Computed Tomography serves as an indispensable tool in clinical workflows, providing non-invasive visualization of internal anatomical structures. Existing CT reconstruction works are limited to small-capacity model architecture and inflexible volume representation. In this work, we present X-GRM (X-ray Gaussian Reconstruction Model), a large feedforward model for reconstructing 3D CT volumes from sparse-view 2D X-ray projections. X-GRM employs a scalable transformer-based architecture to encode sparse-view X-ray inputs, where tokens from different views are integrated efficiently. Then, these tokens are decoded into a novel volume representation, named Voxel-based Gaussian Splatting (VoxGS), which enables efficient CT volume extraction and differentiable X-ray rendering. This combination of a high-capacity model and flexible volume representation, empowers our model to produce high-quality reconstructions from various testing inputs, including in-domain and out-domain X-ray projections. Our codes are available at: https://github.com/CUHK-AIM-Group/X-GRM.
X-ray imaging is indispensable in medical diagnostics, yet its use is tightly regulated due to potential health risks. To mitigate radiation exposure, recent research focuses on generating novel views from sparse inputs and reconstructing Computed Tomography (CT) volumes, borrowing representations from the 3D reconstruction area. However, these representations originally target visible light imaging that emphasizes reflection and scattering effects, while neglecting penetration and attenuation properties of X-ray imaging. In this paper, we introduce X-Field, the first 3D representation specifically designed for X-ray imaging, rooted in the energy absorption rates across different materials. To accurately model diverse materials within internal structures, we employ 3D ellipsoids with distinct attenuation coefficients. To estimate each material's energy absorption of X-rays, we devise an efficient path partitioning algorithm accounting for complex ellipsoid intersections. We further propose hybrid progressive initialization to refine the geometric accuracy of X-Filed and incorporate material-based optimization to enhance model fitting along material boundaries. Experiments show that X-Field achieves superior visual fidelity on both real-world human organ and synthetic object datasets, outperforming state-of-the-art methods in X-ray Novel View Synthesis and CT Reconstruction.
Deep learning has emerged as a powerful tool for solving inverse problems in imaging, including computed tomography (CT). However, most approaches require paired training data with ground truth images, which can be difficult to obtain, e.g., in medical applications. We present TomoSelfDEQ, a self-supervised Deep Equilibrium (DEQ) framework for sparse-angle CT reconstruction that trains directly on undersampled measurements. We establish theoretical guarantees showing that, under suitable assumptions, our self-supervised updates match those of fully-supervised training with a loss including the (possibly non-unitary) forward operator like the CT forward map. Numerical experiments on sparse-angle CT data confirm this finding, also demonstrating that TomoSelfDEQ outperforms existing self-supervised methods, achieving state-of-the-art results with as few as 16 projection angles.
Emerging unsupervised implicit neural representation (INR) methods, such as NeRP, NeAT, and SCOPE, have shown great potential to address sparse-view computed tomography (SVCT) inverse problems. Although these INR-based methods perform well in relatively dense SVCT reconstructions, they struggle to achieve comparable performance to supervised methods in sparser SVCT scenarios. They are prone to being affected by noise, limiting their applicability in real clinical settings. Additionally, current methods have not fully explored the use of image domain priors for solving SVCsT inverse problems. In this work, we demonstrate that imperfect reconstruction results can provide effective image domain priors for INRs to enhance performance. To leverage this, we introduce Self-prior embedding neural representation (Spener), a novel unsupervised method for SVCT reconstruction that integrates iterative reconstruction algorithms. During each iteration, Spener extracts local image prior features from the previous iteration and embeds them to constrain the solution space. Experimental results on multiple CT datasets show that our unsupervised Spener method achieves performance comparable to supervised state-of-the-art (SOTA) methods on in-domain data while outperforming them on out-of-domain datasets. Moreover, Spener significantly improves the performance of INR-based methods in handling SVCT with noisy sinograms. Our code is available at https://github.com/MeijiTian/Spener.
