In this work, edge sets are mapped one to the other by representing these zero area sets as diffuse images which have positive measure supports that can be registered elastically. The driving application for this work is to map a Purkinje fiber network in the endocardium of one heart to the endocardium of another heart. The approach is to register sufficiently accurate diffuse surface representations of two endocardia and then to apply the resulting transformation to the points of the Purkinje fiber network. To create a diffuse image from a given edge set, a region growing method is used to approximate diffusion of brightness from an edge set to a given point. To be minimized is the sum of squared differences of the transformed diffuse images along with a linear elastic penalty for the registration transformation. A Newton iteration is employed to solve the optimality system, and the degree of diffusion is larger in initial iterations while smaller in later iterations so that a desired local minimum is selected by means of vanishing diffusion. Favorable results are shown for registering highly detailed cardiac geometries.