  Physiological motion, such as cardiac and respiratory motion, during Magnetic
Resonance (MR) image acquisition can cause image artifacts. Motion correction
techniques have been proposed to compensate for these types of motion during
thoracic scans, relying on accurate motion estimation from undersampled
motion-resolved reconstruction. A particular interest and challenge lie in the
derivation of reliable non-rigid motion fields from the undersampled
motion-resolved data. Motion estimation is usually formulated in image space
via diffusion, parametric-spline, or optical flow methods. However, image-based
registration can be impaired by remaining aliasing artifacts due to the
undersampled motion-resolved reconstruction. In this work, we describe a
formalism to perform non-rigid registration directly in the sampled Fourier
space, i.e. k-space. We propose a deep-learning based approach to perform fast
and accurate non-rigid registration from the undersampled k-space data. The
basic working principle originates from the Local All-Pass (LAP) technique, a
recently introduced optical flow-based registration. The proposed LAPNet is
compared against traditional and deep learning image-based registrations and
tested on fully-sampled and highly-accelerated (with two undersampling
strategies) 3D respiratory motion-resolved MR images in a cohort of 40 patients
with suspected liver or lung metastases and 25 healthy subjects. The proposed
LAPNet provided consistent and superior performance to image-based approaches
throughout different sampling trajectories and acceleration factors.
