Shape averaging with diffeomorphic flows for atlas creation

Average anatomical shape is important to the spatial normalization process used in both functional and structural studies, as it provides a least-biased reference frame. Linear averaging of transformation fields is often used to construct average shapes from a given population. This type of averaging will not necessarily preserve topology of the anatomy and may not result in an average shape that lies within the permissible shape space, as defined by the continuum mechanical or variational model. These problems may be addressed by using a formulation where one finds the average configuration of the domain in time, as opposed to the average displacement. This is achieved by solving a transport ordinary differential equation of the form g~ = v~(g~), which imposes the diffeomorphism constraint directly on the average transformation, g~. An energy formulation for this problem will be given along with an algorithm for its minimization. The new diffeomorphic shape averaging algorithms are compared with the traditional linear averaging method.