Learning Regularization Parameter-Maps for Variational Image Reconstruction Using Deep Neural Networks and Algorithm Unrolling

We introduce a method for the fast estimation of data-adapted, spatially and temporally dependent regularization parameter-maps for variational image reconstruction, focusing on total variation (TV) minimization. The proposed approach is inspired by recent developments in algorithm unrolling using deep neural networks (NNs) and relies on two distinct subnetworks. The first subnetwork estimates the regularization parameter-map from the input data. The second subnetwork unrolls T iterations of an iterative algorithm which approximately solves the corresponding TV-minimization problem incorporating the previously estimated regularization parameter-map. The overall network is then trained end-to-end in a supervised learning fashion using pairs of clean and corrupted data but crucially without the need for access to labels for the optimal regularization parameter-maps. We first prove consistency of the unrolled scheme by showing that the unrolled minimizing energy functional used for the supervised learning Gamma-converges, as T tends to infinity, to the corresponding functional that incorporates the exact solution map of the TV-minimization problem. Then, we apply and evaluate the proposed method on a variety of large-scale and dynamic imaging problems with retrospectively simulated measurement data for which the automatic computation of such regularization parameters has been so far challenging using the state-of-the-art methods: a 2D dynamic cardiac magnetic resonance imaging (MRI) reconstruction problem, a quantitative brain MRI reconstruction problem, a low-dose computed tomography problem, and a dynamic image denoising problem. The proposed method consistently improves the TV reconstructions using scalar regularization parameters, and the obtained regularization parameter-maps adapt well to imaging problems and data by leading to the preservation of detailed features. Although the choice of the regularization parameter-maps is data-driven and based on NNs, the subsequent reconstruction algorithm is interpretable since it inherits the properties (e.g., convergence guarantees) of the iterative reconstruction method from which the network is implicitly defined.