Iteratively Refined Image Reconstruction with Learned Attentive Regularizers

TL;DR

This work reframes image reconstruction as the variational problem $\arg\min_{\mathbf x} E({\bf H}{\mathbf x},{\mathbf y}) + \lambda \mathcal{R}({\mathbf x})$ and introduces two data-driven, interpretable regularizers. The Majorization-Minimization Regularization (MMR) uses a sequence of convex surrogates with solution-adaptive weights, guaranteeing convergence to a critical point; the Solution-Adaptive Fixed-Point Iterations (SAFI) replace these weights with a learnable mask generator $\tilde{\boldsymbol\Lambda}$, producing a fixed-point operator with existence guarantees under invertible forward operators. The authors provide detailed parameterization and training strategies for learnable linear operators, concave potentials, and activation functions, and demonstrate competitive performance on denoising and MRI reconstruction tasks, while maintaining interpretability and theoretical underpinnings. The results show that learned, spatially adaptive regularizers can approach the performance of strong weakly convex models, and the SAFI framework delivers further improvements by enabling richer, data-driven mask generation without sacrificing convex subproblems per iteration. Overall, the work offers a robust, universal, and interpretable approach to inverse problems with promising extensions to other modalities and theoretical convergence analyses.

Abstract

We propose a regularization scheme for image reconstruction that leverages the power of deep learning while hinging on classic sparsity-promoting models. Many deep-learning-based models are hard to interpret and cumbersome to analyze theoretically. In contrast, our scheme is interpretable because it corresponds to the minimization of a series of convex problems. For each problem in the series, a mask is generated based on the previous solution to refine the regularization strength spatially. In this way, the model becomes progressively attentive to the image structure. For the underlying update operator, we prove the existence of a fixed point. As a special case, we investigate a mask generator for which the fixed-point iterations converge to a critical point of an explicit energy functional. In our experiments, we match the performance of state-of-the-art learned variational models for the solution of inverse problems. Additionally, we offer a promising balance between interpretability, theoretical guarantees, reliability, and performance.

Figures (8)

• Figure 1: Mask-generation architecture of the majorization-minimization (top) and the solution-driven (below) setting. Above each arrow, we denote the signal dimension at the corresponding stage.
• Figure 2: Denoising of the castle image corrupted by additive white Gaussian noise with $\sigma=25/255$.
• Figure 3: Solution path of the MMR method for denoising with $\sigma=25/255$. Each image ($k$, $e_k$, $\text{PSNR}_k$) represents ${\mathbf{x}}_{k+1}$ at the $k$th step of Algorithm \ref{['alg:MajMin']}, with relative error $e_k = \frac{\left\rVert{\mathbf{x}}_{k+1} - {\mathbf{x}}_k\right\rVert_2}{\left\rVert{\mathbf{x}}_k\right\rVert_2}$.
• Figure 4: Solution path of the SAFI scheme for denoising with $\sigma=25/255$. Each image ($k$, $e_k$, $\text{PSNR}_k$) represents ${\mathbf{x}}_{k+1}$ at the $k$th step of Algorithm \ref{['alg:FixIters']}, with relative error $e_k = \frac{\left\rVert{\mathbf{x}}_{k+1} - {\mathbf{x}}_k\right\rVert_2}{\left\rVert{\mathbf{x}}_k\right\rVert_2}$.
• Figure 5: Masks and responses for the learned regularization architectures \ref{['eq:reg_rw']} and \ref{['eq:RegConvMask']}. Black corresponds to lower values and white to higher ones. Note that $\{{\bf{W}}_c\}_{c=1}^{N_C}$ is learned within the MMR and SAFI frameworks for the first and last three figures (from the left), respectively.

Theorems & Definitions (16)

• Theorem 1
• Remark 1
• Theorem 2: Theorem of $\Gamma$-convergence Braides02
• Theorem 3
• proof
• Remark 2
• Lemma 1
• proof
• Proposition 1
• proof