Fidelity metrics between curves and surfaces: Currents, varifolds, and normal cycles

This chapter provides an overview of some mathematical and computational models that have been proposed over the past few years for defining data attachment terms on shape spaces of curves or surfaces. In all these models shapes are seen as elements of a space of generalized distributions, such as currents or varifolds. Then norms are defined through reproducing kernel Hilbert spaces (RKHS), which lead to shape distances that can be conveniently computed in practice. These were originally introduced in conjunction with diffeomorphic methods in computational anatomy and have indeed proved to be very efficient in this field. We provide a basic description of these different models and their practical implementation, then discuss the respective properties and potential advantages or downsides of each of them in diffeomorphic registration problems.