Qingping Zhou

Qingping Zhou, Jiayu Qian, Junqi Tang et al.

Combining the strengths of model-based iterative algorithms and data-driven deep learning solutions, deep unrolling networks (DuNets) have become a popular tool to solve inverse imaging problems. While DuNets have been successfully applied to many linear inverse problems, nonlinear problems tend to impair the performance of the method. Inspired by momentum acceleration techniques that are often used in optimization algorithms, we propose a recurrent momentum acceleration (RMA) framework that uses a long short-term memory recurrent neural network (LSTM-RNN) to simulate the momentum acceleration process. The RMA module leverages the ability of the LSTM-RNN to learn and retain knowledge from the previous gradients. We apply RMA to two popular DuNets -- the learned proximal gradient descent (LPGD) and the learned primal-dual (LPD) methods, resulting in LPGD-RMA and LPD-RMA respectively. We provide experimental results on two nonlinear inverse problems: a nonlinear deconvolution problem, and an electrical impedance tomography problem with limited boundary measurements. In the first experiment we have observed that the improvement due to RMA largely increases with respect to the nonlinearity of the problem. The results of the second example further demonstrate that the RMA schemes can significantly improve the performance of DuNets in strongly ill-posed problems.

Jiayu Qian, Yuanyuan Liu, Jingya Yang et al.

Bayesian inference with deep generative prior has received considerable interest for solving imaging inverse problems in many scientific and engineering fields. The selection of the prior distribution is learned from, and therefore an important representation learning of, available prior measurements. The SA-Roundtrip, a novel deep generative prior, is introduced to enable controlled sampling generation and identify the data's intrinsic dimension. This prior incorporates a self-attention structure within a bidirectional generative adversarial network. Subsequently, Bayesian inference is applied to the posterior distribution in the low-dimensional latent space using the Hamiltonian Monte Carlo with preconditioned Crank-Nicolson (HMC-pCN) algorithm, which is proven to be ergodic under specific conditions. Experiments conducted on computed tomography (CT) reconstruction with the MNIST and TomoPhantom datasets reveal that the proposed method outperforms state-of-the-art comparisons, consistently yielding a robust and superior point estimator along with precise uncertainty quantification.

Guixian Xu, Huihui Wang, Qingping Zhou

Electrical Impedance Tomography (EIT) is widely applied in medical diagnosis, industrial inspection, and environmental monitoring. Combining the physical principles of the imaging system with the advantages of data-driven deep learning networks, physics-embedded deep unrolling networks have recently emerged as a promising solution in computational imaging. However, the inherent nonlinear and ill-posed properties of EIT image reconstruction still present challenges to existing methods in terms of accuracy and stability. To tackle this challenge, we propose the learned half-quadratic splitting (HQSNet) algorithm for incorporating physics into learning-based EIT imaging. We then apply Anderson acceleration (AA) to the HQSNet algorithm, denoted as AA-HQSNet, which can be interpreted as AA applied to the Gauss-Newton step and the learned proximal gradient descent step of the HQSNet, respectively. AA is a widely-used technique for accelerating the convergence of fixed-point iterative algorithms and has gained significant interest in numerical optimization and machine learning. However, the technique has received little attention in the inverse problems community thus far. Employing AA enhances the convergence rate compared to the standard HQSNet while simultaneously avoiding artifacts in the reconstructions. Lastly, we conduct rigorous numerical and visual experiments to show that the AA module strengthens the HQSNet, leading to robust, accurate, and considerably superior reconstructions compared to state-of-the-art methods. Our Anderson acceleration scheme to enhance HQSNet is generic and can be applied to improve the performance of various physics-embedded deep learning methods.

Chen Cheng, Qingping Zhou

This paper investigates the application of unsupervised learning methods for computed tomography (CT) reconstruction. To motivate our work, we review several existing priors, namely the truncated Gaussian prior, the $l_1$ prior, the total variation prior, and the deep image prior (DIP). We find that DIP outperforms the other three priors in terms of representational capability and visual performance. However, the performance of DIP deteriorates when the number of iterations exceeds a certain threshold due to overfitting. To address this issue, we propose a novel method (MCDIP-ADMM) based on Multi-Code Deep Image Prior and plug-and-play Alternative Direction Method of Multipliers. Specifically, MCDIP utilizes multiple latent codes to generate a series of feature maps at an intermediate layer within a generator model. These maps are then composed with trainable weights, representing the complete image prior. Experimental results demonstrate the superior performance of the proposed MCDIP-ADMM compared to three existing competitors. In the case of parallel beam projection with Gaussian noise, MCDIP-ADMM achieves an average improvement of 4.3 dB over DIP, 1.7 dB over ADMM DIP-WTV, and 1.2 dB over PnP-DIP in terms of PSNR. Similarly, for fan-beam projection with Poisson noise, MCDIP-ADMM achieves an average improvement of 3.09 dB over DIP, 1.86 dB over ADMM DIP-WTV, and 0.84 dB over PnP-DIP in terms of PSNR.

Hongji Wang, Hongqiao Wang, Jinyong Ying et al.

Inverse problems governed by partial differential equations (PDEs) play a crucial role in various fields, including computational science, image processing, and engineering. Particularly, Darcy flow equation is a fundamental equation in fluid mechanics, which plays a crucial role in understanding fluid flow through porous media. Bayesian methods provide an effective approach for solving PDEs inverse problems, while their numerical implementation requires numerous evaluations of computationally expensive forward solvers. Therefore, the adoption of surrogate models with lower computational costs is essential. However, constructing a globally accurate surrogate model for high-dimensional complex problems demands high model capacity and large amounts of data. To address this challenge, this study proposes an efficient locally accurate surrogate that focuses on the high-probability regions of the true likelihood in inverse problems, with relatively low model complexity and few training data requirements. Additionally, we introduce a sequential Bayesian design strategy to acquire the proposed surrogate since the high-probability region of the likelihood is unknown. The strategy treats the posterior evolution process of sequential Bayesian design as a Gaussian process, enabling algorithmic acceleration through one-step ahead prior. The complete algorithmic framework is referred to as Sequential Bayesian design for locally accurate surrogate (SBD-LAS). Finally, three experiments based the Darcy flow equation demonstrate the advantages of the proposed method in terms of both inversion accuracy and computational speed.