Stability of Image-Reconstruction Algorithms

Robustness and stability of image-reconstruction algorithms have recently come under scrutiny. Their importance to medical imaging cannot be overstated. We review the known results for the topical variational regularization strategies ( ℓ2 and ℓ1 regularization) and present novel stability results for ℓp -regularized linear inverse problems for p∈(1,∞) . Our results guarantee Lipschitz continuity for small p and Hölder continuity for larger p . They generalize well to the Lp (Ω)  function spaces.

Introduction

Inverse problems are at the core of computational imaging. Medical imaging critically depends on the guarantees provided by established image-reconstruction methodologies to inform diagnostic and treatment decisions. New techniques based on artificial intelligence and deep neural networks offer major average performance improvements in most applications [1], [2], [3], [4], at the cost of poor practical stability [5], [6] and a lack in theoretical guarantees. In particular, seemingly small perturbations of the measurements can produce large errors in the resulting image. Insidiously, these errors may incorporate deceptive patterns that look realistic because they were learnt from the training database (hallucination) [7].