GSNR：图平滑空域表示用于逆问题
GSNR: Graph Smooth Null-Space Representation for Inverse Problems
AI总结：
本文提出GSNR方法，通过在不可见成分中施加结构，解决逆问题中空域信息约束不足的问题，提升重建精度。
摘要：
图像逆问题是病态的，由于传感矩阵的非平凡空域，导致存在无限多与测量一致的解。常见的图像先验在一般图像流形上促进解，如稀疏性、平滑性或得分函数。然而，这些先验不约束空域成分，可能偏移重建。因此，本文旨在将有意义的空域信息纳入重建框架。受图上平滑图像表示的启发，提出图平滑空域表示（GSNR），一种仅对不可见成分施加结构的机制。特别地，给定图拉普拉斯矩阵，我们构建一个空域受限的拉普拉斯矩阵，编码空域信号中相邻像素之间的相似性，并设计一个低维投影矩阵，从$p$-平滑的谱图模式（最低图频率）中获取。该方法在理论和实践中均有重要影响：i）通过空域仅图正则化提升收敛性；ii）更广泛的覆盖，空域方差由$p$模式捕捉多少；iii）高可预测性，这些模式从测量中推断多好。GSNR被纳入已知的逆问题求解器，如PnP、DIP和扩散求解器，在四个场景中：图像模糊消除、压缩感知、去马赛克和图像超分辨率，提供比基线方法提升高达4.3 dB，在PSNR上比端到端学习模型提升高达1 dB。
英文摘要：
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.
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