Bayesian topology-aware reconstruction for difference EIT using a persistent homology prior

Abstract Image reconstruction in electrical impedance tomography (EIT) is a severely ill-posed inverse problem in which conventional local regularization often produces structurally implausible images, such as false merging of nearby targets or loss of lung separation. We propose a topology-aware reconstruction framework for difference EIT that incorporates persistent homology (PH) as a global prior on image connectivity within a linearized Bayesian formulation. The prior penalizes discrepancies between the persistence-diagram features of a candidate image and those of a reference image, thereby enforcing connectivity constraints beyond local smoothness. To make the proposed prior computationally tractable, a local finite-difference surrogate of the PH feature sensitivity is constructed to approximate a topological Jacobian, which is then embedded in a hybrid quadratic regularizer that combines a PH-induced precision matrix with a conventional Laplacian prior. The resulting framework is implemented using both a single-step linearized maximum a posteriori (MAP) reconstruction (a one-step Gauss–Newton solve) and a Gaussian variational inference scheme. The method is evaluated in numerical simulations with two inclusions over multiple noise levels, saline-tank phantom experiments with decreasing inter-target distance, and human thoracic EIT data acquired during tidal breathing and breath-hold. In simulations, the PH-informed prior preserves the intended two-inclusion topology and achieves the highest median correlation with the ground truth across the considered signal-to-noise ratios. In contrast, the Laplace, NOSER, and Tikhonov priors exhibit stronger topology-related artifacts and lower correlations. In phantom experiments, the PH prior maintains the separability of two conductive targets over a range of positions and improves the amplitude-response, position-error, and ringing metrics at the smallest separations. In the human thoracic data, the PH-informed reconstructions consistently preserve bilateral lung separation and yield temporally stable images during dynamic ventilation. These findings indicate that PH-derived priors can improve the topological fidelity of difference EIT reconstructions when a plausible reference topology is available.