The concept of the Logarithm of the Odds (LogOdds) is frequently used in areas such as artificial neural networks, economics, and biology. Here, we utilize LogOdds for a shape representation that demonstrates desirable properties for medical imaging. For example, the representation encodes the shape of an anatomical structure as well as the variations within that structure. These variations are embedded in a vector space that relates to a probabilistic model. We apply our representation to a voxel based segmentation algorithm. We do so by embedding the manifold of Signed Distance Maps (SDM) into the linear space of LogOdds. The LogOdds variant is superior to the SDM model in an experiment segmenting 20 subjects into subcortical structures. We also use LogOdds in the non-convex interpolation between space conditioned distributions. We apply this model to a longitudinal schizophrenia study using quadratic splines. The resulting time-continuous simulation of the schizophrenic aging process has a higher accuracy then a model based on convex interpolation.