  Nonlocal self-similarity within images has become an increasingly popular
prior in deep-learning models. Despite their successful image restoration
performance, such models remain largely uninterpretable due to their black-box
construction. Our previous studies have shown that interpretable construction
of a fully convolutional denoiser (CDLNet), with performance on par with
state-of-the-art black-box counterparts, is achievable by unrolling a
convolutional dictionary learning algorithm. In this manuscript, we seek an
interpretable construction of a convolutional network with a nonlocal
self-similarity prior that performs on par with black-box nonlocal models. We
show that such an architecture can be effectively achieved by upgrading the L1
sparsity prior (soft-thresholding) of CDLNet to an image-adaptive
group-sparsity prior (group-thresholding). The proposed learned
group-thresholding makes use of nonlocal attention to perform spatially varying
soft-thresholding on the latent representation. To enable effective training
and inference on large images with global artifacts, we propose a novel
circulant-sparse attention. We achieve competitive natural-image denoising
performance compared to black-box nonlocal DNNs and transformers. The
interpretable construction of our network allows for a straightforward
extension to Compressed Sensing MRI (CS-MRI), yielding state-of-the-art
performance. Lastly, we show robustness to noise-level mismatches between
training and inference for denoising and CS-MRI reconstruction.
