GSNR: Graph Smooth Null-Space Representation for Inverse Problems

2026-02-23

AI Summary

This work addresses the non-uniqueness of solutions in imaging inverse problems, which arises from the non-trivial null space of the sensing matrix and often leads to reconstruction bias when conventional priors inadequately constrain null-space components. To overcome this limitation, the authors propose Graph-Smoothed Null-space Representation (GSNR), a novel approach that introduces graph signal smoothness into null-space modeling for the first time. By constructing a null-space-constrained graph Laplacian and leveraging the lowest p smooth spectral modes—corresponding to the smallest graph frequencies—to form a low-dimensional projection matrix, GSNR imposes structural constraints exclusively on the unobservable null-space components. This strategy achieves effective null-space regularization, substantially improving both null-space variance coverage and mode predictability. The method seamlessly integrates with mainstream frameworks such as Plug-and-Play (PnP), Deep Image Prior (DIP), and diffusion models, yielding consistent performance gains: across image deblurring, compressive sensing, demosaicing, and super-resolution tasks, it achieves up to a 4.3 dB PSNR improvement over baselines and outperforms end-to-end learned models by up to 1 dB.

Technology Category

• Computer Vision: Representation Learning for Vision
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

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

Inverse problems in imaging are ill-posed, leading to infinitely many solutions consistent with the measurements due to the non-trivial null-space of the sensing matrix. Common image priors promote solutions on the general image manifold, such as sparsity, smoothness, or score function. However, as these priors do not constrain the null-space component, they can bias the reconstruction. Thus, we aim to incorporate meaningful null-space information in the reconstruction framework. Inspired by smooth image representation on graphs, we propose Graph-Smooth Null-Space Representation (GSNR), a mechanism that imposes structure only into the invisible component. Particularly, given a graph Laplacian, we construct a null-restricted Laplacian that encodes similarity between neighboring pixels in the null-space signal, and we design a low-dimensional projection matrix from the $p$-smoothest spectral graph modes (lowest graph frequencies). This approach has strong theoretical and practical implications: i) improved convergence via a null-only graph regularizer, ii) better coverage, how much null-space variance is captured by $p$ modes, and iii) high predictability, how well these modes can be inferred from the measurements. GSNR is incorporated into well-known inverse problem solvers, e.g., PnP, DIP, and diffusion solvers, in four scenarios: image deblurring, compressed sensing, demosaicing, and image super-resolution, providing consistent improvement of up to 4.3 dB over baseline formulations and up to 1 dB compared with end-to-end learned models in terms of PSNR.

Problem

• inverse problems
• null-space
• image reconstruction
• ill-posedness
• graph smoothness

Innovation

• Graph Laplacian
• Null-space regularization
• Inverse problems
• Spectral graph modes
• Image reconstruction