  Partial shapes correspondences is a problem that often occurs in computer
vision (occlusion, evolution in time...). In medical imaging, data may come
from different modalities and be acquired under different conditions which
leads to variations in shapes and topologies. In this paper we use an
asymmetric data dissimilarity term applicable to various geometric shapes like
sets of curves or surfaces, assessing the embedding of a shape into another one
without relying on correspondences. It is designed as a data attachment for the
Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, allowing to
compute a meaningful deformation of one shape onto a subset of the other. We
refine it in order to control the resulting non-rigid deformations and provide
consistent deformations of the shapes along with their ambient space. We show
that partial matching can be used for robust multi-modal liver registration
between a Computed Tomography (CT) volume and a Cone Beam Computed Tomography
(CBCT) volume. The 3D imaging of the patient CBCT at point of care that we call
live is truncated while the CT pre-intervention provides a full visualization
of the liver. The proposed method allows the truncated surfaces from CBCT to be
aligned non-rigidly, yet realistically, with surfaces from CT with an average
distance of 2.6mm(+/- 2.2). The generated deformations extend consistently to
the liver volume, and are evaluated on points of interest for the physicians,
with an average distance of 5.8mm (+/- 2.7) for vessels bifurcations and 5.13mm
(+/- 2.5) for tumors landmarks. Such multi-modality volumes registrations would
help the physicians in the perspective of navigating their tools in the
patient's anatomy to locate structures that are hardly visible in the CBCT used
during their procedures. Our code is available at
https://github.com/plantonsanti/PartialMatchingVarifolds.
