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.
Abstract (translated)
在成像中的逆问题通常是不适定的，这导致由于感知矩阵非平凡零空间的存在，存在无数个与测量结果一致的解。常见的图像先验假设倾向于推广一般的图像流形上的解决方案，例如稀疏性、平滑度或分数函数等。然而，这些先验假设并不能约束零空间的部分，因此可能会偏置重建过程。我们的目标是将有意义的零空间信息纳入到重建框架中。 受图上平滑表示启发，我们提出了一种名为“Graph-Smooth Null-Space Representation”(GSNR) 的机制，该方法仅对不可见部分施加结构约束。具体而言，在给定一个图拉普拉斯算子的情况下，我们可以构建出一种受限的零空间拉普拉斯算子，用来编码在零空间信号中的相邻像素之间的相似性，并设计了一个从 $p$ 光滑谱图模式（最低图频率）映射到低维投影矩阵的机制。这种方法具有强大的理论和实践意义：一、通过一个仅限于零空间的图正则化器改进收敛性；二、更好的覆盖率，即多少零空间方差可以通过 $p$ 模式捕获；三、高预测能力，即这些模式从测量中可以多好地被推断出来。 GSNR 被集成到几种已知的逆问题求解器中，如PnP、DIP 和扩散求解器，在四种场景下进行了测试：图像去模糊化、压缩感知、拜耳插值和超分辨率成像。与基线方法相比，GSNR 在峰值信噪比（PSNR）方面提供了最多达 4.3 分贝的改进；在对比于端到端学习模型时，也有高达1分贝的优势。
URL
https://arxiv.org/abs/2602.20328
