Qingping Zhou (周清平)

Diffusion models have become particularly prominent for generating high-quality samples. Consequently, they have become widely adopted for modeling priors in image reconstruction. However, integrating diffusion priors into nonlinear inverse problems presents significant computational challenges, primarily due to the inherent complexity of nonlinear forward operators. This work presents a new approach for incorporating a diffusion prior into the split Gibbs sampler, denoted as DP-SGS, to sample from the posterior distribution in nonlinear settings. DP-SGS simplifies the complex posterior sampling problem by introducing an auxiliary variable that decomposes it into two simple conditional distributions: a forward model that handles the likelihood term and a prior model implicitly determined by the pre-trained diffusion model.Both conditional distributions are efficiently approximated as Gaussian distributions using first-order Taylor expansion for the prior model and Laplace approximation for the forward model, thereby ensuring computational tractability despite the nonlinear forward operator.Experiments on the challenging electrical impedance tomography (EIT) imaging problem demonstrate that our method enhances solution accuracy while achieving rapid convergence rates, confirming its effectiveness for solving nonlinear inverse problems.

In this paper, we study Bayesian approach for solving large scale linear inverse problems arising in various scientific and engineering fields. We propose a fused L_1/2 prior with edge-preserving and sparsity-promoting properties and show that it can be formulated as a Gaussian mixture Markov random field. Since the density function of this family of prior is neither log-concave nor Lipschitz, gradient-based Markov chain Monte Carlo methods can not be applied to sample the posterior. Thus, we present a Gibbs sampler in which all the conditional posteriors involved have closed form expressions. The Gibbs sampler works well for small size problems but it is computationally intractable for large scale problems due to the need for sample high dimensional Gaussian distribution. To reduce the computation burden, we construct a Gibbs bouncy particle sampler (Gibbs-BPS) based on a piecewise deterministic Markov process. This new sampler combines elements of Gibbs sampler with bouncy particle sampler and its computation complexity is an order of magnitude smaller. We show that the new sampler converges to the target distribution. With computed tomography examples, we demonstrate that the proposed method shows competitive performance with existing popular Bayesian methods and is highly efficient in large scale problems.

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. Although DuNets have been successfully applied to many linear inverse problems, their performance tends to be impaired by nonlinear problems. 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.