Learning effective regularization is crucial for solving ill-posed inverse problems, which arise in a wide range of scientific and engineering applications. While data-driven methods that parameterize regularizers using deep neural networks have demonstrated strong empirical performance, they often result in highly nonconvex formulations that lack theoretical guarantees. Recent work has shown that incorporating structured nonconvexity into neural network-based regularizers, such as weak convexity, can strike a balance between empirical performance and theoretical tractability. In this paper, we demonstrate that a broader class of nonconvex functions, difference-of-convex (DC) functions, can yield improved empirical performance while retaining strong convergence guarantees. The DC structure enables the use of well-established optimization algorithms, such as the Difference-of-Convex Algorithm (DCA) and a Proximal Subgradient Method (PSM), which extend beyond standard gradient descent. Furthermore, we provide theoretical insights into the conditions under which optimal regularizers can be expressed as DC functions. Extensive experiments on computed tomography (CT) reconstruction tasks show that our approach achieves strong performance across sparse and limited-view settings, consistently outperforming other weakly supervised learned regularizers. Our code is available at \url{https://github.com/YasminZhang/ADCR}.
Diffusion model have been successfully applied to many inverse problems, including MRI and CT reconstruction. Researchers typically re-purpose models originally designed for unconditional sampling without modifications. Using two different posterior sampling algorithms, we show empirically that such large networks are not necessary. Our smallest model, effectively a ResNet, performs almost as good as an attention U-Net on in-distribution reconstruction, while being significantly more robust towards distribution shifts. Furthermore, we introduce models trained on natural images and demonstrate that they can be used in both MRI and CT reconstruction, out-performing model trained on medical images in out-of-distribution cases. As a result of our findings, we strongly caution against simply re-using very large networks and encourage researchers to adapt the model complexity to the respective task. Moreover, we argue that a key step towards a general diffusion-based prior is training on natural images.
Objective: X-ray computed tomography employing sparse projection views has emerged as a contemporary technique to mitigate radiation dose. However, due to the inadequate number of projection views, an analytic reconstruction method utilizing filtered backprojection results in severe streaking artifacts. Recently, deep learning strategies employing image-domain networks have demonstrated remarkable performance in eliminating the streaking artifact caused by analytic reconstruction methods with sparse projection views. Nevertheless, it is difficult to clarify the theoretical justification for applying deep learning to sparse view CT reconstruction, and it has been understood as restoration by removing image artifacts, not reconstruction. Approach: By leveraging the theory of deep convolutional framelets and the hierarchical decomposition of measurement, this research reveals the constraints of conventional image- and projection-domain deep learning methodologies, subsequently, the research proposes a novel dual-domain deep learning framework utilizing hierarchical decomposed measurements. Specifically, the research elucidates how the performance of the projection-domain network can be enhanced through a low-rank property of deep convolutional framelets and a bowtie support of hierarchical decomposed measurement in the Fourier domain. Main Results: This study demonstrated performance improvement of the proposed framework based on the low-rank property, resulting in superior reconstruction performance compared to conventional analytic and deep learning methods. Significance: By providing a theoretically justified deep learning approach for sparse-view CT reconstruction, this study not only offers a superior alternative to existing methods but also opens new avenues for research in medical imaging.
Uncertainty quantification is crucial to account for the imperfect predictions of machine learning algorithms for high-impact applications. Conformal prediction (CP) is a powerful framework for uncertainty quantification that generates calibrated prediction intervals with valid coverage. In this work, we study how CP intervals are affected by bias - the systematic deviation of a prediction from ground truth values - a phenomenon prevalent in many real-world applications. We investigate the influence of bias on interval lengths of two different types of adjustments -- symmetric adjustments, the conventional method where both sides of the interval are adjusted equally, and asymmetric adjustments, a more flexible method where the interval can be adjusted unequally in positive or negative directions. We present theoretical and empirical analyses characterizing how symmetric and asymmetric adjustments impact the "tightness" of CP intervals for regression tasks. Specifically for absolute residual and quantile-based non-conformity scores, we prove: 1) the upper bound of symmetrically adjusted interval lengths increases by $2|b|$ where $b$ is a globally applied scalar value representing bias, 2) asymmetrically adjusted interval lengths are not affected by bias, and 3) conditions when asymmetrically adjusted interval lengths are guaranteed to be smaller than symmetric ones. Our analyses suggest that even if predictions exhibit significant drift from ground truth values, asymmetrically adjusted intervals are still able to maintain the same tightness and validity of intervals as if the drift had never happened, while symmetric ones significantly inflate the lengths. We demonstrate our theoretical results with two real-world prediction tasks: sparse-view computed tomography (CT) reconstruction and time-series weather forecasting. Our work paves the way for more bias-robust machine learning systems.
