Title: Geometrical Growth Models for Computational Anatomy
Advisor: Alain Trouvé
Keywords: computational anatomy, growth model, shape spaces, Riemannian metrics, group of diffeomorphisms, large deformations, variational methods, optimal control.
In the field of computational anatomy, the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework has proved to be highly efficient for addressing the problem of modeling and analysis of the variability of populations of shapes, allowing for the direct comparison and quantization of diffeomorphic morphometric changes. However, the analysis of medical imaging data also requires the processing of more complex changes, which especially appear during growth or aging phenomena. The observed organisms are subject to transformations over time that are no longer diffeomorphic, at least in a biological sense. One reason might be a gradual creation of new material uncorrelated to the preexisting one. The evolution of the shape can then be described by the joint action of a deformation process and a creation process.
For this purpose, we offer to extend the LDDMM framework to address the problem of non diffeomorphic structural variations in longitudinal data. We keep the geometric central concept of a group of deformations acting on embedded shapes. The necessity for partial mappings leads to a time-varying dynamic that modifies the action of the group of deformations. Ultimately, growth priors are integrated into a new optimal control problem for assimilation of time-varying surface data, leading to an interesting problem in the field of the calculus of variations where the choice of the attachment term on the data, current or varifold, plays an unexpected role.
The underlying minimization problem requires an adapted framework to consider a new set of cost functions (penalization term on the deformation). This new model is inspired by the deployment of animal horns and will be applied to it.