Abstract. The dose of X-ray radiation and the scanning time are crucial factors in computed tomography (CT) for clinical applications. In this work, we introduce a multisource static CT (MSCT) imaging system designed to rapidly acquire sparse view and limited angle data in CT imaging, addressing these critical factors. This linear imaging inverse problem is solved by a conditional generation process within the denoising diffusion image reconstruction framework. The noisy volume data sample generated by the reverse time diffusion process is projected onto the affine set to ensure its consistency with the measured data. To enhance the quality of the reconstruction, the 3D phantom’s orthogonal space projector is parameterized implicitly by a neural network. Then, a self-supervised learning algorithm is adopted to optimize the implicit neural representation. Through this multistage conditional generation process, we obtain a new approximate posterior sampling strategy for MSCT volume reconstruction. Numerical experiments are implemented with various imaging settings to verify the effectiveness of our methods for incomplete data MSCT volume reconstruction.

Verifiable formal languages like Lean have profoundly impacted mathematical reasoning, particularly through the use of large language models (LLMs) for automated reasoning. A significant challenge in training LLMs for these formal languages is the lack of parallel datasets that align natural language with formal language proofs. To address this challenge, this paper introduces a novel framework for translating the Mathlib4 corpus (a unified library of mathematics in formal language Lean 4) into natural language. Building upon this, we employ a dual augmentation strategy that combines tactic-based and informal-based approaches, leveraging the Lean-jixia system, a Lean 4 analyzer. We present the results of this pipeline on Mathlib4 as Herald (Hierarchy and Retrieval-based Translated Lean Dataset). We also propose the Herald Translator, which is fine-tuned on Herald. Herald translator achieves a 96.7\% accuracy (Pass@128) on formalizing statements in the miniF2F-test and a 23.5\% accuracy on our internal graduate-level textbook dataset, outperforming InternLM2-Math-Plus-7B (73.0\% and 7.5\%) and TheoremLlama (50.1\% and 4.0\%). Furthermore, we propose a section-level translation framework for real-world applications. As a direct application of Herald translator, we have successfully translated a template section in the Stack project, marking a notable progress in the automatic formalization of graduate-level mathematical literature. Our model, along with the datasets, are open-sourced to the public.

Abstract. Image restoration is critical across many fields, as it addresses the challenge of recovering clear images from degraded data. Two prominent approaches to this problem are wavelet-based methods and partial differential equation (PDE) models. Wavelet methods can be viewed as discrete analogs of PDE models, and through asymptotic analysis, wavelet models often converge to PDE-based approaches such as the total variation model. Wavelet methods are known for their simple implementation and multiscale time-frequency analysis, while PDE models provide a geometric interpretation of image structures, particularly edges. The relationship between these two approaches offers a comprehensive framework for image restoration. This paper designs a wavelet frame-based image restoration model, focusing on enhancing edge preservation and regularity. More importantly, we establish a connection to the [Formula: see text] version of the Mumford–Shah model, showing that the wavelet model converges to this variational model. This connection is significant, as it combines the geometric explanation of edges in the Mumford–Shah model with the simplicity of wavelet-based implementation. The primary contribution of this paper lies in the asymptotic analysis and proof of convergence of the two-scale wavelet model to the [Formula: see text] Mumford–Shah model, providing both theoretical insights and practical wavelet models for image restoration with enhanced edge detection.

Real-world medical image segmentation has tremendous long-tailed complexity of objects, among which tail conditions correlate with relatively rare diseases and are clinically significant. A trustworthy medical AI algorithm should demonstrate its effectiveness on tail conditions to avoid clinically dangerous damage in these out-of-distribution (OOD) cases. In this paper, we adopt the concept of object queries in Mask Transformers to formulate semantic segmentation as a soft cluster assignment. The queries fit the feature-level cluster centers of inliers during training. Therefore, when performing inference on a medical image in real-world scenarios, the similarity between pixels and the queries detects and localizes OOD regions. We term this OOD localization as MaxQuery. Furthermore, the foregrounds of real-world medical images, whether OOD objects or inliers, are lesions. The difference between them is less than that between the foreground and background, possibly misleading the object queries to focus redundantly on the background. Thus, we propose a query-distribution (QD) loss to enforce clear boundaries between segmentation targets and other regions at the query level, improving the inlier segmentation and OOD indication. Our proposed framework is tested on two real-world segmentation tasks, i.e., segmentation of pancreatic and liver tumors, outperforming previous state-of-the-art algorithms by an average of 7.39% on AUROC, 14.69% on AUPR, and 13.79% on FPR95 for OOD localization. On the other hand, our framework improves the performance of inlier segmentation by an average of 5.27% DSC when compared with the leading baseline nnUNet.

