Xue-Cheng Tai

Blood flow imaging provides important information for hemodynamic behavior within the vascular system and plays an essential role in medical diagnosis and treatment planning. However, obtaining high-quality flow images remains a significant challenge. In this work, we address the problem of denoising flow images that may suffer from artifacts due to short acquisition times or device-induced errors. We formulate this task as an optimization problem, where the objective is to minimize the discrepancy between the modeled velocity field, constrained to satisfy the Navier–Stokes equations, and the observed noisy velocity data. To solve this problem, we decompose it into two subproblems: a fluid subproblem and a geometry subproblem. The fluid subproblem leverages a physics-informed neural network (PINN) to reconstruct the velocity field from noisy observations, assuming a fixed domain. The geometry subproblem aims to infer the underlying flow region by optimizing a quasi-conformal mapping that deforms a reference domain. These two subproblems are solved in an alternating Gauss–Seidel fashion, iteratively refining both the velocity field and the domain. Upon convergence, the framework yields a high-quality reconstruction of the flow image. We validate the proposed method through experiments on synthetic flow data in a converging channel geometry under varying levels of Gaussian noise and on real-life flow data in an aortic geometry with signal-dependent noise. The results demonstrate the effectiveness and robustness of the approach. Additionally, ablation studies are conducted to assess the influence of key hyperparameters.

The Transformer architecture has revolutionized the field of sequence modeling and underpins the recent breakthroughs in large language models (LLMs). However, a comprehensive mathematical theory that explains its structure and operations remains elusive. In this work, we propose a novel continuous framework that rigorously interprets the Transformer as a discretization of a structured integro-differential equation. Within this formulation, the self-attention mechanism emerges naturally as a nonlocal integral operator, and layer normalization is characterized as a projection to a time-dependent constraint. This operator-theoretic and variational perspective offers a unified and interpretable foundation for understanding the architecture’s core components, including attention, feedforward layers, and normalization. Our approach extends beyond previous theoretical analyses by embedding the entire Transformer operation in continuous domains for both token indices and feature dimensions. This leads to a principled and flexible framework that not only deepens on theoretical insight but also offers new directions for architecture design, analysis, and control-based interpretations. This new interpretation provides a step toward bridging the gap between deep learning architectures and continuous mathematical modeling, and contributes a foundational perspective to the ongoing development of interpretable and theoretically grounded neural network models.

Existing research highlights the crucial role of topological priors in image segmentation, particularly in preserving essential structures, such as connectivity and genus. Accurately capturing these topological features often requires incorporating width-related information, including the thickness and length inherent to the image structures. However, traditional mathematical definitions of topological structures lack this dimensional width information, limiting methods like persistent homology from fully addressing practical segmentation needs. To overcome this limitation, we propose a novel mathematical framework that explicitly integrates width information into the characterization of topological structures. This method leverages persistent homology, complemented by smoothing concepts from PDEs, to modify local extrema of upper level sets. This approach enables the resulting topological structures to inherently capture width properties. We incorporate this enhanced topological description into variational image segmentation models. Using some proper loss functions, we are also able to design neural networks that can segment images with the required topological and width properties. Through variational constraints on the relevant topological energies, our approach successfully preserves essential topological invariants, such as connectivity and genus counts, simultaneously ensuring that segmented structures retain critical width attributes, including line thickness and length. Numerical experiments demonstrate the effectiveness of our method, showcasing its capability to maintain topological fidelity while explicitly embedding width characteristics into segmented image structures.

The TV-Stokes model is a two-step variational method for image denoising that combines the estimation of a divergence-free tangent field with total variation regularization in the first step and then uses that to reconstruct the image in the second step. Although effective in practice, its mathematical structure and potential for parallelization have remained unexplored. In this work, we establish a rigorous functional-analytic foundation for the TV-Stokes model. We formulate both steps in appropriate infinite-dimensional function spaces, derive their dual formulations, and analyze the compatibility and mathematical consistency of the coupled system. In particular, we identify analytical inconsistencies in the original formulation and demonstrate how an alternative model resolves them. We also examine the orthogonal projection onto the divergence-free subspace, proving its existence in a continuous setting and establishing consistency with its discrete counterpart. Building on this theoretical framework, we develop the first domain decomposition method for TV-Stokes by applying overlapping Schwarz-type iterations to the duals of both steps. Although the divergence-free constraint gives rise to a global projection operator in the continuous model, we show that it becomes locally computable in the discrete setting. This insight enables a fully parallelizable algorithm suitable for large-scale image processing in memory-constrained environments. Numerical experiments demonstrate the correctness of the domain decomposition approach and its usability in parallel image reconstruction.

