$${\widehat{\mathit{x}}}_0\left({\mathit{x}}_t\right)=\frac{1}{\sqrt{{\bar{\alpha }}_t}}\left({\mathit{x}}_t-\sqrt{1-{\bar{\alpha }}_t}{\mathit{s}}_{\mathit{\theta }}\left({\mathit{x}}_t,t\right)\right).$$ (8)

Unlike DDPM, DDIM [21] introduces a deterministic sampling procedure by using a reparameterized formulation for forward and reverse diffusion processes. This deterministic nature allows DDIM to sample from the model with significantly fewer time steps while maintaining comparable generation quality. Moreover, by tracing a non-Markovian trajectory through latent space, DDIM enables precise, order-agnostic interpolation between data point—facilitating controllable edits and smooth latent walks that are not possible with purely stochastic samplers. The reduced sampling complexity also directly translates into substantial inference-time speedups, making DDIM particularly attractive for real-time or resource-constrained applications without sacrificing the fidelity of the generated images.

C. Conditional Diffusion Models for CT Reconstruction

The CT reconstruction problem seeks to recover an unknown object x from partially observed and noisy measurements y . Traditionally, this inverse problem is formulated as the minimization of an objective that balances a data fidelity term and a regularization term like Eq.(2). Recently, pretrained diffusion models [28] have emerged as powerful generative priors for CT reconstruction in a zero-shot framework. Owing to their capacity to model complex image distributions, efficient iterative denoising process, and amenability to conditional sampling, diffusion priors offer a compelling alternative to classical regularizers.

Given measurements y with a system matrix A, we can use diffusion models to solve CT reconstruction problem by replacing the score function in Eq.(6) with a conditional score function. According to Bayes’ theorem, the score function of the posterior distribution can be computed as:

$${\nabla }_{\mathit{x}}\mathrm{l}\mathrm{o}\mathrm{g}{p}_t(\mathit{x}\mid \mathit{y})={\nabla }_{\mathit{x}}\mathrm{l}\mathrm{o}\mathrm{g}{p}_t(\mathit{x})+{\nabla }_{\mathit{x}}\mathrm{l}\mathrm{o}\mathrm{g}{p}_t(\mathit{y}\mid \mathit{x}),$$ (9)

where the likelihood part ${\nabla }_{\mathit{x}}\mathrm{l}\mathrm{o}\mathrm{g}{p}_t(\mathit{y}\mid \mathit{x})$ enforces the data consistency to satisfy $\mathit{y}=\mathit{A}\mathit{x}$. There are three type methods to address data consistency as follows.

1). Subspace Decomposition:

By assuming that the system matrix A is full-rank, ScoreMed [18] introduces a decomposition method for A within the sampling process by injecting noise into y to synthesize ${\mathit{y}}_t$ to build a relationship. Meanwhile, the Denoising Diffusion Null-Space Models (DDNM) [29] employ a range–null space split for zeroshot reconstruction. In these frameworks, the range subspace enforces measurement consistency, whereas the null space improves image realism via learned priors. Both methods rely on back-projection tricks [30] to maintain fidelity in noise-free measurement scenarios.

2). Direct Projection:

Early diffusion-based reconstruction methods enforce data consistency by directly projecting the noisy estimate ${\mathit{x}}_t$ onto the measurement subspace after one sampling step [31], [19], [20]:

$${\mathit{x}}_{t-1}^{\prime }\leftarrow \mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}\mathrm{e}\left({\mathit{x}}_{t-1},{\mathit{s}}_{\mathit{\theta }}\right),$$ (10)

$${\mathit{x}}_t\leftarrow P\{\mathit{x}\mid \mathit{A}\mathit{x}=\mathit{y}\}\left({\mathit{x}}_{t-1}^{\prime }\right),$$ (11)

where ”solve” represents a numerical solver of Eq.(10), and P denotes the projection operator by following the Projection Onto Convex Sets (POCS) principle to satisfy the data constraint. However, Eq.(11) is disconnected from Eq.(9). Note that the data fidelity term comes from ${\nabla }_{\mathit{x}}\mathrm{l}\mathrm{o}\mathrm{g}{p}_t(\mathit{x}\mid \mathit{y})$, which is the probability of measurement y conditioned on the noisy variable ${\mathit{x}}_t$ , not ${\mathit{x}}_0$ . As a result, enforcing consistency directly on noisy variable ${\mathit{x}}_t$ is less effective and requires more sampling steps to converge [32].

