Abstract
Linear and non-linear inverse problems in medical imaging, such as low-dose computed tomography (CT) and accelerated magnetic resonance imaging (MRI), are inherently ill-posed, presenting significant challenges in reconstructing high-fidelity clinical volumes from under-sampled or noisy measurements. Classical variational approaches guarantee stability and mathematical interpretability through explicit regularizers like Total Variation, yet frequently introduce over-smoothing or staircasing artifacts. Conversely, purely data-driven deep learning models deliver superior empirical performance but often lack mathematical convergence guarantees and generalizability. In this paper, we propose a rigorous variational unrolling framework that embeds deep convolutional proximal operators within an unrolled primal-dual hybrid gradient optimization scheme. By parameterizing the regularization functional via weakly convex neural networks, we guarantee the existence and stability of the minimizers while learning rich, data-adaptive image priors. We evaluate the proposed architecture on both sparse-view CT and sub-sampled multi-coil MRI benchmarks. Our results demonstrate that the variational deep learning framework significantly outperforms standard iterative and purely end-to-end architectures in terms of structural similarity (SSIM) and peak signal-to-noise ratio (PSNR), while retaining robust performance under adversarial perturbations and out-of-distribution noise regimes.
