GSNR: Graph Smooth Null-Space Representation for Inverse Problems

TL;DR

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

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.

Figures (24)

• Figure 1: For image SR task with $SRF=4, n= 3 \cdot 128^2$, we show ground-truth, adjoint reconstruction, NS projection, projection onto graph-smooth NS with $\mathbf{L}_{4nn}$ and $\mathbf{L}_{8nn}$.
• Figure 2: Coverage and spectral analysis for SR task. (a)$\mathbf{x^\ast} = \mathbf{H}^\top \mathbf{Hx^\ast+S}^\top\mathbf{Sx^\ast}$ for $\mathbf{S}$ given by \ref{['eq:S_definition']} using $\mathbf{L}=\mathbf{I}$ (orange), $\mathbf{L = L}_{4nn}$ (green) and $\mathbf{L=L}_{8nn}$ (blue) for different values of $p$. (b) RNSD representation error, varying $p$. (c) Variation of $\mathbf{T}$ normalized eigenvalues with respect to their index.
• Figure 3: Per-mode predictability for each case when $\mathbf{L} \in \{ \mathbf{I},\mathbf{L}_{8nn},\mathbf{L}_{4nn} \}$ for SR case with $SRF = 4$ with $n=64^2$.
• Figure 4: a) Coverage and b) Predictability vs. $p$ for CS with CIFAR-10 dataset. In this case, $m/n=0.1$ and $p = 1 \cdots n-m$.
• Figure 5: Results of DM-based solvers (DPS dps & DiffPIR DiffPIR) for Baseline, NPN Neurips, and GSNR with $\mathbf{L_{4nn}}$ and $\mathbf{L_{8nn}}$. Here, $p=0.1n$.

Theorems & Definitions (11)

• Definition 1: Gaussian Markov Random Field rue2005gaussian
• Theorem 1: Coverage for graph-smooth null-space
• Theorem 2: Minimax optimality
• Remark 1
• Proposition 1: Per–mode predictability bound
• Remark 2
• Theorem 1: Coverage for graph smooth null-space
• Corollary 1
• Theorem 2: Minimax optimality
• Remark 3