Medical image registration aligns two images by finding a smooth deformation that maps one to the other. The standard mathematical framework — Large Deformation Diffeomorphic Metric Mapping — requires the deformation to be a diffeomorphism: smooth, invertible, with smooth inverse. The smoothness constraint is the theoretical foundation. It is also the practical limitation.
Lungs slide against the chest wall. Organs slide against each other during breathing. The deformation field across a sliding boundary is discontinuous — the tissue on one side moves independently of the tissue on the other. Enforcing smoothness across a sliding boundary forces the algorithm to smear the discontinuity, producing anatomically incorrect registrations.
Diffeomorphism groupoids extend the framework to handle discontinuities (arXiv:2603.11806). A groupoid generalizes a group by allowing composition only between compatible elements — transformations that share a domain boundary. The deformation is diffeomorphic within each homogeneous region but can be discontinuous across region boundaries. The sliding boundary is not a violation of the framework; it is encoded in the groupoid structure.
The associated Lie algebroid and its dual provide the variational machinery. The Euler-Arnold equations — which describe geodesic flows on diffeomorphism groups in standard LDDMM — generalize to the groupoid setting, producing optimal discontinuous flows. The optimization finds the shortest path through deformation space while respecting the sliding constraints.
The mathematical upgrade is from groups to groupoids — from a structure where any two elements can compose to a structure where composition respects boundaries. The anatomy demanded this distinction; the mathematics followed.