Computed tomography is a widely used imaging modality with applications ranging from medical imaging to material analysis. One major challenge arises from the lack of scanning information at certain angles, leading to distorted CT images with artifacts. This results in an ill-posed problem known as the Limited Angle Computed Tomography (LACT) reconstruction problem. To address this problem, we propose Residual Null-Space Diffusion Stochastic Differential Equations (RN-SDEs), which are a variant of diffusion models that characterize the diffusion process with mean-reverting (MR) stochastic differential equations. To demonstrate the generalizability of RN-SDEs, our experiments are conducted on two different LACT datasets, i.e., ChromSTEM and C4KC-KiTS. Through extensive experiments, we show that by leveraging learned Mean-Reverting SDEs as a prior and emphasizing data consistency using Range-Null Space Decomposition (RNSD) based rectification, RN-SDEs can restore high-quality images from severe degradation and achieve state-of-the-art performance in most LACT tasks. Additionally, we present a quantitative comparison of computational complexity and runtime efficiency, highlighting the superior effectiveness of our proposed approach.
Computed Tomography (CT) scans are the standard-of-care for the visualization and diagnosis of many clinical ailments, and are needed for the treatment planning of external beam radiotherapy. Unfortunately, the availability of CT scanners in low- and mid-resource settings is highly variable. Planar x-ray radiography units, in comparison, are far more prevalent, but can only provide limited 2D observations of the 3D anatomy. In this work we propose DIFR3CT, a 3D latent diffusion model, that can generate a distribution of plausible CT volumes from one or few (<10) planar x-ray observations. DIFR3CT works by fusing 2D features from each x-ray into a joint 3D space, and performing diffusion conditioned on these fused features in a low-dimensional latent space. We conduct extensive experiments demonstrating that DIFR3CT is better than recent sparse CT reconstruction baselines in terms of standard pixel-level (PSNR, SSIM) on both the public LIDC and in-house post-mastectomy CT datasets. We also show that DIFR3CT supports uncertainty quantification via Monte Carlo sampling, which provides an opportunity to measure reconstruction reliability. Finally, we perform a preliminary pilot study evaluating DIFR3CT for automated breast radiotherapy contouring and planning -- and demonstrate promising feasibility. Our code is available at https://github.com/yransun/DIFR3CT.
Rapid and accurate diagnosis of pneumothorax, utilizing chest X-ray and computed tomography (CT), is crucial for assisted diagnosis. Chest X-ray is commonly used for initial localization of pneumothorax, while CT ensures accurate quantification. However, CT scans involve high radiation doses and can be costly. To achieve precise quantitative diagnosis while minimizing radiation exposure, we proposed X-Recon, a CT ultra-sparse reconstruction network based on ortho-lateral chest X-ray images. X-Recon integrates generative adversarial networks (GANs), including a generator with a multi-scale fusion rendering module and a discriminator enhanced by 3D coordinate convolutional layers, designed to facilitate CT reconstruction. To improve precision, a projective spatial transformer is utilized to incorporate multi-angle projection loss. Additionally, we proposed PTX-Seg, a zero-shot pneumothorax segmentation algorithm, combining image processing techniques with deep-learning models for the segmentation of air-accumulated regions and lung structures. Experiments on a large-scale dataset demonstrate its superiority over existing approaches. X-Recon achieved a significantly higher reconstruction resolution with a higher average spatial resolution and a lower average slice thickness. The reconstruction metrics achieved state-of-the-art performance in terms of several metrics including peak signal-to-noise ratio. The zero-shot segmentation algorithm, PTX-Seg, also demonstrated high segmentation precision for the air-accumulated region, the left lung, and the right lung. Moreover, the consistency analysis for the pneumothorax chest occupancy ratio between reconstructed CT and original CT obtained a high correlation coefficient. Code will be available at: https://github.com/wangyunpengbio/X-Recon
Cone-Beam Computed Tomography (CBCT) is an indispensable technique in medical imaging, yet the associated radiation exposure raises concerns in clinical practice. To mitigate these risks, sparse-view reconstruction has emerged as an essential research direction, aiming to reduce the radiation dose by utilizing fewer projections for CT reconstruction. Although implicit neural representations have been introduced for sparse-view CBCT reconstruction, existing methods primarily focus on local 2D features queried from sparse projections, which is insufficient to process the more complicated anatomical structures, such as the chest. To this end, we propose a novel reconstruction framework, namely DIF-Gaussian, which leverages 3D Gaussians to represent the feature distribution in the 3D space, offering additional 3D spatial information to facilitate the estimation of attenuation coefficients. Furthermore, we incorporate test-time optimization during inference to further improve the generalization capability of the model. We evaluate DIF-Gaussian on two public datasets, showing significantly superior reconstruction performance than previous state-of-the-art methods.
