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 \textit{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.
Abstract (translated)
图像逆问题的目标是从欠采样且含有噪声的测量中恢复高维信号，这是一个本质上无法确定的问题，在感知算子的零空间中有无限多个解。为了消除这种不确定性，通常通过手工设计的正则化器或学习模型来整合先验信息以约束解的空间范围。然而，这些先验信息往往忽略了该零空间中的任务特定结构。在本文中，我们提出了“非线性零空间投影”（NPN），这是一种新的正则化方法，它不通过图像域施加结构性限制，而是利用神经网络来促进那些位于感知矩阵的低维零空间投影上的解。我们的方法有两大优势： 1. **可解释性**：通过关注零空间结构，我们设计了与感知过程本质盲目的信号成分正交的信息特异于感知矩阵的先验。 2. **灵活性**：NPN能够适应各种逆问题，可以与现有的重建框架兼容，并且是对传统图像域先验的一种补充。我们在插件式方法中提供了关于收敛性和重构准确性的理论保证。 实验结果表明，在多种不同的感知矩阵下，NPN在压缩感知、去模糊化、超分辨率、计算机断层扫描和磁共振成像等各类成像逆问题中的重构保真度上始终有所提升。这些改进适用于插件式方法、展卷网络、深度图像先验以及扩散模型等多种技术框架内。
URL
https://arxiv.org/abs/2510.01608
