A computational model of multidimensional shape

We develop a computational model of shape that extends existing Riemannian models of shape of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. The model employs a representation of shape based on the discrete exterior derivative of parametrizations over a finite simplicial complex. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a local-global formulation that accounts for regional shape differences and integrates them into a global measure of dissimilarity. The Riemannian shape spaces provide a common framework to treat numerous problems such as the statistical modeling of shapes, the comparison of shapes associated with different individuals and groups, and modeling and simulation of dynamical shapes. We give multiple examples of geodesic interpolations and illustrations of the use of the model in brain mapping, particularly, the analysis of anatomical variation based on neuroimaging data.