Though achieving excellent performance in some cases, current unsupervised learning methods for single image denoising usually have constraints in applications. In this paper, we propose a new approach which is more general and applicable to complicated noise models. Utilizing the property of score function, the gradient of logarithmic probability, we define a solving system for denoising. Once the score function of noisy images has been estimated, the denoised result can be obtained through the solving system. Our approach can be applied to multiple noise models, such as the mixture of multiplicative and additive noise combined with structured correlation. Experimental results show that our method is comparable when the noise model is simple, and has good performance in complicated cases where other methods are not applicable or perform poorly.

As one of the most challenging and practical segmentation tasks, open-world semantic segmentation requires the model to segment the anomaly regions in the images and incrementally learn to segment out-of-distribution (OOD) objects, especially under a few-shot condition. The current state-of-the-art (SOTA) method, Deep Metric Learning Network (DMLNet), relies on pixel-level metric learning, with which the identification of similar regions having different semantics is difficult. Therefore, we propose a method called region-aware metric learning (RAML), which first separates the regions of the images and generates region-aware features for further metric learning. RAML improves the integrity of the segmented anomaly regions. Moreover, we propose a novel meta-channel aggregation (MCA) module to further separate anomaly regions, forming high-quality sub-region candidates and thereby improving the model performance for OOD objects. To evaluate the proposed RAML, we have conducted extensive experiments and ablation studies on Lost And Found and Road Anomaly datasets for anomaly segmentation and the CityScapes dataset for incremental few-shot learning. The results show that the proposed RAML achieves SOTA performance in both stages of open world segmentation. Our code and appendix are available at https://github.com/czifan/RAML.

In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physics-informed neural networks (PINNs)method emerges to be a promising method for solving both forward and inverse PDE problems. PDEs with a point source that is expressed as a Dirac delta function in the governing equations are mathematical models of many physical processes. However, they cannot be solved directly by conventional PINNs method due to the singularity brought by the Dirac delta function. In this paper, we propose a universal solution to tackle this problem by proposing three novel techniques. Firstly the Dirac delta function is modeled as a continuous probability density function to eliminate the singularity at the point source; secondly a lower bound constrained uncertainty weighting algorithm is proposed to balance the physics-informed loss terms of point source area and the remaining areas; and thirdly a multi-scale deep neural network with periodic activation function is used to improve the accuracy and convergence speed. We evaluate the proposed method with three representative PDEs, and the experimental results show that our method outperforms existing deep learning based methods with respect to the accuracy, the efficiency and the versatility.

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc. Recently, building learning-based numerical solvers for parametric PDEs has become an emerging new field. One category of methods such as the Deep Galerkin Method (DGM) and Physics-Informed Neural Networks (PINNs) aim to approximate the solution of the PDEs. They are typically unsupervised and mesh-free, but require going through the time-consuming network training process from scratch for each set of parameters of the PDE. Another category of methods such as Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) try to approximate the solution mapping directly. Being fast with only one forward inference for each PDE parameter without retraining, they often require a large corpus of paired input-output observations drawn from numerical simulations, and most of them need a predefined mesh as well. In this paper, we propose Meta-Auto-Decoder (MAD), a mesh-free and unsupervised deep learning method that enables the pre-trained model to be quickly adapted to equation instances by implicitly encoding (possibly heterogenous) PDE parameters as latent vectors. The proposed method MAD can be interpreted by manifold learning in infinite-dimensional spaces, granting it a geometric insight. Extensive numerical experiments show that the MAD method exhibits faster convergence speed without losing accuracy than other deep learning-based methods.

Unsupervised domain adaption has recently been used to reduce the domain shift, which would ultimately improve the performance of the semantic segmentation on unlabeled real-world data. In this paper, we follow the trend to propose a novel method to reduce the domain shift using strategies of discriminator attention and self-training. The discriminator attention strategy contains a two-stage adversarial learning process, which explicitly distinguishes the well-aligned (domain-invariant) and poorly-aligned (domain-specific) features, and then guides the model to focus on the latter. The self-training strategy adaptively improves the decision boundary of the model for the target domain, which implicitly facilitates the extraction of domain-invariant features. By combining the two strategies, we find a more effective way to reduce the domain shift. Extensive experiments demonstrate the effectiveness of the proposed method on numerous benchmark datasets.

