NPN: Non-Linear Projections of the Null-Space for Imaging Inverse Problems

TL;DR

This work tackles ill-posed imaging inverse problems by introducing Non-Linear Projections of the Null-Space (NPN), a regularization strategy that constrains reconstructions to a low-dimensional, learned subspace of the sensing matrix null-space. A projection matrix $\mathbf{S}$ with rows in $\operatorname{Null}(\mathbf{H})$ and a neural network $\mathrm{G}^*: \mathbb{R}^m\to\mathbb{R}^p$ predict $\mathbf{Sx}$ from measurements $\mathbf{y}$, enabling a data-adaptive prior via the term $\gamma\|\mathrm{G}^*(\mathbf{y})-\mathbf{S}\tilde{\mathbf{x}}\|_2^2$ integrated into plug-and-play, unrolled, DIP, and diffusion-model solvers. The authors provide convergence guarantees for PnP-NPN under RIP-like conditions and Lipschitz assumptions, and demonstrate improved reconstruction fidelity across CS, SR, CT, MRI, and deblurring, along with enhanced convergence speed in practice. The framework is designed to be adaptable to various sensing matrices and reconstruction pipelines, offering interpretability by targeting the null-space directions that are fundamentally unobservable by the measurements. Overall, NPN advances regularization for imaging inverse problems by coupling model-aware null-space structure with learning, yielding robust gains across traditional and data-driven solvers with theoretical and empirical validation.

Abstract

Imaging inverse problems aim 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 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.

Figures (12)

• Figure 1: Geometric comparison of subspace–prior learning versus direct reconstruction in a $\mathbb{R}^3$ toy example. (a) In the low–dimensional projection space, the learned mapping $\mathrm{G}^*(\mathbf{y})$ trained on points inside the unit circle, closely matches the true null–space projection $\mathbf{S}\mathbf{x}^*$ for both training (solid) and test (semi-transparent) inputs, whereas the direct–reconstruction estimate $\tilde{\mathbf{x}}_0$ projected into $\mathbf{S}$ is significantly inaccurate. (b) In the original signal domain, the measurements $\mathbf{H x^* = y}$ define two intersecting planes.
• Figure 2: PnP-FISTA convergence analysis in CS. (a) Reconstruction error. (b) Null-space prediction error for (red) Initialization $\mathbf{\tilde{S}=\mathrm{QR}(\mathbf{H})}$ from Algorithm \ref{['alg:generate_rows_qr']}, and (blue) Designed $\mathbf{S}$ with Eq. \ref{['eq:joint_opt']} and $m/n=p/n=0.1$. In this case, the CIZ from Definition \ref{['def:error']} is highlighted in light red and light blue. (c) Acceleration ratio of signal convergence; here, the CIZ is defined as the empirical convergence ratio of the proposed solution that is lower than the baseline (black).
• Figure 3: Convergence curves for $\sigma \in \{2, 5, 10\}$ in deblurring, $\text{AF} \in \{4, 8, 12\}$ in MRI.
• Figure 3: Ablation over $\gamma$ for two methods. Best results are bold teal; second-best are underline orange.
• Figure 4: Deblurring and MRI reconstruction results for PnP and PnP-NPN using a DnCNN prior, with 5 dB SNR measurement noise. The measurements are denoted by $\mathbf{y}$, the nonlinear approximation of the null-space of $\mathbf{H}$ by $\hat{\mathbf{y}}_s$, and the true null-space by $\mathbf{y}_s^*$. The estimation $\hat{\mathbf{x}}_0$ is obtained with \ref{['eq:base']}, and $\hat{\mathbf{x}}$ is obtained with the proposed regularization with \ref{['eq:opt_prop']}. The ground truth signal is $\mathbf{x}^*$.

Theorems & Definitions (7)

• Definition 1: Null-Space
• Definition 2: Restricted Isometry Property shah2011iterative
• Definition 3: Convergence Improvement Zone (CIZ) by $\phi(\mathbf{x}^\ell)$
• Theorem 1: PnP-NPN Convergence
• Theorem 2: Convergence of NPN Regularization
• Theorem 1: PnP-NPN Convergence
• Theorem 2: Convergence of NPN Regularization