Xiang Cao (曹翔)

The regularized D-bar method is a popular method for solving electrical impedance tomography (EIT) problems due to its efficiency and simplicity. It utilizes the low-pass truncated scattering data in the non-linear Fourier domain to solve the associated D-bar integral equations, yielding a smooth conductivity approximation. However, the D-bar reconstruction often presents low contrast and resolution due to the absence of accurate high-frequency information and the ill-posedness of the problem. In this paper, we propose a deep learning-based supervised approach for real-time EIT reconstruction. Based on the D-bar method, we propose to utilize both multi-scale frequency enhancement and spatial consistency for a high-image-quality reconstruction. Additionally, we propose a fixed-point iteration for solving discrete D-bar systems on GPUs for fast computation. Numerical results are performed for both the continuum model and complete electrode model simulation on KIT4 and ACT4 datasets to demonstrate notable improvements in absolute EIT imaging quality.

Ultrasound Computed Tomography (USCT) constitutes a nonlinear inverse problem with inherent ill-posedness that can benefit from regularization through diffusion generative priors. However, traditional approaches for solving Helmholtz equation-constrained USCT face three fundamental challenges when integrating these priors: PDE-constrained gradient computation, discretization-induced approximation errors, and computational imbalance between neural networks and numerical PDE solvers. In this work, we introduce Diff-ANO (Diffusion-based Models with Adjoint Neural Operators), a novel framework that combines conditional consistency models with adjoint operator learning to address these limitations. Our two key innovations include: (1) a conditional consistency model that enables measurement-conditional few-step sampling by directly learning a self-consistent mapping from diffusion trajectories, and (2) an adjoint operator learning module that replaces traditional PDE solvers with neural operator surrogates for efficient adjoint-based gradient computation. To enable practical deployment, we introduce the batch-based Convergent Born Series (BCBS)–a memory-efficient strategy for online generation of neural operator training pairs. Comprehensive experiments demonstrate that Diff-ANO significantly improves both computational efficiency and reconstruction quality, especially under sparse-view and partial-view measurement scenarios.

Diffusion models have been widely studied as effective generative tools for solving inverse problems. The main ideas focus on performing the reverse sampling process conditioned on noisy measurements, using well-established numerical solvers for gradient updates. Although diffusion-based sampling methods can produce high-quality reconstructions, challenges persist in nonlinear PDE-based inverse problems and sampling speed. In this work, we explore solving PDE-based travel-time tomography based on subspace diffusion generative models. Our main contributions are twofold: first, we propose a posterior sampling process for PDE-based inverse problems by solving the associated adjoint-state equation in a plug-and-play fashion. Second, we present a subspace-based dimension reduction technique, enabling solving PDE-based inverse problems from coarse to refined grids, for conditional sampling acceleration. Our numerical experiments showed satisfactory advancements in improving the travel-time imaging quality and reducing the sampling time for reconstruction.