Registration typically proceeds in several steps. First, one decides how to measure
similarity between images. One may include the similarity among pixel intensity
values, as well as the proximity of predefined image features such as implanted
fiducials or anatomical landmarks10. Next, one looks for a transformation which
maximizes similarity when applied to one of the images. Often this transformation
is given as the solution of an optimization problem where the transformations to
be considered are constrained to be of a predetermined class C. In the case of rigid
registration (Section 5.2.1), C is the set of Euclidean transformations. Soft tissues
in the human body typically do not deform rigidly. For example, physical deformation
of the brain occurs during neurosurgery as a result of swelling, cerebrospinal
fluid loss, hemorrhage and the intervention itself. Therefore a more realistic and
more challenging problem is elastic registration (Section 5.2.2) where C is the set
of smooth diffeomorphisms. Due to anatomical variability, non-rigid deformation
maps are also useful when comparing images from different patients.