3). Indirect Projection:

To resolve the inconsistency between projections carried out in the noisy space and the clean image subspace, DPS [32] adopted a posterior sampling scheme that uses a Laplacian approximation of the likelihood without a strict measurement consistency projection step, yielding a more desirable generative path in noisy settings. In the following work of DDS [33], they applied the ADMM-CG to the predicted ${\mathit{x}}_0$ instead of ${\mathit{x}}_t$, which achieved dramatic improvements in quality and reconstruction speed. In Resample [34], based on latent diffusion [35], it incorporates data consistency in latent space by solving an optimization problem to the predicted ${\mathit{x}}_0$ during the reverse sampling process. In both of their works, they all use the Tweedie formula to obtain the predicted ${\mathit{x}}_0$ . By using estimated x₀, we could obtain a more accurate alignment with the measurement constraints and reduce the number of required sampling steps.

D. Workflow

The key for CT image reconstruction is how to design the data consistency part in the content of conditional diffusion models. To establish a meaningful relationship between the noisy variable and the projection, we adopt a workflow using an indirect projection method that employs the Tweedie formula to estimate the underlying clean data. In addition, we incorporate the RDLS method [24] to enforce measurement consistency. The overall framework is summarized as follows.

$${\widehat{ϵ}}_t\leftarrow s_{\theta }(x_t,t),$$ (12)

$${\widehat{x}}_0(x_t)\leftarrow \text{Tweedie}(x_t,{\widehat{ϵ}}_t),$$ (13)

$$x_t^{\prime }\leftarrow P\{x\mid Ax=y\}({\widehat{x}}_0(x_t)),$$ (14)

$$x_{t-1}\leftarrow \text{DDIM}(x_t^{\prime }).$$ (15)

In each of the above reverse diffusion step, we start with a noisy intermediate state ${\mathit{x}}_t$ and apply the trained score-based diffusion model ${\mathit{s}}_{\mathit{\theta }}\left({\mathit{x}}_t,t\right)$ to estimate noise ${\widehat{ϵ}}_t$. Then, we use the Tweedie formula to obtain an estimated clean image ${\widehat{\mathit{x}}}_0\left({\mathit{x}}_t\right)$. Next, to ensure that the reconstruction matches the measured data y under the system matrix A, we project ${\widehat{\mathit{x}}}_0\left({\mathit{x}}_t\right)$ onto the data-consistent solution space $\{\mathit{x}\mid A\mathit{x}=\mathit{y}\}$ using the RDLS formulation, producing a refined estimate ${\mathit{x}}_t^{\prime }$. This data-consistent result is then re-injected into the reverse diffusion sampling step to determine the subsequent latent variable ${\mathit{x}}_{t-1}$ for the next iteration. By alternating repeatedly between diffusion refinement (using ${\mathit{s}}_{\mathit{\theta }}$) and data consistency enforcement (via RDLS), the method converges to a high-quality CT image that is faithful to the measured data and guided by the learned diffusion priors. However, it is computationally intensive to strictly enforce data consistency at every reverse sampling iteration t. To mitigate this overhead, we first observe that, at the beginning of the denoising process, the estimated image ${\widehat{\mathit{x}}}_0\left({\mathit{x}}_t\right)$ does not yet exhibit a meaningful structure that is comparable to the reference. Furthermore, previous work has identified a three-stage phenomenon in diffusion sampling [36], demonstrating that data consistency is most beneficial during the semantic and refinement stages. Similarly, the Resample approach [37] substantiates the efficiency of this selective strategy. Motivated by these findings and through extensive empirical evaluation, we therefore apply data consistency only during the final 20 steps of the 50-step sampling procedure.

