Split Gibbs diffusion posterior sampling for nonlinear inverse problems with application to electrical impedance tomography

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.