  Deformable image registration (DIR) is an active research topic in biomedical
imaging. There is a growing interest in developing DIR methods based on deep
learning (DL). A traditional DL approach to DIR is based on training a
convolutional neural network (CNN) to estimate the registration field between
two input images. While conceptually simple, this approach comes with a
limitation that it exclusively relies on a pre-trained CNN without explicitly
enforcing fidelity between the registered image and the reference. We present
plug-and-play image registration network (PIRATE) as a new DIR method that
addresses this issue by integrating an explicit data-fidelity penalty and a CNN
prior. PIRATE pre-trains a CNN denoiser on the registration field and "plugs"
it into an iterative method as a regularizer. We additionally present PIRATE+
that fine-tunes the CNN prior in PIRATE using deep equilibrium models (DEQ).
PIRATE+ interprets the fixed-point iteration of PIRATE as a network with
effectively infinite layers and then trains the resulting network end-to-end,
enabling it to learn more task-specific information and boosting its
performance. Our numerical results on OASIS and CANDI datasets show that our
methods achieve state-of-the-art performance on DIR.