Existing research highlights the crucial role of topological priors in image segmentation, particularly in preserving essential structures such as connectivity and genus. Accurately capturing these topological features often requires incorporating width-related information, including the thickness and length inherent to the image structures. However, traditional mathematical definitions of topological structures lack this dimensional width information, limiting methods like persistent homology from fully addressing practical segmentation needs. To overcome this limitation, we propose a novel mathematical framework that explicitly integrates width information into the characterization of topological structures. This method leverages persistent homology, complemented by smoothing concepts from partial differential equations (PDEs), to modify local extrema of upper-level sets. This approach enables the resulting topological structures to inherently capture width properties. We incorporate this enhanced topological description into variational image segmentation models. Using some proper loss functions, we are also able to design neural networks that can segment images with the required topological and width properties. Through variational constraints on the relevant topological energies, our approach successfully preserves essential topological invariants such as connectivity and genus counts, simultaneously ensuring that segmented structures retain critical width attributes, including line thickness and length. Numerical experiments demonstrate the effectiveness of our method, showcasing its capability to maintain topological fidelity while explicitly embedding width characteristics into segmented image structures.

Positron Emission Tomography (PET) images reflect the metabolic rate of tracers in different tissues of the human body, crucial for early cancer diagnosis and treatment. Accurate tumor segmentation is essential to aid clinicians in determining drug dosages. Due to the low resolution of PET images, prior information (such as CT, MRI or distance information) are often incorporated to assist PET segmentation. In this paper, we propose an automatic 3D PET tumor segmentation framework assisted by geodesic sequences. Specifically, considering the intrinsic characteristics of PET images, we first construct geodesic prior, which effectively enhances the contrast between the tumor and background while suppressing noise and the influence of other tissues. To address the need for seed points in the geodesic prior, an automatic marking strategy is designed that identifies all suspected lesion regions and uses their central points as a series of seeds to generate the corresponding geodesic sequences. Subsequently, we develop a three-branch network architecture to simultaneously process PET images, geodesic sequences, and background geodesic information. To enhance image features, a distance attention mechanism is introduced at the end of the network encoder to effectively measure the similarity between different geodesic features, refining the image features. Finally, the network incorporates spatial regularization and local PET intensity information into the activation function via the Soft Threshold Dynamics with Local Intensity Fitting (STDLIF) module, further improving segmentation accuracy. Experimental results demonstrate that, compared to existing state-of-the-art algorithms, the proposed method shows better segmentation performance on both clinical and public datasets.

Selective segmentation has been a notable increase in interest in the use of interactive deep learning-based methods. However, developing an efficient and accurate segmentation method for medical applications remains a formidable challenge, primarily due to the inherent complexity and diversity of medical image structures. To address these challenges, we introduce a novel deep learning approach, termed the double-domain driven Unet method (DDUM), designed for the selective segmentation of medical images. Our approach utilizes a threshold geodesic distance in conjunction with the original images as input to construct a parallel Unet architecture that captures information from both the image domain and the geodesic distance domain. To further enhance the accuracy and efficacy of the segmentation process, we employ soft threshold dynamics as a replacement for the sigmoid activation function in the final layer. The efficacy of our proposed DDUM is substantiated through extensive experiments conducted on multiple medical image datasets. In particular, the DDUM method exhibits exceptional performance in terms of both segmentation accuracy and robustness.

Convex shapes (CSs) are common priors for image segmentation. It is important to design proper techniques to represent CS. So far, it remains a challenge to guarantee that the output objects from deep convolution neural networks (DCNNs) are CS. In this work, we propose a convex shape technique that can be easily integrated into the commonly used DCNN architectures and guarantee that outputs are CS. This method is flexible, and it can handle multiple objects and allow some of them to be convex. Our method is based on the dual representation of the sigmoid activation function in DCNNs. Moreover, our method can integrate spatial regularization and other shape priors by using a soft threshold dynamics (STD) method. This regularization can make the boundary curves of the segmented objects simultaneously smooth and convex. We design a very stable active set projection algorithm to solve our model numerically. This algorithm can form a new plug-and-play DCNN layer called CS-STD, whose outputs must be a nearly binary segmentation of convex objects. In the CS-STD block, the convexity information can be propagated to guide the DCNN in both forward and backward propagation during the training and prediction. As an application example, we apply the convexity prior layer to the segmentation of the retinal fundus image using the popular U-Net, BCDU, MedSAM and DeepLabV3+ as the backbone networks. Experimental results on several public datasets show that our method is efficient and outperforms classic DCNN segmentation methods.