Ensemble Kalman inversion (EKI) is an adaption of the ensemble Kalman filter (EnKF) for the numerical solution of inverse problems. Both EKI and EnKF suffer from the 'subspace property', i.e. the EKI and EnKF solutions are linear combinations of the initial ensembles. The subspace property implies that the ensemble size should be larger than the problem dimension to ensure EKI's convergence to the correct solution. This scaling of ensemble size is impractical and prevents the use of EKI in high-dimensional problems. 'Localization' has been used for many years in EnKF to break the subspace property in a way that a localized EnKF can solve high-dimensional problems with a modest ensemble size, independently of the number of unknowns. Here, we study localization of the EKI and demonstrate how a localized EKI (LEKI) can solve high-dimensional inverse problems with a modest ensemble size. Our analysis is mathematically rigorous and applies to the continuous time limit of the EKI. Specifically, we can prove an intended ensemble collapse and convergence guarantees with an ensemble size that is less than the number of unknowns, which sets this work apart from the current state-of-the-art. We illustrate our theory with numerical experiments where some of our mathematical assumptions may only be approximately valid.

Dictionary learning, aiming at representing a signal in terms of the atoms of a dictionary, has gained popularity in a wide range of applications, including, but not limited to, image denoising, face recognition, remote sensing, medical imaging and feature extraction. Dictionary learning can be seen as a possible data-driven alternative to solve inverse problems by identifying the data with possible outputs that are either generated numerically using a forward model or the results of earlier observations of controlled experiments. Sparse dictionary learning is particularly interesting when the underlying signal is known to be representable in terms of a few vectors in a given basis. In this paper, we propose to use hierarchical Bayesian models for sparse dictionary learning that can capture features of the underlying signals, e.g. sparse representation and nonnegativity. The same framework can be employed to reduce the dimensionality of an annotated dictionary through feature extraction, thus reducing the computational complexity of the learning task. Computed examples where our algorithms are applied to hyperspectral imaging and classification of electrocardiogram data are also presented.

Low-dose CT (LDCT) imaging attracted a considerable interest for the reduction of the object's exposure to x-ray radiation. In recent years, supervised deep learning (DL) has been extensively studied for LDCT image reconstruction, which trains a network over a dataset containing many pairs of normal-dose and low-dose images. However, the challenge on collecting many such pairs in the clinical setup limits the application of supervised-learning-based methods for LDCT image reconstruction in practice. Aiming at addressing the challenges raised by the collection of a training dataset, this paper proposed an unsupervised DL method for LDCT image reconstruction, which does not require any external training data. The proposed method is built on a re-parametrization technique for Bayesian inference via a deep network with random weights, combined with additional total variational regularization. The experiments show that the proposed method noticeably outperforms existing dataset-free image reconstruction methods on the test data.

Deep learning-based image reconstruction approaches have demonstrated impressive empirical performance in many imaging modalities. These approaches usually require a large amount of high-quality paired training data, which is often not available in medical imaging. To circumvent this issue we develop a novel unsupervised knowledge-transfer paradigm for learned reconstruction within a Bayesian framework. The proposed approach learns a reconstruction network in two phases. The first phase trains a reconstruction network with a set of ordered pairs comprising of ground truth images of ellipses and the corresponding simulated measurement data. The second phase fine-tunes the pretrained network to more realistic measurement data without supervision. By construction, the framework is capable of delivering predictive uncertainty information over the reconstructed image. We present extensive experimental results on low-dose and sparse-view computed tomography showing that the approach is competitive with several state-of-the-art supervised and unsupervised reconstruction techniques. Moreover, for test data distributed differently from the training data, the proposed framework can significantly improve reconstruction quality not only visually, but also quantitatively in terms of PSNR and SSIM, when compared with learned methods trained on the synthetic dataset only.