NPN: Non-Linear Projections of the Null-Space for Imaging Inverse Problems

2025-10-01

AI Summary

Imaging inverse problems under undersampling and noise suffer from fundamental ill-posedness, leading to non-unique solutions. To address this, we propose a neural regularization framework grounded in nullspace structural priors. Our core innovation lies in employing a neural network to learn a low-dimensional nonlinear projection onto the nullspace of the forward operator, which is then embedded into the reconstruction pipeline to enforce task-adaptive and interpretable structural constraints. Unlike conventional methods, our approach does not require explicit image-domain priors and is compatible with diverse paradigms—including plug-and-play, unrolled networks, deep image priors, and diffusion models. We demonstrate significant improvements in reconstruction accuracy across multiple imaging modalities: compressed sensing, deblurring, super-resolution, computed tomography (CT), and magnetic resonance imaging (MRI). Experimental results validate the effectiveness and generality of nullspace modeling as a novel regularization mechanism for ill-posed inverse problems.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Structured Learning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Imaging inverse problems aims to recover high-dimensional signals from undersampled, noisy measurements, a fundamentally ill-posed task with infinite solutions 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 extit{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, super-resolution, computed tomography, and magnetic resonance imaging, with plug-and-play methods, unrolling networks, deep image prior, and diffusion models.

Problem

• Recovering high-dimensional signals from undersampled noisy measurements
• Addressing ill-posed inverse problems with infinite null-space solutions
• Incorporating task-specific null-space structure into regularization methods

Innovation

• Neural network projects null-space to low dimensions
• Sensing-matrix-specific priors capture orthogonal information
• Flexible regularization compatible with existing reconstruction frameworks