In this paper, we present XctDiff, an algorithm framework for reconstructing CT from a single radiograph, which decomposes the reconstruction process into two easily controllable tasks: feature extraction and CT reconstruction. Specifically, we first design a progressive feature extraction strategy that is able to extract robust 3D priors from radiographs. Then, we use the extracted prior information to guide the CT reconstruction in the latent space. Moreover, we design a homogeneous spatial codebook to improve the reconstruction quality further. The experimental results show that our proposed method achieves state-of-the-art reconstruction performance and overcomes the blurring issue. We also apply XctDiff on self-supervised pre-training task. The effectiveness indicates that it has promising additional applications in medical image analysis. The code is available at:https://github.com/qingze-bai/XctDiff
Diffusion models can learn strong image priors from underlying data distribution and use them to solve inverse problems, but the training process is computationally expensive and requires lots of data. Such bottlenecks prevent most existing works from being feasible for high-dimensional and high-resolution data such as 3D images. This paper proposes a method to learn an efficient data prior for the entire image by training diffusion models only on patches of images. Specifically, we propose a patch-based position-aware diffusion inverse solver, called PaDIS, where we obtain the score function of the whole image through scores of patches and their positional encoding and utilize this as the prior for solving inverse problems. First of all, we show that this diffusion model achieves an improved memory efficiency and data efficiency while still maintaining the capability to generate entire images via positional encoding. Additionally, the proposed PaDIS model is highly flexible and can be plugged in with different diffusion inverse solvers (DIS). We demonstrate that the proposed PaDIS approach enables solving various inverse problems in both natural and medical image domains, including CT reconstruction, deblurring, and superresolution, given only patch-based priors. Notably, PaDIS outperforms previous DIS methods trained on entire image priors in the case of limited training data, demonstrating the data efficiency of our proposed approach by learning patch-based prior.
X-ray Computed Tomography (CT) is one of the most important diagnostic imaging techniques in clinical applications. Sparse-view CT imaging reduces the number of projection views to a lower radiation dose and alleviates the potential risk of radiation exposure. Most existing deep learning (DL) and deep unfolding sparse-view CT reconstruction methods: 1) do not fully use the projection data; 2) do not always link their architecture designs to a mathematical theory; 3) do not flexibly deal with multi-sparse-view reconstruction assignments. This paper aims to use mathematical ideas and design optimal DL imaging algorithms for sparse-view tomography reconstructions. We propose a novel dual-domain deep unfolding unified framework that offers a great deal of flexibility for multi-sparse-view CT reconstruction with different sampling views through a single model. This framework combines the theoretical advantages of model-based methods with the superior reconstruction performance of DL-based methods, resulting in the expected generalizability of DL. We propose a refinement module that utilizes unfolding projection domain to refine full-sparse-view projection errors, as well as an image domain correction module that distills multi-scale geometric error corrections to reconstruct sparse-view CT. This provides us with a new way to explore the potential of projection information and a new perspective on designing network architectures. All parameters of our proposed framework are learnable end to end, and our method possesses the potential to be applied to plug-and-play reconstruction. Extensive experiments demonstrate that our framework is superior to other existing state-of-the-art methods. Our source codes are available at https://github.com/fanxiaohong/MVMS-RCN.
Computed Tomography (CT) image reconstruction is crucial for accurate diagnosis and deep learning approaches have demonstrated significant potential in improving reconstruction quality. However, the choice of loss function profoundly affects the reconstructed images. Traditional mean squared error loss often produces blurry images lacking fine details, while alternatives designed to improve may introduce structural artifacts or other undesirable effects. To address these limitations, we propose Eagle-Loss, a novel loss function designed to enhance the visual quality of CT image reconstructions. Eagle-Loss applies spectral analysis of localized features within gradient changes to enhance sharpness and well-defined edges. We evaluated Eagle-Loss on two public datasets across low-dose CT reconstruction and CT field-of-view extension tasks. Our results show that Eagle-Loss consistently improves the visual quality of reconstructed images, surpassing state-of-the-art methods across various network architectures. Code and data are available at \url{https://github.com/sypsyp97/Eagle_Loss}.
