How to Register a Live onto a Liver ? Partial Matching in the Space of Varifolds

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.