Typically, inversion algorithms assume that a forward model, which relates a source to its resulting measurements, is known and fixed. Using collected indirect measurements and the forward model, the goal becomes to recover the source. When the forward model is unknown, or imperfect, artifacts due to model mismatch occur in the recovery of the source. In this paper, we study the problem of blind inversion: solving an inverse problem with unknown or imperfect knowledge of the forward model parameters. We propose DeepGEM, a variational Expectation-Maximization (EM) framework that can be used to solve for the unknown parameters of the forward model in an unsupervised manner. DeepGEM makes use of a normalizing flow generative network to efficiently capture complex posterior distributions, which leads to more accurate evaluation of the source's posterior distribution used in EM. We showcase the effectiveness of our DeepGEM approach by achieving strong performance on the challenging problem of blind seismic tomography, where we significantly outperform the standard method used in seismology. We also demonstrate the generality of DeepGEM by applying it to a simple case of blind deconvolution.

When the forward model is unknown, or imperfect, artifacts due to model mismatch occur in the recovery of the source.

Typically, inversion algorithms assume that a forward model, which relates a source to its resulting measurements, is known and fixed. Using collected indirect measurements and the forward model, the goal becomes to recover the source. When the forward model is unknown, or imperfect, artifacts due to model mismatch occur in the recovery of the source. In this paper, we study the problem of blind inversion: solving an inverse problem with unknown or imperfect knowledge of the forward model parameters. We propose DeepGEM, a variational Expectation-Maximization (EM) framework that can be used to solve for the unknown parameters of the forward model in an unsupervised manner.

In numerous healthcare and other contemporary applications, the variables of primary interest are obtained through indirect measurements, such as in the case of Magnetic Resonance Imaging (MRI) and Computed Tomography (CT). For some of these applications, the reliability of the results is of particular importance. The accuracy and trustworthiness of the outcomes obtained through indirect measurements are significantly influenced by two critical factors: the degree of uncertainty associated with the measuring instrument and the appropriateness of the (forward) model used for the reconstruction of the parameters of interest (measurand). In this paper, we consider Bayesian inversion to obtain the measurand from signals measured by the instrument and a noise model that mimics both the instrument noise and the error of the forward model.