Structured Deformation Modeling with Implicit Deformation Modules

This paper presents a comprehensive theoretical exploration of implicit deformation modules and their application in shape analysis, especially registration problems. The implicit deformation module framework enables the generation of advanced structured deformations in a user-friendly and efficient manner. We establish the well-posedness of deformations generated by these modules and demonstrate the existence of solutions to the resulting registration problem. To achieve this, we introduce the notion of featured landmarks, which enables the definition of point deformations carrying features under a flow of diffeomorphisms, even when the features themselves cannot be directly transformed via a group action of diffeomorphisms. These contributions bridge the gap left by the IMODAL1 software paper [42], which provides an implementation of some of the deformation modules presented here (explicit and implicit of order one) but lacks advanced theoretical guarantees. Finally, we demonstrate how complex structured deformation models can be estimated from data, supported by practical examples.