In the recent years, deep learning techniques have shown great success in various tasks related to inverse problems, where a target quantity of interest can only be observed through indirect measurements of a forward operator. Common approaches apply deep neural networks in a post-processing step to the reconstructions obtained by classical reconstruction methods. However, the latter methods can be computationally expensive and introduce artifacts that are not present in the measured data and, in turn, can deteriorate the performance on the given task.

To overcome these limitations, we propose a class of equivariant neural networks that can be directly applied to the measurements to solve the desired task. To this end, we build appropriate network structures by developing layers that are equivariant with respect to data transformations induced by symmetries in the domain of the forward operator. We rigorously analyze the relation between the measurement operator and the resulting group representations and prove a representer theorem that characterizes the class of linear operators that translate between a given pair of group actions.

Based on this theory, we extend the existing concepts of Lie group equivariant deep learning to inverse problems and introduce new representations that are the results of the involved measurement operations. This allows us to efficiently solve classification, regression or even reconstruction tasks based on indirect measurements also for very sparse data problems, where a classical reconstruction based approach may be hard or even impossible. To illustrate the effectiveness of our approach, we perform numerical experiments on selected inverse problems and compare our results to existing methods.

The application of generative models in MRI reconstruction is shifting researchers' attention from the unrolled reconstruction networks to the probabilistic methods which can be used for unsupervised medical image reconstruction [1-4]. We formulate the image reconstruction problem from the perspective of Bayesian inference, which enables efficient sampling from the learned posterior probability distributions [1-2]. Different from conventional deep learning-based MRI reconstruction techniques, samples are drawn from the posterior distribution given the measured k-space using the Markov chain Monte Carlo (MCMC) method. Because the generative model can be learned from an image database independently from the forward operator, the same pre-trained models can be applied to k-space acquired with different sampling schemes or receive coils. Here, we present additional results in terms of the uncertainty of reconstruction, the transferability of learned information, and the comparison using data from the fastMRI challenge.

[1] G. Luo, M. Heide, M. Uecker. Using data-driven Markov chains for MRI reconstruction with Joint Uncertainty Estimation. Proc. Intl. Soc. Mag. Reson. Med. 30: 0298.

[2] G. Luo, M. Blumenthal, M. Heide, M. Uecker. Bayesian MRI reconstruction with joint uncertainty estimation using diffusion models. Magn. Reson. Med. 90: 295- 311, 2023.

[3] A. Jalal, M. Arvinte, G. Daras, E. Price, A. Dimakis, J. Tamir. Robust Compressed Sensing MRI with Deep Generative Priors. Neural Information Processing Systems 34: 14938–14954, 2021.

[4] H. Chung, C. Ye. Score-based diffusion models for accelerated MRI, Medical Image Analysis 80: 102479, 2022.

Photoacoustic tomography (PAT) is an imaging modality based on the photoacoustic effect. In the inverse problem of PAT, an initial pressure distribution induced by absorption of an externally introduced light is estimated from measured photoacoustic data. In the recent years, utilisation of machine learning in the inverse problem of PAT has gained significant interest. However, many of these machine learning-based methods do not provide information regarding the uncertainty of the reconstructed image.

In this work, we proposed a machine learning-based framework for the Bayesian inverse problem of PAT. The approach is based on the variational autoencoder (VAE) and the recently proposed uncertainty quantification variational autoencoder (UQ-VAE). In the VAE and UQ-VAE, an approximation of the true underlying posterior distribution is estimated by minimizing a divergence between the true and estimated posterior distributions using a neural network. The approach is evaluated using numerical simulations both in full and limited view measurement geometries with multiple levels of measurement noise.

The last decade brought us substantial progress in computational imaging techniques for current and next-generation interferometric telescopes, such as the SKA. Imaging methods have exploited sparsity and more recent deep learning architectures with remarkable results.  Despite good reconstruction quality, obtaining reliable uncertainty quantification (UQ) remains a common pitfall of most imaging methods. The UQ problem can be addressed by reformulating the inverse problem in the Bayesian framework. The posterior probability density function provides a comprehensive understanding of the uncertainties. However, computing the posterior in high-dimensional settings is an extremely challenging task. Posterior probabilities are often computed with sampling techniques, but these cannot yet cope with the high-dimensional settings from radio imaging.

This work proposes a method to address uncertainty quantification in radio-interferometric imaging with data-driven (learned) priors for very high-dimensional settings. Our model uses an analytic physically motivated model for the likelihood and exploits a data-driven prior learned from data. The proposed prior can encode complex information learned implicitly from training data and improves results from handcrafted priors (e.g., wavelet-based sparsity-promoting priors). We exploit recent advances in neural-network-based convex regularisers for the prior that allow us to ensure the log-concavity of the posterior while still being expressive. We leverage probability concentration phenomena of log-concave posterior functions that let us obtain information about the posterior avoiding the use of sampling techniques. Our method only requires the maximum-a-posteriori (MAP) estimation and evaluations of the likelihood and prior potentials. We rely on convex optimisation methods to compute the MAP estimation, which are known to be much faster and better scale with dimension than sampling strategies. The proposed method allows us to compute local credible intervals, i.e., Bayesian error bars, and perform hypothesis testing of structure on the reconstructed image. We demonstrate our method by reconstructing simulated radio-interferometric images and carrying out fast and scalable uncertainty quantification.