Low-dose computed tomography (LDCT) is desirable for both diagnostic imaging and image-guided interventions. Denoisers are widely used to improve the quality of LDCT. Deep learning (DL)-based denoisers have shown state-of-the-art performance and are becoming mainstream methods. However, there are two challenges to using DL-based denoisers: 1) a trained model typically does not generate different image candidates with different noise-resolution tradeoffs, which are sometimes needed for different clinical tasks; and 2) the model's generalizability might be an issue when the noise level in the testing images differs from that in the training dataset. To address these two challenges, in this work, we introduce a lightweight optimization process that can run on top of any existing DL-based denoiser during the testing phase to generate multiple image candidates with different noise-resolution tradeoffs suitable for different clinical tasks in real time. Consequently, our method allows users to interact with the denoiser to efficiently review various image candidates and quickly pick the desired one; thus, we termed this method deep interactive denoiser (DID). Experimental results demonstrated that DID can deliver multiple image candidates with different noise-resolution tradeoffs and shows great generalizability across various network architectures, as well as training and testing datasets with various noise levels.

X-ray Computed Tomography (CT) is widely used in clinical applications such as diagnosis and image-guided interventions. In this paper, we propose a new deep learning based model for CT image reconstruction with the backbone network architecture built by unrolling an iterative algorithm. However, unlike the existing strategy to include as many data-adaptive components in the unrolled dynamics model as possible, we find that it is enough to only learn the parts where traditional designs mostly rely on intuitions and experience. More specifically, we propose to learn an initializer for the conjugate gradient (CG) algorithm that involved in one of the subproblems of the backbone model. Other components, such as image priors and hyperparameters, are kept as the original design. Since a hypernetwork is introduced to inference on the initialization of the CG module, it makes the proposed model a certain meta-learning model. Therefore, we shall call the proposed model the meta-inversion network (MetaInv-Net). The proposed MetaInv-Net can be designed with much less trainable parameters while still preserves its superior image reconstruction performance than some state-of-the-art deep models in CT imaging. In simulated and real data experiments, MetaInv-Net performs very well and can be generalized beyond the training setting, i.e., to other scanning settings, noise levels, and data sets.

Missing data recovery is an important and yet challenging problem in imaging and data science. Successful models often adopt certain carefully chosen regularization. Recently, the low dimensional manifold model (LDMM) was introduced by [S. Osher, Z. Shi, and W. Zhu, Low Dimensional Manifold Model for Image Processing, Technical report, cam report 16-04, UCLA, Los Angeles, CA, 2016] and shown to be effective in image inpainting. The authors of [ Low Dimensional Manifold Model for Image Processing, Technical report, cam report 16-04, UCLA, Los Angeles, CA, 2016] observed that enforcing low dimensionality on the image patch manifold serves as a good image regularizer. In this paper, we observe that having only the low dimensional manifold regularization is not enough sometimes, and we need smoothness as well. For that, we introduce a new regularization by combining the low dimensional manifold regularization with a higher order \bf CUrvature \bf REgularization, which we call new regularization CURE for short. The key step of CURE is to solve a biharmonic equation on a manifold. We further introduce a weighted version of CURE, called WeCURE, in a similar manner as the weighted nonlocal Laplacian (WNLL) method [Z. Shi, S. Osher, and W. Zhu, Weighted nonlocal Laplacian on interpolation from sparse data, J. Sci. Comput., 73 (2017), pp. 1164--1177]. Numerical experiments for image inpainting and semisupervised learning show that the proposed CURE and WeCURE significantly outperform LDMM and WNLL, respectively.

CT image reconstruction from incomplete data, such as sparse views and limited angle reconstruction, is an important and challenging problem in medical imaging. This work proposes a new deep convolutional neural network (CNN), called JSR-Net, that jointly reconstructs CT images and their associated Radon domain projections. JSR-Net combines the traditional model based approach with deep architecture design of deep learning. A hybrid loss function is adopted to improve the performance of the JSR-Net making it more effective in protecting important image structures. Numerical experiments demonstrate that JSR-Net outperforms some latest model based reconstruction methods, as well as a recently proposed deep model.

