Unrolled NESTA: Constructing stable, accurate and efficient neural networks for gradient-sparse imaging problems

Compressive imaging is vital in computational science, engineering and medicine. Its aim is to perform the challenging task of reconstructing images from highly undersampled physical measurements. Deep learning has shown substantial potential to outperform standard techniques for compressive imaging, with empirical evidence indicating superior accuracy. However, deep learning approaches are fraught with many key issues, including hallucinations, instabilities and unpredictable generalization. This motivates a growing body of research to construct accurate neural networks with stability guarantees. In this thesis, we construct stable, accurate and efficient neural networks designed to tackle Fourier imaging problems under a gradient-sparse image model. The networks are constructed by unrolling a novel optimization algorithm based on NESTA, which reconstructs images from undersampled Fourier measurements via TV minimization. To enable fast image reconstruction, we apply a restart scheme which leads to the number of network layers growing logarithmically in the desired image error. Finally, we validate and explore our findings in a series of numerical experiments. The main impact of our work is the construction of neural networks that achieve the same performance guarantees as state-of-the-art handcrafted methods for gradient-sparse imaging.