Unrolling Plug-and-Play Gradient Graph Laplacian Regularizer for Image Restoration

TL;DR

This paper introduces interpretable, unrolled graph-based optimization networks for image restoration, combining mathematical rigor with competitive performance and robustness to data shifts.

Contribution

It formulates a new graph Laplacian regularizer for image reconstruction and unrolls ADMM algorithms into trainable neural networks with adaptive graph learning modules.

Findings

Competitive restoration quality with fewer parameters

Enhanced robustness to covariate shift

Effective integration of graph learning in unrolled networks

Abstract

Generic deep learning (DL) networks for image restoration like denoising and interpolation lack mathematical interpretability, require voluminous training data to tune a large parameter set, and are fragile in the face of covariate shift. To address these shortcomings, we build interpretable networks by unrolling variants of a graph-based optimization algorithm of different complexities. Specifically, for a general linear image formation model, we first formulate a convex quadratic programming (QP) problem with a new $\ell_2$-norm graph smoothness prior called gradient graph Laplacian regularizer (GGLR) that promotes piecewise planar (PWP) signal reconstruction. To solve the posed unconstrained QP problem, instead of computing a linear system solution straightforwardly, we introduce a variable number of auxiliary variables and correspondingly design a family of ADMM algorithms. We then unroll them into variable-complexity feed-forward networks, amenable to parameter tuning via back-propagation. More complex unrolled networks require more labeled data to train more parameters, but have better overall performance. The unrolled networks have periodic insertions of a graph learning module, akin to a self-attention mechanism in a transformer architecture, to learn pairwise similarity structure inherent in data. Experimental results show that our unrolled networks perform competitively to generic DL networks in image restoration quality while using only a fraction of parameters, and demonstrate improved robustness to covariate shift.