A Time-Dependent Joint Segmentation and Registration Model: Application to Longitudinal Registration of Hepatic DCE-MRI Sequences

While segmentation consists in partitioning a given image into meaningful constituents in order to identify relevant structures such as homogeneous regions or edges, registration, given two images, aims at finding an optimal orientationpreserving one-to-one deformation aligning the structures visible in an image into the corresponding ones in the other. Recently, intertwining both tasks into a single framework has proven to yield better results in terms of accuracy -in particular when the images exhibit weak boundary definition -and increase of reliability of the encoded structure matching -since now, not only based on intensity distribution comparison but also on geometrical and topological features -. In line with this idea, we propose going a step further by adding explicitly some dynamics in the modelling, i.e., by making the minimization problem both space and time-dependent so that the correlation between both tasks is achieved through the process, connecting thus the problem to an interpolation one. The shapes to be matched are viewed as Saint Venant-Kirchhoff materials, a special instance of hyperelastic ones, and are implicitly modelled by level-set functions. These are evolved in order to minimize a functional including both a nonlinear-elasticity-based regularizer prescribing the physical nature of the deformation and a term penalizing the shape misalignment, thus promoting structure matching rather than intensity pairing. Theoretical results emphasizing the mathematical soundness of the model are provided, among which the existence of minimizers and the existence of a weak viscosity solution to the related evolution problem. The model is then applied to the longitudinal registration of hepatic dynamic contrast-enhanced MRI sequences and shows good performance. This application has an important impact on the computer-aided follow-up of patients suffering from liver cancers.