NPN: Non-Linear Projections of the Null Space for Imaging Inverse Problems
Henry Arguello, Roman Jacome, Romario Gualdrón-Hurtado, León Suárez-Rodríguez
Imaging inverse problems aim to recover high-dimensional signals from undersampled, noisy measurements, a fundamentally ill-posed task with infinite solutions lying in the null space of the sensing operator. To resolve this ambiguity, prior information is typically incorporated through handcrafted regularizers or learned models that constrain the solution space. However, these priors typically ignore the task-specific structure of that null-space.  In this work, we propose Non-linear Projections of the Null Space (NPN), a novel class of regularization that, instead of enforcing structural constraints in the image domain, promotes solutions that lie in a low-dimensional projection of the sensing matrix's null space with a neural network.Our approach has two key advantages: (1) Interpretability: by focusing on the structure of the null space, we design sensing-matrix-specific priors that capture information orthogonal to the signal components that are fundamentally blind to the sensing process. (2) Flexibility: NPN is adaptable to various inverse problems, compatible with existing reconstruction frameworks, and complementary to conventional image-domain priors. We provide theoretical guarantees on convergence and reconstruction accuracy when used within plug-and-play methods. Empirical results across diverse sensing matrices demonstrate that NPN priors consistently enhance reconstruction fidelity in various imaging inverse problems, such as compressive sensing, deblurring, and magnetic resonance imaging, with plug-and-play methods and unrolling networks.