  Convolutional neural networks (CNNs) have been tremendously successful in
solving imaging inverse problems. To understand their success, an effective
strategy is to construct simpler and mathematically more tractable
convolutional sparse coding (CSC) models that share essential ingredients with
CNNs. Existing CSC methods, however, underperform leading CNNs in challenging
inverse problems. We hypothesize that the performance gap may be attributed in
part to how they process images at different spatial scales: While many CNNs
use multiscale feature representations, existing CSC models mostly rely on
single-scale dictionaries. To close the performance gap, we thus propose a
multiscale convolutional dictionary structure. The proposed dictionary
structure is derived from the U-Net, arguably the most versatile and widely
used CNN for image-to-image learning problems. We show that incorporating the
proposed multiscale dictionary in an otherwise standard CSC framework yields
performance competitive with state-of-the-art CNNs across a range of
challenging inverse problems including CT and MRI reconstruction. Our work thus
demonstrates the effectiveness and scalability of the multiscale CSC approach
in solving challenging inverse problems.