In this study, we also employ an anisotropic TV regularizer. It applies a Huber-like loss function to the differences between adjacent samples in 3D space. The number of neighboring voxels for the TV calculation is set to 26. The anisotropic TV functional with the Huber-like loss function is defined as

$$R(x):=\sum _u\sum _{v\in N_u}\Vert u-v{\Vert }^{-1}h(x_u-x_v),$$ (16)

where

$$h(\tau )=\{\begin{array}{ll}\frac{1}{2}{\tau }^2,\hfill & \text{if}\left|\tau \right|\le \delta ,\hfill \\ {\delta }^{2-p}|\tau {|}^p+{\delta }^2(\frac{1}{2}-\frac{1}{p}),\hfill & \text{if}\left|\tau \right|>\delta ,\hfill \end{array}$$ (17)

Here, $N_u=26$ is the neighborhood around the voxel index $u=\left(u_1,u_2,u_3\right)$, x_u, x_v are the intensity values at voxels u and v, respectively; $\Vert u-v\Vert$ is the Euclidean distance between voxel indices u and v, h(τ) is the Huber-like loss applied to the difference $\tau =x_u-x_v$; δ > 0 is a threshold parameter, and p is a parameter to control the behavior of the loss function for large values of $\left|\tau \right|$. Fig.1 shows the architecture of the proposed model.

Fig. 1.

Architecture of the APDM image reconstruction framework. In the initial phase, a score-based diffusion model is employed to generate a preliminary image. It is followed by Tweedie denoising and the implementation of a data consistency constraint by integrating the RDLS optimization technique. N is the number of sampling step.

III. EXPERIMENT AND RESULTS

In this section, we begin by presenting an overview of the training process and XCAT simulation dataset for the proposed method. Subsequently, we introduce the real-world clinical data from GE. Finally, we perform a comparative analysis to evaluate the proposed approach against several fully supervised and diffusion-based methods.

A. Model Training and Sampling Detail

In our experiment, the APDM is trained using guided-diffusion from OpenAI [16]. Specifically, the learning rate is set to 1 × 10⁻⁴ for the simulation dataset (256 × 256) and 1 × 10⁻⁷ for the real clinical dataset (512 × 512). The implementation is developed in Python using PyTorch frameworks, and RDLS reconstruction and 3D TV regularizer are implemented by using the LivermorE AI Projector (LEAP) toolbox [38]. All experiments are executed on a machine equipped with an NVIDIA A100-40GB GPU. This study utilizes the GPU resources provided by the Delta system in the NSF ACCESS program [39] for computation. During the sampling stage, the DDIM sampling step is configured to 50. In each sampling step, data consistency is maintained through the application of the RDLS algorithm, which is iterated 100 times for simulated dataset and real clinical dataset.

B. Data Specification

XCAT Simulated Dataset:

The 4D extended cardiactorso (XCAT) phantom [40], version 2, developed at Duke University, provides a highly detailed multimodal model that can simulate cardiac CT imaging, including dynamic and motion-captured scenarios of a beating heart. We use XCAT to create patient-specific phantoms with realistic attenuation coefficients, generating projection data to simulate cardiac CT procedures. Our setup employs the parameters of a GE CT scanner to mimic realistic imaging conditions with specific factors. The source-detector distance is set to 950 mm, source-object distance is set to 538.5 mm, the curved detector array has 835 units, a full scan includes 984 views, and the field of view diameter is 500 mm. For dynamic imaging, each cardiac cycle is conceptualized as one second and divided into 200 phases, sampled at equal time intervals. We employ an empirically calibrated tube current-time product of 0.25 mAs per projection to align simulated image noise levels with those seen in real cardiac CT scans [3]. Projections are generated to simulate 10 patient-specific datasets, yielding images with 256 × 256 pixel resolution. Each simulated patient has 352 image slices, resulting a total of 3,520 slices. Nine of these patient datasets are used for training and the remaining one is reserved for testing. This process is repeated 10 times, with each patient dataset being used exactly once as the testing set.

