Non-parametric volumetric registration

In non-parametric volumetric registration (NVR), the entire spatial transformation is treated as an unknown function subject to a set of constraints, such as smoothness and invertibility. NVR has led to elegant mathematical solutions that tie together elements of group theory, differential geometry, and probability theory. This chapter reviews key mathematical concepts in NVR, including the concept of diffeomorphic transformations and ways in which such transformations can be generated by composition of small transformations. Next, the chapter describes the optical flow algorithm, which can be used to compute the velocity of moving objects in image sequences, and uses this algorithm as a building block for a set of fast VR algorithms that yield diffeomorphic transformations. The latter part of the chapter is dedicated to large deformation diffeomorphic metric mapping (LDDMM), a VR technology that also yields diffeomorphic transformations, but additionally provides a well-defined concept of distance between medical images and can be used as a tool for performing statistics on the manifold of medical images.