| Abstract: |
| In medical image analysis, constructing an atlas, i.e. a mean representative of an ensemble of images, is a critical task for practitioners to estimate variability of shapes inside a population, and to characterise how structural shape changes have an impact on health. This involves identifying significant shape constituents of a set of images, a process called segmentation, and mapping this group of images to an unknown mean image, a task called registration, making a statistical analysis of the image population possible. To achieve this goal, we propose treating these operations jointly to leverage their positive mutual influence, in a hyperelasticity setting, by viewing the shapes to be matched as Ogden materials. The approach is complemented by novel hard constraints on the $L^\infty$ norm of both the Jacobian and its inverse, ensuring the deformation is a bi-Lipschitz homeomorphism. Segmentation is based on the Potts model, which allows for a partition into more than one shape. The connection to registration is ensured by the dissimilarity measure aiming to align the segmented shapes. A representation of the deformation field in a linear space is then computed to perform a geometry-driven Principal Component Analysis (PCA) and to extract the main modes of variations inside the image population. |
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