Clinical Cardiac Dataset:

To evaluate the efficiency of the proposed APDM in clinical cardiac applications, a high-quality human cardiac dataset was acquired using a GE scanner with the approval of the Institutional Review Boards at the Vanderbilt University Medical Center and the University of Massachusetts Lowell. The patient was imaged in axial mode with a clockwise gantry rotation. The source-to-isocenter and source-to-detector distances were 625.61 mm and 1,097.6 mm, respectively. The detector array comprises 828 cells, with each detector bin measuring 1.09 mm × 1.09 mm. A total of 984 projections were acquired over a complete angular range of [0, 2π], with a field-of-view (FOV) of 500 mm. The imaging system featured a 256-channel data acquisition system and a high-performance X-ray tube, ensuring optimal image quality for cardiac imaging applications. Reconstruction was performed using Feldkamp-Davis-Kress algorithm (FDK) in a cone-beam geometry, with an image resolution of 512 × 512 pixels. The study used data from six patients, resulting in a total of 840 slices from five patients for training. An independent testing set was obtained from one patient, consisting of 200 slices.

C. Simulation Reconstruction Result

For comparison, we select the RDLS-TV method [24], a transformer-based approach (Restormer) [41], and a diffusion-based method (DDS) [33]. Fig. 2 shows representative reconstruction results from the testing dataset, utilizing different methods for 120° limited-angle reconstruction. Under 120° condition, traditional methods such as FBP and RDLS-TV are significantly affected by severe limited-angle artifacts. The Restormer method demonstrates the ability to suppress most of these artifacts. However, edge structures exhibit deformation and certain vessels appear distorted. In contrast, the diffusion-based DDS method achieves relatively favorable results. However, in the magnified regions of interest marked by red and yellow boxes, the vessels look less clear than they do with our proposed method. Fig. 3 shows representative results for 90° limited-angle reconstruction. The result of Restormer and DDS become worse in severe condition. However, our proposed method still can reconstruct relatively good results, especially in the key cardiac areas as enlarged in the red and yellow boxes.

Fig. 2.

Representative reconstruction results of the XCAT dataset from different methods with a scanning angular range of 120°. The display window is [−540 1000] HU.

Fig. 3.

Same as Fig. 2 but for another slice with a scanning angular range of 90°.

To further evaluate the performance of our algorithm, Fig. 4 plots the profile curve distributions of different algorithms for 120° limited-angle reconstruction along the yellow line, which spans across the cardiac vessel region. We also magnify the cardiac region to highlight that our proposed method excels at processing vessel information. The curve distribution reveals that the profile curve of the reconstruction result from the proposed approach in this study is smoother and aligns more closely to the reference image. This observation suggests that its reconstruction accuracy surpasses that of other competing algorithms. Fig. 5 presents the profile curve distributions of different algorithms for 90° limited-angle reconstruction along the yellow line. The magnified cardiac edge region highlights that our proposed method excels at processing edge information. The resulting curve distributions demonstrates that the profile curve of our reconstruction is more refined and closely matches the reference image, indicating superior edge-handling accuracy compared to other competing algorithms.

Fig. 4.

Profile intensity curve distributions of different algorithms for 120° limited-angle reconstruction.

Fig. 5.

Profile intensity curve distributions of different algorithms for 90° limited-angle reconstruction.

The reconstruction results are also quantitatively evaluated under two limited-angle cases of 120° and 90°, with the results summarized in Table I. Consistent with our visual assessments, APDM outperforms RDLS-TV, Restormer, and DDS, achieving the highest PSNR and SSIM scores. These results confirm the advantages of APDM and underscore its superior performance for limited-angle image reconstruction under the tested conditions.

TABLE I.