Deep learning achieves state-of-the-art results in many tasks in computer vision and natural language processing. However, recent works have shown that deep networks can be vulnerable to adversarial perturbations which raised a serious robustness issue of deep networks. Adversarial training, typically formulated as a robust optimization problem, is an effective way of improving the robustness of deep networks. A major drawback of existing adversarial training algorithms is the computational overhead of the generation of adversarial examples, typically far greater than that of the network training. This leads to unbearable overall computational cost of adversarial training. In this paper, we show that adversarial training can be cast as a discrete time differential game. Through analyzing the Pontryagin’s Maximum Principle (PMP) of the problem, we observe that the adversary update is only coupled with the parameters of the first layer of the network. This inspires us to restrict most of the forward and back propagation within the first layer of the network during adversary updates. This effectively reduces the total number of full forward and backward propagation to only one for each group of adversary updates. Therefore, we refer to this algorithm YOPO (\textbf{Y}ou \textbf{O}nly \textbf{P}ropagate \textbf{O}nce). Numerical experiments demonstrate that YOPO can achieve comparable defense accuracy with \textbf{approximately 1/5 $\sim$ 1/4 GPU time} of the projected gradient descent (PGD) algorithm~\cite{kurakin2016adversarial}.

Quantitative susceptibility mapping (QSM) uses the phase data in magnetic resonance signals to visualize a three-dimensional susceptibility distribution by solving the magnetic field to susceptibility inverse problem. Due to the presence of zeros of the integration kernel in the frequency domain, QSM is an ill-posed inverse problem. Although numerous regularization-based models have been proposed to overcome this problem, incompatibility in the field data, which leads to deterioration of the recovery, has not received enough attention. In this paper, we show that the data acquisition process of QSM inherently generates a harmonic incompatibility in the measured local field. Based on this discovery, we propose a novel regularization-based susceptibility reconstruction model with an additional sparsity-based regularization term on the harmonic incompatibility. Numerical experiments show that the proposed method achieves better performance than existing approaches.

Diffusion magnetic resonance imaging (DMRI) suffers from lower signal-to-noise-ratio (SNR) due to MR signal attenuation associated with the motion of water molecules. To improve SNR, the non-local means (NLM) algorithm has demonstrated state-of-the-art performance in noise reduction. However, existing NLM algorithms do not take into account explicitly the fact that DMRI signal can vary significantly with local fiber orientations. Applying NLM naïvely can hence blur subtle structures and aggravate partial volume effects. To overcome this limitation, we improve NLM by performing neighborhood matching in non-flat domains and removing noise with information from both x -space (spatial domain) and q -space (wavevector domain). Specifically, we first encode the q -space sampling domain using a graph. We then perform graph framelet transforms to extract robust rotation-invariant features for each sampling point in x-q space. The resulting features are employed for robust neighborhood matching to locate recurrent information. Finally, we remove noise via an NLM framework. To adapt to the various types of noise in multi-coil MR imaging, we transform the signal before denoising so that it is Gaussian-distributed, allowing noise removal to be carried out in an unbiased manner. Our method is able to more effectively locate recurrent information in white matter structures with different orientations, avoiding the blurring effects caused by naïvely applying NLM. Experiments on synthetic, repetitively-acquired, and infant DMRI data demonstrate that our method is able to preserve subtle structures while effectively removing noise.

Partial differential equations (PDEs) play a prominent role in many disciplines of science and engineering. PDEs are commonly derived based on empirical observations. However, with the rapid development of sensors, computational power, and data storage in the past decade, huge quantities of data can be easily collected and efficiently stored. Such vast quantity of data offers new opportunities for data-driven discovery of physical laws. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to uncover the underlying hidden PDE models. Comparing with existing approaches, our approach has the most flexibility by learning both differential operators and the nonlinear response function of the underlying PDE model. A special feature of the proposed PDE-Net is that all filters are properly constrained, which enables us to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network. These constrains are carefully designed by fully exploiting the relation between the orders of differential operators and the orders of sum rules of filters (an important concept originated from wavelet theory). Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

