  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 contain 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 tiny
fraction of parameters, and demonstrate improved robustness to covariate shift.
