“…They typically either restrict the space of
possible pair of transformations (resulting in fewer degrees of freedom), or
penalize those values of the two transformations that move the mid-space away from
the native spaces. The most common such constraints, proposed in the mid-space
registration and atlas construction literature, are restrictions on the two
transformations to have opposite displacement fields (Aljabar et al, 2008; Bhatia et al,
2004; Bouix et al, 2010; Fonov et al, 2011; Guimond et al, 2000; Miller et al, 1997; Noblet et al,
2012; Studholme and Cardenas,
2004; Yang et al, 2008) or velocity
fields (Ashburner and Ridgway, 2013; Grenander and Miller, 1998; Lorenzi et al, 2013). In large deformation models,
geodesic averaging of the deformations has also been proposed, which preserves the
desired properties of the transformations (Avants and
Gee, 2004; Joshi et al, 2004;
Lorenzen et al, 2006).…”