Abstract
The problem of registration of images is a classical one, with a variety of solutions proposed in the past. Motivated by recent progress in functional data analysis, we approach this problem from a novel viewpoint and develop a notion of phase and amplitude for image data. We define amplitude of an image as a property that is invariant to diffeomorphic deformations of the background space; these deformations are said to change only the relative phases. Consequently, image registration is viewed as separation of phase and amplitude components in given images, and we derive a formal, metric-based framework for this separation. The main advantage of this approach is that the objective function used here for registration satisfies some important properties - (1) it is invariant under identical deformations of images being compared, and (2) it induces a proper metric on the amplitude space of images. The key is to use a mathematical representation, termed q-map and used previously in shape analysis, such that the L2-norm in the range space of this map forms the objective function. This is also called the amplitude distance, and is used to align images. This alignment effectively results in a separation of phase and amplitude in image data. We demonstrate this framework using examples from a variety of image types and compare performances with some recent methods.
