Estimation of a growth development with partial diffeomorphic mappings

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 analyzing of the variability of populations of shapes, allowing for the direct comparison and quantization of diffeomorphic morphometric changes. However, with the progress achieved in medical imaging analysis, the interest for longitudinal data set has substantially increased in the last years and 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. The evolution of the shape can then be described by the joint action of a deformation process and a creation process.

In this paper, we extend the LDDMM framework to address the problem of nondiffeomorphic structural variations in longitudinal data. We keep the geometric central concept of a group of deformations acting on embedded shapes. The need for partial mappings leads to a time-varying dynamic that modifies the action of the group of deformations. We develop a theoretical framework and two algorithms to estimate realistic individual growth scenarios from a set of observations sparsely distributed in time. We present few numerical experiments on animal horns where the shapes are modeled by oriented varifolds. Each computed scenario is parametrized by low-dimensional variables providing the support for statistical analysis.