Quantitative evaluation results of XCAT dataset. Bold: Best, under: second best.

Method	120°		90°
	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑
RDLS-TV[24]	24.33 ± 1.25	0.79 ± 0.0007	20.2 ± 0.60	0.69 ± 0.0007
Restormer[41]	26.23 ± 2.12	0.89 ± 0.0005	23.89 ± 2.05	0.85 ± 0.0005
DDS[33]	28.04 ± 2.49	0.87 ± 0.0003	26.81 ± 1.11	0.84 ± 0.0004
Proposed	32.59 ± 2.63	0.95 ± 0.0001	28.34 ± 1.98	0.91 ± 0.0003

D. Clinical Cardiac Result

In this subsection, we present the results of real clinical dataset. Reconstruction from real cardiac data presents challenges due to the inherent complexity of anatomical structures and the substantial noise typically in authentic sinogram measurements. Moreover, the limited availability of relevant datasets further complicates accurate reconstruction compared to simulation-based approaches. Fig. 6 presents representative reconstruction results from the testing dataset to compare different methods for 120° limited-angle reconstruction. Our analysis reveals that while Restormer and DDS effectively reduce most of the limited-angle artifacts, they introduce additional deformations in the missing regions. Moreover, these methods often lose fine details, particularly in the area highlighted by the red circle. In contrast, the proposed method is the only one that successfully reconstructs the bone region. And as indicated by the blue arrow in the sagittal view, Restormer exhibits some pseudo-structures, which are absent in both DDS and our method. In the coronal view, the yellow arrow highlights residual streak-like artifacts on the surface of DDS, whereas Restormer and our method produce smoother results. In addition, in the cardiac area, our method reconstructs sharper results. Quantitative metrics, including PSNR and SSIM, confirm that the proposed method achieves the best image quality.

Fig. 6.

Reconstruction results from GE dataset with 120° projections using different methods.The rows on the right side of the figure show the sagittal results, while the rows at the bottom display the coronal results. The display window is [−400 1500] HU.

To provide region-specific evaluation, we computed SSIM values within the cardiac region of interest using representative deep learning–based reconstruction methods for comparison. In Fig. 7, the first image shows the ground-truth cardiac ROI, while the subsequent three images present the SSIM maps of the red-circled region for Restormer, DDS, and the proposed method. It can be seen that the proposed method provides the most distinct visualization of coronary vessels, with sharper boundaries and improved contrast, highlighting its effectiveness in reconstructing cardiac CT from limited-angle data. In addition, it achieves the highest SSIM score (0.88) among all compared approaches, further confirming its superior ability to preserve structural details in cardiac ROI and ensuring that the reconstructed images are not only visually clearer but also quantitatively more reliable.

Fig. 7.

Region-specific evaluation in the cardiac ROI. The first image shows the ground-truth cardiac ROI, while the following three images present SSIM maps of the red-circled region for Restormer, DDS, and the proposed method. The display window is [−400 1500] HU.

E. Ablation Study

1). Effectiveness of the 3D TV:

We evaluate the effectiveness of the 3D TV term and analyze how the number of RDLS iteration steps affects the quality of the final results and the overall computation time. Fig. 8 presents a comparative analysis of reconstruction results with and without the TV term. In addition, two different neighborhood sizes for the TV term are evaluated. Quantitative metrics demonstrate that the TV term improves both the perceptual quality and fidelity of the reconstructed images, with the best performance achieved using a neighborhood size of 26. Furthermore, as shown in Fig. 9, increasing the number of RDLS iterations significantly improves both PSNR and SSIM metrics.

Fig. 8.

Ablation study for the effectiveness of TV term with 120° projections using APDM. (a) w/o TV prior, (b) w/ 3D TV prior (TV neighbour=6), (c) w/ 3D TV prior (TV neighbour=26), and (d) Ground truth. The values displayed above each image represent PSNR (left) and SSIM (right), three columns represent different transaxial image slices. The display window is [−540 1000] HU.

Fig. 9.

