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Deformable image registration is a crucial component in the analysis of motion in time series. In medical data, the deformation fields are often predictable to a certain degree: the muscles and other tissues causing the motion-of-interest form shapes that may be used as a geometric prior. Using an Implicit Neural Representation to parameterize a deformation field allows the coordinate space to be chosen arbitrarily. We propose to curve this coordinate space around anatomical structures that influence the motion in our time series, yielding a space that is aligned with the expected motion. The geometric information is therefore explicitly encoded into the neural representation, reducing the complexity of the optimized deformation function. We design and evaluate this concept using an abdominal 3D cine-MRI dataset, where the motion of interest is bowel motility. We align the coordinate system of the neural representations with automatically extracted centerlines of the small intestine. We show that explicitly encoding the intestine geometry in the neural representations improves registration accuracy for bowel loops with active motility when compared to registration using neural representations in the original coordinate system. Additionally, we show that registration accuracy can be further improved using a model that combines a neural representation in image coordinates with a separate neural representation that operates in the proposed tangent coordinate system. This approach may improve the efficiency of deformable registration when describing motion-of-interest that is influenced by the shape of anatomical structures.
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