Deep neural networks have become the state-of-the-art models in numerous machine learning tasks. However, general guidance to network architecture design is still missing. In our work, we bridge deep neural network design with numerical differential equations. We show that many effective networks, such as ResNet, PolyNet, FractalNet and RevNet, can be interpreted as different numerical discretizations of differential equations. This finding brings us a brand new perspective on the design of effective deep architectures. We can take advantage of the rich knowledge in numerical analysis to guide us in designing new and potentially more effective deep networks. As an example, we propose a linear multi-step architecture (LM-architecture) which is inspired by the linear multi-step method solving ordinary differential equations. The LM-architecture is an effective structure that can be used on any ResNet-like networks. In particular, we demonstrate that LM-ResNet and LM-ResNeXt (i.e. the networks obtained by applying the LM-architecture on ResNet and ResNeXt respectively) can achieve noticeably higher accuracy than ResNet and ResNeXt on both CIFAR and ImageNet with comparable numbers of trainable parameters. In particular, on both CIFAR and ImageNet, LM-ResNet/LM-ResNeXt can significantly compress (>50%) the original networks while maintaining a similar performance. This can be explained mathematically using the concept of modified equation from numerical analysis. Last but not least, we also establish a connection between stochastic control and noise injection in the training process which helps to improve generalization of the networks. Furthermore, by relating stochastic training strategy with stochastic dynamic system, we can easily apply stochastic training to the networks with the LM-architecture. As an example, we introduced stochastic depth to LM-ResNet and achieve significant improvement over the original LM-ResNet on CIFAR10.

High-density implants such as metals often lead to serious artifacts in reconstructed computerized tomographic (CT) images, which hampers the accuracy of image-based diagnosis and treatment planning. In this paper, we propose a novel wavelet frame--based CT image reconstruction model to reduce metal artifacts. This model is built on a joint spatial and Radon (projection) domain (JSR) image reconstruction framework with a built-in weighting and reweighting mechanism in the Radon domain to repair degraded projection data. The new weighting strategy used in the proposed model makes the regularization in the Radon domain by wavelet frame transform more effective. The proposed model, which will be referred to as the reweighted JSR model, combines the ideas of the recently proposed wavelet frame--based JSR model [B. Dong, J. Li, and Z. Shen, J. Sci. Comput., 54 (2013), pp. 333--349] and the normalized metal artifact reduction model [E. Meyer, R. Raupach, M. Lell, B. Schmidt, and M. Kachelriess, Med. Phys., 37 (2010), pp. 5482--5493.] and manages to achieve noticeably better CT reconstruction quality than both methods. To solve the proposed reweighted JSR model, an efficient alternative iteration algorithm is proposed with guaranteed convergence. Numerical experiments on both simulated and real CT image data demonstrate the effectiveness of the reweighted JSR model and its advantage over some state-of-the-art methods.

In diffusion MRI, the outcome of estimation problems can often be improved by taking into account the correlation of diffusion-weighted images scanned with neighboring wavevectors in q -space. For this purpose, we propose in this paper to employ tight wavelet frames constructed on non-flat domains for multi-scale sparse representation of diffusion signals. This representation is well suited for signals sampled regularly or irregularly, such as on a grid or on multiple shells, in q -space. Using spectral graph theory, the frames are constructed based on quasi-affine systems (i.e., generalized dilations and shifts of a finite collection of wavelet functions) defined on graphs, which can be seen as a discrete representation of manifolds. The associated wavelet analysis and synthesis transforms can be computed efficiently and accurately without the need for explicit eigen-decomposition of the graph Laplacian, allowing scalability to very large problems. We demonstrate the effectiveness of this representation, generated using what we call tight graph framelets , in two specific applications: denoising and super-resolution in q -space using \(\ell _{0}\) regularization. The associated optimization problem involves only thresholding and solving a trivial inverse problem in an iterative manner. The effectiveness of graph framelets is confirmed via evaluation using synthetic data with noncentral chi noise and real data with repeated scans. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Ultrasound video segmentation is a challenging task due to low contrast, shadow effects, complex noise statistics, and the need for high precision and efficiency in real time applications such as operation navigation and therapy planning. In this paper, we propose a wavelet frame based video segmentation framework incorporating different noise statistics and sequential distance shape priors. The proposed individual frame nonconvex segmentation model is solved by a proximal alternating minimization algorithm, and the convergence of the scheme is established based on the recently proposed Kurdyka--Łojasiewicz property. The performance of the overall method is demonstrated through numerical results on two real ultrasound video data sets. The proposed method is shown to achieve better results compared to the related level sets models and edge indicator shape priors, in terms of both segmentation quality and computational time.