PSNR (left) and SSIM (right) of APDM reconstructions with respect to inner iteration count (10, 20, 50, 100). Markers denote mean values and error bars show ±1 standard deviation—PSNR at true scale and SSIM error bars magnified by a factor of K = 30 for clarity.

2). Effectiveness of the Number of Inner Iterations:

In Table. II we reproduce the DDS results for four ADMM–CG configurations, where each curve converges to nearly the same PSNR beyond 50 inner-loop CG iterations. This indicates that varying the number of ADMM splits or inner CG steps has little impact on the final image quality, but the combination of ADMM = 5 and CG = 10 yields the highest peak PSNR of the group. Hence, we fix ADMM = 5 and CG = 10 as our reference setting. In contrast, our APDM reconstructions showed in Fig. 9 exhibit a clear, monotonic increase in both PSNR and SSIM as the inner-iteration count rises from 10 to 100. Because only a single “inner-iteration” parameter must be tuned, our method is both easier to control and more predictable in its quality gains.

TABLE II.

Number of Inner Iterations in DDS

ADMM	CG = 10		CG = 20		CG = 50		CG = 100
	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑	PSNR ↑	SSIM ↑
1	27.93 ± 2.34	0.87 ± 0.0003	27.76 ± 2.47	0.87 ± 0.0003	27.83 ± 2.33	0.87 ± 0.0003	27.79 ± 2.35	0.86 ± 0.0003
2	28.04 ± 2.49	0.87 ± 0.0003	27.82 ± 2.44	0.87 ± 0.0003	27.85 ± 2.08	0.86 ± 0.0003	27.98 ± 2.03	0.86 ± 0.0003
3	27.97 ± 2.59	0.87 ± 0.0003	27.79 ± 2.44	0.86 ± 0.0003	27.76 ± 2.12	0.85 ± 0.0003	27.63 ± 1.97	0.84 ± 0.0002
4	28.00 ± 2.56	0.87 ± 0.0003	27.79 ± 2.37	0.86 ± 0.0002	27.73 ± 2.04	0.84 ± 0.0002	27.68 ± 1.94	0.84 ± 0.0002
5	28.10 ± 2.55	0.87 ± 0.0003	27.79 ± 2.35	0.86 ± 0.0003	27.69 ± 1.99	0.84 ± 0.0002	27.51 ± 1.85	0.83 ± 0.0002

3). Effectiveness of the Indirect Projection:

In Section II-C, direct and indirect projection strategies are compared. Fig. 10 clearly shows that, at low NFE (Number of Function Evaluations, i.e. the total number of times the denoising network is evaluated during the diffusion sampling process), direct projection without the Tweedie denoising step results in poor image quality and only improves when the NFE becomes large. Hence, the 1,000–2,000 denoising steps are required as earlier methods [19], [20]. In contrast, indirect projection already delivers a reasonable reconstruction at NFE = 10. Moreover, our APDM approach achieves a nearly linear gain in PSNR as NFE increases, consistently outperforming DDS across the entire range. We observed that performance converges for NFE values between 50 and 100. Although increasing NFE to 100 yields an approximate 1 dB improvement in PSNR, it also doubles the runtime relative to NFE = 50. Consequently, we adopt NFE = 50 for both DDS and the proposed APDM to balance image quality and computational efficiency.

Fig. 10.

XCAT limited-angle (120°) CT reconstruction evaluation in terms of PSNR vs. NFE (log scale).

4). Computational Efficiency:

In the data-consistency module, APDM employs a lightweight RDLS algorithm that is particularly well optimized for GPU execution. At NFE = 50 on the simulation dataset (32 × 256 × 256), APDM completes reconstruction in 1 min 36 s, compared to DDS’s 2 min 56 s. While this 1.8× speedup is modest for small volumes, the clinical dataset, reconstructing a 512 × 512 × 100 volume to match typical medical imaging resolutions, reveals greater speedup: APDM takes only 10 min 11 s versus DDS’s 34 min 46 s, a 3.4× improvement. Moreover, as shown in Fig. 10, APDM maintains a PSNR above 30 even when reducing NFE to 20, indicating that further acceleration is feasible without sacrificing reconstruction quality. Together, these results highlight APDM’s superior scalability and its potential for real-time clinical applications.