X-ray is widely applied for transmission imaging due to its stronger penetration than natural light. When rendering novel view X-ray projections, existing methods mainly based on NeRF suffer from long training time and slow inference speed. In this paper, we propose a 3D Gaussian splatting-based framework, namely X-Gaussian, for X-ray novel view synthesis. Firstly, we redesign a radiative Gaussian point cloud model inspired by the isotropic nature of X-ray imaging. Our model excludes the influence of view direction when learning to predict the radiation intensity of 3D points. Based on this model, we develop a Differentiable Radiative Rasterization (DRR) with CUDA implementation. Secondly, we customize an Angle-pose Cuboid Uniform Initialization (ACUI) strategy that directly uses the parameters of the X-ray scanner to compute the camera information and then uniformly samples point positions within a cuboid enclosing the scanned object. Experiments show that our X-Gaussian outperforms state-of-the-art methods by 6.5 dB while enjoying less than 15% training time and over 73x inference speed. The application on sparse-view CT reconstruction also reveals the practical values of our method. Code is publicly available at https://github.com/caiyuanhao1998/X-Gaussian . A video demo of the training process visualization is at https://www.youtube.com/watch?v=gDVf_Ngeghg .
Low-Dose computer tomography (LDCT) is an ideal alternative to reduce radiation risk in clinical applications. Although supervised-deep-learning-based reconstruction methods have demonstrated superior performance compared to conventional model-driven reconstruction algorithms, they require collecting massive pairs of low-dose and norm-dose CT images for neural network training, which limits their practical application in LDCT imaging. In this paper, we propose an unsupervised and training data-free learning reconstruction method for LDCT imaging that avoids the requirement for training data. The proposed method is a post-processing technique that aims to enhance the initial low-quality reconstruction results, and it reconstructs the high-quality images by neural work training that minimizes the $\ell_1$-norm distance between the CT measurements and their corresponding simulated sinogram data, as well as the total variation (TV) value of the reconstructed image. Moreover, the proposed method does not require to set the weights for both the data fidelity term and the plenty term. Experimental results on the AAPM challenge data and LoDoPab-CT data demonstrate that the proposed method is able to effectively suppress the noise and preserve the tiny structures. Also, these results demonstrate the rapid convergence and low computational cost of the proposed method. The source code is available at \url{https://github.com/linfengyu77/IRLDCT}.
Deep neural networks used for reconstructing sparse-view CT data are typically trained by minimizing a pixel-wise mean-squared error or similar loss function over a set of training images. However, networks trained with such pixel-wise losses are prone to wipe out small, low-contrast features that are critical for screening and diagnosis. To remedy this issue, we introduce a novel training loss inspired by the model observer framework to enhance the detectability of weak signals in the reconstructions. We evaluate our approach on the reconstruction of synthetic sparse-view breast CT data, and demonstrate an improvement in signal detectability with the proposed loss.
Estimating and disentangling epistemic uncertainty, uncertainty that is reducible with more training data, and aleatoric uncertainty, uncertainty that is inherent to the task at hand, is critically important when applying machine learning to high-stakes applications such as medical imaging and weather forecasting. Conditional diffusion models' breakthrough ability to accurately and efficiently sample from the posterior distribution of a dataset now makes uncertainty estimation conceptually straightforward: One need only train and sample from a large ensemble of diffusion models. Unfortunately, training such an ensemble becomes computationally intractable as the complexity of the model architecture grows. In this work we introduce a new approach to ensembling, hyper-diffusion models (HyperDM), which allows one to accurately estimate both epistemic and aleatoric uncertainty with a single model. Unlike existing single-model uncertainty methods like Monte-Carlo dropout and Bayesian neural networks, HyperDM offers prediction accuracy on par with, and in some cases superior to, multi-model ensembles. Furthermore, our proposed approach scales to modern network architectures such as Attention U-Net and yields more accurate uncertainty estimates compared to existing methods. We validate our method on two distinct real-world tasks: x-ray computed tomography reconstruction and weather temperature forecasting.