IV. Discussion & Conclusion

In this study, we explore the application of DDIM integrated with the RDLS algorithm to address the challenges of 3D limited-angle reconstruction in cardiac CT imaging. Our findings highlight the potential of this approach to mitigate limited-angle artifacts, a persistent issue in enhancing temporal resolution to capture the dynamics of a beating heart. However, while the DDIM-accelerated strategy offers notable improvements over traditional DDPM, its speed performance in real-world, high-resolution datasets still lags behind CNNs in “on-the-fly” reconstruction speed for high-resolution real datasets, as it starts from Gaussian noise and requires multiple iterations, unlike CNNs’ single-pass approach.

Importantly, our experiments on XCAT simulations indicate that the integration of RDLS with a 3D-TV regularizer yields a more stable convergence behavior and better preservation of fine anatomical structures compared to conventional projection-based priors. Nevertheless, the semi-local nature of RDLS can lead to uneven reconstruction quality in regions with highly heterogeneous attenuation properties, such as at the boundaries of cardiac chambers and coronary vessels. More work is needed to adaptively tune the regularization weight λ or to employ spatially varying step-sizes in the RDLS updates, thereby balancing artifact suppression with edge fidelity.

From a practical standpoint, the computational burden of iterative diffusion sampling remains a barrier to clinical deployment. Beyond the diffusion bridge strategy discussed below, hybrid schemes that combine a coarse CNN initialization with a few DDIM-RDLS refinement steps may offer a promising compromise, dramatically reducing the total number of diffusion iterations without sacrificing reconstruction quality. Moreover, leveraging hardware accelerators and exploring low-rank approximations of A could further narrow the performance gap to single-pass CNN methods.

For future work, we will extend our evaluation to incorporate a wide range of real clinical cardiac datasets, including 4D cardiac CT scans, to evaluate robustness across varying noise levels and motion artifacts. We also plan to evaluate our method on next-generation helical photon-counting CT data which offer superior spectral resolution and dose efficiency, thereby demonstrating the pipeline’s broader clinical utility. Although the proposed reconstruction pipeline yields higher PSNR and SSIM compared to the baseline algorithms, further evaluation is necessary to determine whether these quantitative improvements can be translated into improved diagnostic performance. We plan to conduct a task-driven reader study involving experienced cardiovascular radiologists to quantify lesion detectability, stenosis grading accuracy, and inter-reader agreement. This study will directly answer the question whether the image quality improvements from our method can be translated into meaningful clinical impacts in diagnostic accuracy and patient care. Moreover, we intend to incorporate a flow-based ordinary differential equation (Flow ODE) framework into the diffusion sampling procedure. Although SDE approaches, even when accelerated via DDIM, typically require the order of 50 discretization steps, deterministic flow ODE solvers can reduce this requirement to approximately ten steps or fewer. Notably, recent innovations such as diffusion bridge models[42], [43] have achieved high-fidelity reconstructions with only ten function evaluations, and mean-flow methodologies[44] have advanced this paradigm further, edging closer to true one-step sampling. Finally, our goal is to integrate motion estimation modules directly into the diffusion process, enabling simultaneous reconstruction and motion correction for free-breathing or irregular-heartbeat acquisitions. These developments will be critical steps toward translating APDM into a clinically viable tool for real-time and artifact-free cardiac CT imaging.

In conclusion, we introduce APDM, a novel framework for 3D limited-angle cardiac CT reconstruction, which synergistically combines DDIM sampling with the RDLS-3DTV projection method. Validation in XCAT phantoms and clinical datasets demonstrates that our approach effectively reduces limited-angle artifacts and preserves critical anatomical detail under high temporal sampling constraints.